Zero -- Even or Not

[Algot Runeman]

http://www.bbc.co.uk/news/magazine-20559052

Math logic takes time to develop, even historically...

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Algot Runeman
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[Tanton, James]

I found this irresistable. I made it the theme of my December curriculum letter. http://www.jamestanton.com/?p=1072. (Top of the list.) Zero is just troublesome!

- J
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From: mathf...@googlegroups.com [mathf...@googlegroups.com] On Behalf Of Algot Runeman [algot....@verizon.net]
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To: mathf...@googlegroups.com
Subject: [Math 2.0] Zero -- Even or Not

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[Bradford Hansen-Smith]

James, you stated; "There’s something worthwhile in returning to and questioning the basics. After all, true business genius often comes from taking an old and simple idea and pushing it in a completely new direction.We can work to foster that thinking"

My observations suggest the circle is a symbol for everything and nothing is the negation of everything, which conveniently leaves us with parts and without any context, which is an impossibility. Conveniently because we can then decide what things are and are not as we wish, as we love to argue about these abstract concepts.
Brad

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[Joshua Zucker]

There's also a lovely blog (mostly about cosmology) that mentions recent widely-heard confusions about the evenness of 0.

  http://blog.richmond.edu/physicsbunn/2012/12/04/a-missed-opportunity-to-teach-some-mathematics/

Enjoy,

--Joshua

[Bradford Hansen-Smith]

Zero is not a number, it is a symbol for origin of both positive and negative numbers. If we could see a way to consider the circle zero as nothing and everything at the same time it would clear up a lot of confusion. There is something about the circle and zero that is so ubiquitous that we dare not question what we have been taught about circles and that we have for centuries been drawing pictures of them.  
Brad

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[Joshua Zucker]

James, thanks for the excellent newsletter.  You left off one of my favorite conundrums about zero, though.  It follows nicely from your question about zero giraffes.

Algebraically, is 0x the same as 0y?  In that case, is zero apples the same as zero airplanes?  And if so, then what about the sentence "Oh, darn it, I'm really hungry and I have zero airplanes!"

I think the answer is that when I say "I have (stuff)" I'm giving one component of an infinite-dimensional vector ... maybe?

Or maybe the answer is just that, as mathematicians, we don't use or think about numbers with units (like apples or feet) enough.  I mean, we say "you can't add apples and oranges" and then we write x^2 + x like there's no problem with that.

On Thu, Dec 6, 2012 at 4:54 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Zero is not a number, it is a symbol for origin of both positive and negative numbers. If we could see a way to consider the circle zero as nothing and everything at the same time it would clear up a lot of confusion. There is something about the circle and zero that is so ubiquitous that we dare not question what we have been taught about circles and that we have for centuries been drawing pictures of them.  
Brad

There are certainly lots of possible definitions of number, but I don't know of any in which there are positive and negative numbers but no number zero.  Is it not true that if you add two numbers, the result is a number?

Also, what do you think about sets?  Is the cardinality of a set a number?  What is the cardinality of the empty set?

--Joshua

[David Wees]

Joshua,

I think those are a problem of units though, not a problem with zero itself. If you say you have zero of something, you've indicated the units of the zero. Perhaps that is all that is necessary to avoid some interesting contradictions?

David

--

[Joshua Zucker]

Yeah, I mentioned units in the paragraph at the end of that part of my reply.  But I don't think we understand units well enough, or at least we don't teach it well enough in our math classes.  Everyone would agree that 0 inches = 0 feet, so why not 0 apples = 0 airplanes?  There's something deeper going on there.  And a lot of people think of units as a sort of multiplication: 0 miles means 0 * (1 mile), which is of course just 0, because 0 times anything is 0.  So isn't 0*(1 apple) = 0*(1 airplane) = 0?  I think this shows that there's more to units than we commonly appreciate.  They're not just extra factors to multiply together.  But what are they?

I'd love a pointer to somewhere to think about units (or, ideally, teach them to 8th graders) in a way that would make sense of these kinds of questions.

--Joshua

On Thu, Dec 6, 2012 at 10:35 PM, David Wees <davi...@googlemail.com> wrote:

I think those are a problem of units though, not a problem with zero itself. If you say you have zero of something, you've indicated the units of the zero. Perhaps that is all that is necessary to avoid some interesting contradictions?

[Sue VanHattum]

Not in relation to 0, but I often say to students (in beginning algebra mostly), "You can't add apples and oranges, unless you call them fruit." My attempt to explain common denominators...

---

From: joshua...@gmail.com
Date: Thu, 6 Dec 2012 23:03:25 -0800
Subject: Re: [Math 2.0] Zero -- Even or Not
To: mathf...@googlegroups.com

[Linda Stojanovska]

Hiya Sue –

I too love the “fruit” denominator.  Linda

[kirby urner]

I think it's useful to stay conscious of the "not math meanings" of these words, like even and odd.

"Even" tends to mean "level" but also in the sense of "fair" as in "level playing field" and "even handed".

"Odd" tends to mean "peculiar" and/or "off" and/or "weird".

We're rediscovering on math-teach how we teach the number line and coordinate systems by relating "positive" to "right" and "up", "negative" to "left" and "down".

These are such relative terms though. 

My right is your left when we're facing each other (theater:  stage left vs. house left).

We think of "right" and "upright" as in "righteous" and/or "normal" -- "orthodox" and "orthogonal" go together (orthogonality is the prevailing orthodoxy).

Mathematics is riddled with moralisms, not just truisms.  One may say they're just connotations, not denotations, so not "real". 

That's whistling in the dark for sure.  These ghost-meanings are everywhere.

Kirby

[Bradford Hansen-Smith]

Yes, it is as important to be conscious of the effect common language has on math, and how math effects common language., Logic is relative to the frame, of which each is only partial truth.
Kirby, can you talk more about the morals in math, we need more discussion about this aspect of mathematics.

Isn't whistling in the dark a reaction to fear of something being there we don't know.
Brad

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[David Chandler]

In vector spaces the additive identity is not zero, but the zero vector.  In groups, the additive identity must be an element of the group.  Zero airplanes is in a set which contains sets of airplanes.  Zero apples is in a set which contains sets of apples.  The two zeros are not identical because they refer to different sets.  Someone can stand and complain that he has zero airplanes while munching on an apple.
--David Chandler

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[Bradford Hansen-Smith]

Joshua,

Sets are a way of segregating through separation by eliminating connections. An empty set is the potential of unexpressed ideas, grouping the unimaginable.
Brad

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[Jon Adie - Sky]

Dystopia - Utopia .... I am a bit lost.. in this  void! - found Indian .. Devanagari - Sanskrit and Arabic concepts.. Mahavira  ..dividing Zero   SHUNYA /SIFRE ... CYPHERS AND CHIFFRE'S in Romance languages

I see (intuit) approximate ... not like Maths , which demands proof..

Is this a case of where two fields meet  two sets merge creating a sub set  Art & Maths

 

I am a Visual artist , liked your leads on paper plate folding ... 'the circle' ..  Buckminster Fuller's Geodesics C60 ...  Space time and Ideas of the sacred 'temenos'... or templum..   Asylum ..Safe Inside.. a place  Out of time and Space .. read the mail re-Zero visual artists Using (Mandala/Icons) for Centuries.. and yet the enigma .. of Zero is huge ..  as are it's  concepts and products.

 

Is this fuzzy logic .. or just a blurred idsea badly presented ? Jon A  

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of David Chandler
Sent: 08 December 2012 03:25
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Zero -- Even or Not

 

In vector spaces the additive identity is not zero, but the zero vector.  In groups, the additive identity must be an element of the group.  Zero airplanes is in a set which contains sets of airplanes.  Zero apples is in a set which contains sets of apples.  The two zeros are not identical because they refer to different sets.  Someone can stand and complain that he has zero airplanes while munching on an apple.
--David Chandler

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[michel paul]

On Dec 7, 2012 7:25 PM, "David Chandler" <david...@gmail.com> wrote:

> In vector spaces the additive identity is not zero, but the zero vector. 

This is a beautiful example of why CS should be integrated with math education. It makes reasoning about things like groups and sets a practical matter. These are classes. If you have a Vector class, then it's clear that the additive identity will have to be a Vector object. If you try to add the integer 0 to a Vector object, you'll get an error.

On Thu, Dec 6, 2012 at 10:32 PM, Joshua Zucker <joshua...@gmail.com> wrote:

> Algebraically, is 0x the same as 0y?

It depends on what we mean by 'same'. They are equivalent values, so 0x = 0y, but they are definitely different ideas.

> maybe the answer is just that, as mathematicians, we don't use or think about numbers with units (like apples or feet) enough.

I think that's definitely true. I've always loved contemplating the old question, "What is a number?" Then one day I shifted to, "What is a quantity?" I found that very useful, as it's an easier question to answer, and it immediately sheds light on how we think about numbers. We can say that a 'quantity' is a 'number of units'. Now, the nature of the units we use can vary tremendously. Some can be subdivided, and others cannot be.

-- Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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[Juan]

Bradford,

With all due respect, I do not know what you mean by "number," or "a number," or "being a number," or by the word "is." You write "Zero is not a number" as if everybody understood and agreed with you on the meaning (whatever it is) you assign to those words. I just want to point out the fact that many mathematicians in the last two centuries have agreed on the notion that Zero is a real number, a complex number, a rational number, and an integer as well. So I would say "Zero is a number" is kind of the currently accepted wisdom. If you say otherwise, it would be nice of you to provide some context as to what you mean by that. Why is Zero not a number, in your opinion? The debate on whether Zero is a number or not is an old one. It was very meaningful during the classic Greek period. I like to believe we have moved on since then, and at least mathematicians have gotten over that debate, and left it behind for good, having collectively decided, a long time ago, that Zero is, indeed, a number, with all equal membership rights to "numbership" as Pi, e, One, Negative One, One-Half, Square-Root-of-Two, the Imaginary Unit, and at least all other complex numbers. No caveats, no exceptions, no exclusionary clauses, no nothing. Zero is as number as a number can be, at least in my opinion, and I believe many mathematicians share that opinion.

Juan

[Algot Runeman]

On 12/07/2012 02:03 AM, Joshua Zucker wrote:

Yeah, I mentioned units in the paragraph at the end of that part of my reply.  But I don't think we understand units well enough, or at least we don't teach it well enough in our math classes.  Everyone would agree that 0 inches = 0 feet, so why not 0 apples = 0 airplanes?  There's something deeper going on there.  And a lot of people think of units as a sort of multiplication: 0 miles means 0 * (1 mile), which is of course just 0, because 0 times anything is 0.  So isn't 0*(1 apple) = 0*(1 airplane) = 0?  I think this shows that there's more to units than we commonly appreciate.  They're not just extra factors to multiply together.  But what are they?

I'd love a pointer to somewhere to think about units (or, ideally, teach them to 8th graders) in a way that would make sense of these kinds of questions.

--Joshua

Math is, I think, an abstraction of our commonplace reality.
(Red Rome apple image from: statesymbolsusa.org)
plus equals two apples.

But how equally true, in terms of commonplace reality is this?
(crispin apple image from: nyapplecountry.com)
plus equals two apples.

I think I'm okay with this equation.

Math abstracts away the issue of units from our commonplace reality.
I can accept both of the apple math abstraction statements to be "true", but it would stretch my understanding of the world to think that a Red Rome tasted like a crispin. Often two red delicious apples don't taste much alike, especially if one is a week or more older and sitting on a kitchen counter.

1 apple + one orange = two fruit. (Unit conversion isn't much challenge in this case.)
1 apple + one airplane = two ... objects? Commonplace reality makes these units seem 'silly' to combine. Depending on the degree of math abstraction, though, the units can be mixed.

However, my ability to extract meaning from the math breaks down if I try to do some unit combinations.
1 gram of water + 1 degree of Celsius temperature = 2 ... what?
Can such a level of abstraction from reality work?

--Algot

[Sue Hellman]

Algo -- thanks very much for this your metaphor I'm working on a workshop in active learning strategies for math educators and would like to use it if I may. David Tall has had a lot to say about difficulties in math and the development of mathematical thinking (see for example: http://digilander.libero.it/leo723/materiali/algebra/dot1995b-pme-plenary.pdf). As I understand what he says, struggling students' difficulties come not as much from difficulty with abstraction, but from a reluctance or inability to let go of primitive processes or to reconceptualize them as objects which can be manipulated in their own right. He explains jump more as a 'compression' than an abstraction -- i.e. critical in order to reduce the cognitive load on working memory. Unfortunately he doesn't offer much to help educators to get the students who cling desperately to reproducing processes when solving more complex problems (i.e. counting to multiply large numbers instead of compressing that process into a known fact which has a use of its own independent of the original meaning or concrete situation from which it sprang).
 
BTW: Here's my variation on your 2 apples problem:
 
  plus    =  
 
(My additions are adapted from images in wikimedia commons.)
 
Sue 

--

[kirby urner]

On Fri, Dec 7, 2012 at 7:19 AM, Bradford Hansen-Smith
<wholem...@gmail.com> wrote:
> Yes, it is as important to be conscious of the effect common language has on
> math, and how math effects common language., Logic is relative to the frame,
> of which each is only partial truth.
> Kirby, can you talk more about the morals in math, we need more discussion
> about this aspect of mathematics.
>

Probably it would help strengthen the mathematics curriculum if we
spent more time investing in off-the-beaten-track approaches that take
us in another direction.

On the issue of positive and negative, I think it'd make sense to
bring in more discussion of particles and anti-particles e.g.
electrons and positrons, to to have some discussion of why anti-matter
is not equally prevalent. There's asymmetry we might want to discuss.

What I do more, in getting off the beaten track, is I stop showing
multiplication as exclusively a right angles affair. A triangle with
all edges n and an area of n makes enough sense to support an
internally consistent logic. A tetrahedron of edges n has volume n^3.
So we don't say 'squared' and 'cubed' for 2nd and 3rd powering
respectively. That's not the knee-jerk reflex it becomes for your
standardized math-head.

Rather that just say "live and let live" and "we should tolerate
differences" it's more eye-opening to push back a little and suggest
the way we do it now is the less intelligent way i.e. not only are
there other ways to design maths, but some of these maths lead to
better thinking than we currently access, thanks to our collective
closed-mindedness.

I do this with my unit volume tetrahedrons, dividing them evenly into
other shapes with easy relationships. So many more wholesome whole
numbers that the unit volume cubes give us. The connotation that
cubes are "relatively awkward, stupid, slow, overbuilt" (relative to
tetrahedrons) is allowed to seep through the lines.

Such a math is "not normal" i..e "not orthodox" and so doesn't get
much discussion, even though the literature is well developed (not all
my invention by a long shot).

My intention is to show how mathematics is laced with conventions and
agreements that are anything but "proved". A way to show this is to
lace a mathematics with different conventions and agreements.

> Isn't whistling in the dark a reaction to fear of something being there we
> don't know.

Yes. Seeing a rather different math growing up, based on different
conventions and assumptions, can be eerie to witness, perhaps somewhat
scary, like witnessing the growth of that alien plant (Audrey 2) in
'Little Shop of Horrors' maybe.

Kirby

> Brad

[michel paul]

On Sun, Dec 9, 2012 at 10:54 AM, kirby urner <kirby...@gmail.com> wrote:

Probably it would help strengthen the mathematics curriculum if we
spent more time investing in off-the-beaten-track approaches that take
us in another direction.

Absolutely. I'd say the biggest obstacle to this getting to happen is high stakes testing. It's deadly. People are very hesitant to veer off the beaten track, at least in US public ed, since everything is so heavily tied to test scores. Why don't people teach Euclid's Algorithm? Well, because it isn't tested. The fact that it's a beautiful piece of reasoning that is a doorway into number theory and computational thinking isn't as important as the fact that it's not tested.  

-- Michel

[michel paul]

Very nice - this discussion of units turns out to be important as well for discussing the difference between addition and multiplication! I totally agree that thinking about what units are should be done more in math. They tend to get neglected if the focus is on the bottom line computation. 

A sum only makes sense when we add units of the same type, and a sum will be composed of the same units as its addends, but a product will always involve units of different types, as in square units from linear units, or as in rate*time. In one way or another a product involves a combination of units, but a sum always involves units of the same type.

A repeated addition model might think in this fashion: a * (b units) = (b units) + (b units) + (b units) ... a times. Here the a apparently has no units. It is a scalar. Someone might think that here we have an example of a product where there is only one type of unit involved. However, it is still the case that a is a different kind of mathematical object than the b units.

We could also express the product as (a groups) * (b units/group). And again, ratio appears as an essential part of what multiplication is.

- Michel

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[kirby urner]

On Sun, Dec 9, 2012 at 11:18 AM, michel paul <python...@gmail.com> wrote:
> On Sun, Dec 9, 2012 at 10:54 AM, kirby urner <kirby...@gmail.com> wrote:
>>
>>
>> Probably it would help strengthen the mathematics curriculum if we
>> spent more time investing in off-the-beaten-track approaches that take
>> us in another direction.
>
>
> Absolutely. I'd say the biggest obstacle to this getting to happen is high
> stakes testing. It's deadly. People are very hesitant to veer off the beaten
> track, at least in US public ed, since everything is so heavily tied to test
> scores. Why don't people teach Euclid's Algorithm? Well, because it isn't
> tested. The fact that it's a beautiful piece of reasoning that is a doorway
> into number theory and computational thinking isn't as important as the fact
> that it's not tested.
>
> -- Michel
>

Yes, high stakes testing has a huge impact.

I wonder if the response should be alternative high stakes testing.

On the other side of the tests, you need a network of academies that
offer worthwhile training and experience.

One need not compete for huge numbers of student in a high profile
way. It's not like ETS (SAT, AP...) will go out of business.

Sure we need to cultivate some sense of "recruiters" or "talent
scouts" but when it comes to underdog subcultures, smaller is better.

You get to be elitist more quickly.

If your biggest challenge is school pride and snobbery, that's likely
not a bad place to be.

Kirby

[Bradford Hansen-Smith]

Jaun, thank you for asking.

You have asked; "If you say otherwise, it would be nice of you to provide some context as to what you mean by that. Why is Zero not a number, in your opinion? "

Zero is a symbol, and yes, there have been many very bright and respected people over centuries that have worked to clarify mathematical issues. I do not discount any of them, but must go by my own observations and understanding as well. Using the circle as a symbol for zero, nothing, does not make the circle a zero. The zero is not a number; what we call numbers are different symbols with different values. The zero and numbers are only by common agreement. I see the circle symbol as an image that represent a circle disk in space, as well meaning zero,

I function in s 3-D world as well as in mind. There must be a balance between the physical, mental, and yes the spiritual. For understanding I need a larger context than what has been agreed upon. We have left out too much. It is important to establish context since everything is in fact multifunctional with more than one meaning. The generalization about abstract objects does not take into consideration the context, rather we talk about isolated objects, units in separation; this does not hold up in understanding the complexities of interrelationships withing a larger context. 

I must side with Kirby, for there are other math and logic systems equally important as what we have agreed upon. One is not better, they all hold some degree of understanding truth, which is the reason to put out all this mental energy in the first place. The symbol of the circle represents a 3-D circle as well abstractions of the zero concept. The properties of this circle disk object has five congruent circles, four more than a 2-D circle. When a circle is folded in half a tetrahedron pattern of movement is generated showing the traditional properties of a tetrahedron. This is observable but not obvious until pointed out "how" we fold the circle. This is another understanding that while very different in approach does not deny what else we have discovered and accepted about circles, but it gives a greater context of understanding making clear some of the inconsistencies that we discuss and protect.

The circle is a symbol of nothing, zero, empty set, but is also an image of a comprehensive whole, a set of everything known and unknown, it is unity of everything. The circle functions as unit and unity simultaneously; no other object can demonstrate this concept. Why disadvantage ourselves and say it is only one thing when we are only talking about an image. It can represent what ever we want it to, but that does not make it so. Every round coin is a circle disk and each has different value depending on size and surface design because of agreement, not because of the object. There are many ways to calculate the abstraction of "money" value, as there are many ways to understand images used in mathematics. I only suggest we expand our understanding of what we know and not get stuck in what someone said about something a long time ago. I do not discount the value and importance of their contributions, but I do not overly value yesterday more that what is dynamically taking place today. We are expanding our understanding about where we are and realizing things may not be what we once thought.  
Brad

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[kirby urner]

On Mon, Dec 10, 2012 at 8:09 AM, Bradford Hansen-Smith
<wholem...@gmail.com> wrote:

<< snip >>

> I must side with Kirby, for there are other math and logic systems equally
> important as what we have agreed upon. One is not better, they all hold some
> degree of understanding truth, which is the reason to put out all this
> mental energy in the first place. The symbol of the circle represents a 3-D
> circle as well abstractions of the zero concept. The properties of this
> circle disk object has five congruent circles, four more than a 2-D circle.
> When a circle is folded in half a tetrahedron pattern of movement is
> generated showing the traditional properties of a tetrahedron. This is
> observable but not obvious until pointed out "how" we fold the circle. This
> is another understanding that while very different in approach does not deny
> what else we have discovered and accepted about circles, but it gives a
> greater context of understanding making clear some of the inconsistencies
> that we discuss and protect.
>

I take the concept of "namespaces" from computer science and mix it
with "language games" from Wittgenstein.

When designing computer languages, people have needed to work out how
to allow different "meaning" of the same word to co-exist peacefully
and get work done. People have a propensity to use the same words for
different objects.

Not that "meaning" has to be seen as "pointing to objects" but that's
the crass nominalism we inherit, and object oriented programming makes
it worse, so I go with the flow sometimes.

Hence Martian Math where I set up the namespaces on either side of a
canyon. The Martians have a D^3 unit volume tetrahedron while the
Earthlings have an R^3 cube. R+R =D in length. They both share a
common reference sphere, at the center of which we can talk about Zero
(both civs agree to some extent, though the Earthlings use XYZ and the
Martians use IVM for coordinate systems / scaffolding).

The antidote to "one and only one math" is "local variables" i.e. in
talking about "lines" or "powering", we're local to a namespace.
There's no requirement to establish the global meanings of terms, as
if "zero" could not anchor in language unless it had "one true
meaning" uber alles. We don't need for it to have that. All we need
are local constructions in which Zero has a role. There may be family
resemblance among them.

Kirby

[Jon Adie - Sky]

Non - plussed... I like your   'description'.. of the symbol, and the extended circle of meanings .. with the emphasis on context ..  I agree although I am reminded of Alan Turin and Buckminster Fuller who both looked back in time while breaking ground moving forward

Jon A

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Bradford Hansen-Smith
Sent: 10 December 2012 16:09
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Zero -- Even or Not

 

Jaun, thank you for asking.

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[Juan]

On Monday, December 10, 2012 10:18:36 AM UTC-8, kirby urner wrote:

The antidote to "one and only one math" is "local variables" i.e. in
talking about "lines" or "powering", we're local to a namespace.

Kirby,
Just to let you know, there we are at polar opposite points of the spectrum. "Antidote'? I see no need for an antidote. I do love "one and only one math." I believe in "one and only one math." I support "one and only one math." I am an enthusiast of "one and only one math." That is the reason I studied math in the first place, instead of, say, political science, or linguistics. Inside the one and only one math, there is room for an infinite number of mathematical branches and sub-branches. There are many maths inside The One Math, and that makes It more fascinating. I find math's uniqueness makes it perfect, in sharp contrast with the maddening cacophony of the real world.

[Juan]

Bradford,
You are welcome. I see you are referring to the number-numeral difference. The circle is not the only numeral for Zero. The Mayans used a symbol that looks more like a football, or an eye, for Zero, rather than a circle. In any case, I really don't care about the numeral. You can use any symbol you want to represent the value zero. You could choose a flower, or a galaxy-looking spiral, a tetrahedron, or anything else. The numeral is just a symbol. In that much I agree with you. However, in your previous post, when you say:
"Zero is not a number, it is a symbol for origin of both positive and negative numbers,"
you make it sound like you are making a distinction between values like One, and Negative-One, on one hand, considering them as numbers; and the value Zero, on the other hand, considering it as a non-number, "just a symbol."
When I talk about Zero, I am referring to Zero the number, Zero the value, Zero the additive identity of the complex numbers, Zero the cardinality of the empty set. For me, the number Zero in and of itself has nothing to do with the circle, nothing whatsoever. The circle is a plane geometrical figure, a subset of the Euclidean plane, a set of points, not a number. Zero is a number, not a circle. The fact that the numeral we use for Zero looks kind of like a circle (more like a vertical ellipse, I would say), does not "make the circle Zero." We are not supposed to confuse the number with its numeral. That is common knowledge since grade school. I know some students sometimes have difficulties separating the number from its numeral but most of them are done with that by the time they go to high school.
The reason why Zero is even is because it is an integer multiple of Two, not because its numeral looks like a circle, and visually, the circle looks "evenly distributed in all directions." That would be a very loose, informal, and imprecise association, not at all a mathematical fact.
Also, Zero is not nothing. "Nothing" is a word that may represent several things but is not a mathematical object. The formal mathematical equivalent for the concept of "nothing," if there is one, it would be the empty set, or maybe "the contents of the empty set." That would be really nothing. Zero is a number, it's not nothing. Zero is the number of things you have when you don't have any - thing. When you don't have anything, nothing is what you have but zero is the number of things you have then. It's different. Zero is not the same as nothing. "Nothing" is not a number. Zero is a number.
Now, the balance between the physical, the mental, and the spiritual, is not a mathematical object, nor a mathematical discipline, or a mathematical problem. It is a human problem, a problem of life, a very important problem but not a mathematical one. Doing math too intensely carries the risk of throwing off your mental, emotional, and/or spiritual balance but that risk has to be managed by the individual, or by the group, it cannot be part of math itself. There is no such a thing as "Spiritual Math," ancient Greek philosopher's requirement "Do not enter this temple if you do not know mathematics" notwithstanding.

Juan

[Bradford Hansen-Smith]

Jaun, thank you. I am as yet unclear about number-numeral difference. I see them all as symbols to mathematical understanding of abstracted values and relationships represented by spatial objects as well as imaginary objects. My present understanding of one and negative one have difference by distinction of symbols used and functional relationships represented between the point of intersection of the xy axis as origin or place of no number symbol, maybe nothing. Possibly an empty set of potential that is realizable, number by number location moving outward revealing relationships of infinite set combinations. I have to consider the point symbol as a small circle since I do not see points referred to as squares. The concentric nature of the circle suggest to me that with expanding scale the number lines increase possibilities for realized potential of greater expanded alignment. The circle, commonly used as a symbol for zero might also function as an empty set of nothing and of all sets realizable as everything yet unrealizable because of conditional limitations of present sequential linear development.

In that objects are represented by symbols that can function in many different ways I am not altogether clear on what a mathematical object is. Where do mathematical problems come from and what is the point of all this expended energy if math is not a human problem. Many of the problems today are the result of used and misused mathematics that happens when we separate our tools from how they are used and for whose benefit. If the point is a mathematical object is it not then a point of balance between all human experience a mathematical problem of human consciousness as we view the relationship between subjective and objective reality? How is math not a human problem?
Brad

--
Bradford Hansen-Smith
www.wholemovement.com

[Juan]

Bradford,
You are welcome. Numerals are the symbols used to represent numbers.
We write the number Five with the numeral 5. The Romans wrote it with
the letter V (to represent an open hand with five fingers). The Mayan
numeral for Five is a long horizontal dash, kind of our underscore
character _ . In binary notation the numeral for Five is 101, that
represents the result of the sum 1x1 + 0x2 + 1x4. So, all these 5, V,
_, 101 are numerals for the number Five but they themselves are not
Five, they are just symbols for Five. So we say Five is the number and
5 is one of its numerals. There are many numerals for Five but only
one number Five. That is true as far as distinguishing numbers from
their numerals.
Now, if we get technical, there are several mathematical objects that
also represent the number Five. These we can consider as some sort of
"abstract structure numerals", each one being akin to a computer-
language object from a different computer-language class. In that
sense we could say there are different "types of number Five" because
you can technically distinguish between 5 as a cardinal number, 5 as a
natural number, 5 as an integer, 5 as a rational number, 5 as a real
number, and 5 as a complex number.
The cardinal number 5 is that which is common to all sets with five
objects; or the proper class of all such sets.
The natural number 5 is the successor of the natural number 4.
Five the integer is the set of all pairs (a,b) of natural numbers a
and b, such that a = b+5 as the sum of two natural numbers.
Five as a rational number is the set of all pairs (m,n) of integers m
and n, where n is not Zero, and m = 5n as the product of two
integers.
There are a few known constructions of the real number system based on
the rational numbers. In one of these, the real number Five is a set
of rational-number Cauchy sequences. In another such construction, the
real number Five is a Dedekin cut, a special type of ordered bi-
partition of the rational numbers.
The complex number 5 can be seen as the linear combination 5+0i, where
"i" is the numeral for the Imaginary Unit, the main square root of the
real number Negative-One.
So, looking at all these number systems, one could say there are many
"different number fives." However, since the complex numbers have a
subset that is isomorphic to the real numbers; the real numbers have a
subset isomorphic to the rational numbers; the rational numbers have a
subset isomorphic to the integers; the integers have a subset
isomorphic to the natural numbers; and the natural numbers can be put
in a one-to-one correspondence with the set of cardinalities of
denumerable sets; because of all this, it makes more sense to think
that there is only one number Five that can take all of these forms,
it can show up as a cardinal number, a natural number, an integer, a
rational, a real number, or a complex number. Technically they are all
different kind of mathematical objects but they keep consistently
exhibiting the same mathematical relationships inside each of these
number systems.

Now, there is no "XY axis." There is the X-axis, the Y-axis, and the
XY-plane, but no such thing as an "XY axis." Axes are supposed to be
straight lines, not planes.

I hope the above clarifies a little the difference between numbers and
their numerals, and that it somehow illustrates what I mean by
mathematical objects. Later I will comment on some of the other topics
you mention. Have a great day.

Juan




On Dec 13, 9:31 am, Bradford Hansen-Smith <wholemovem...@gmail.com>
wrote:

[kirby urner]

On Thu, Dec 13, 2012 at 1:26 PM, Juan <here...@gmail.com> wrote:

Bradford,

You are welcome. Numerals are the symbols used to represent numbers.
We write the number Five with the numeral 5. The Romans wrote it with
the letter V (to represent an open hand with five fingers). The Mayan
numeral for Five is a long horizontal dash, kind of our underscore
character _ . In binary notation the numeral for Five is 101, that
represents the result of the sum 1x1 + 0x2 + 1x4. So, all these 5, V,
_, 101 are numerals for the number Five but they themselves are not
Five, they are just symbols for Five. So we say Five is the number and
5 is one of its numerals. There are many numerals for Five but only
one number Five. That is true as far as distinguishing numbers from
their numerals.

This is what I was taught as a kid, but I no longer believe it having studied Wittgenstein's philosophy of language (and mathematics (more language)) in college.

Numerals are like tools, are in fact tools, and their meaning stems from the way they are used, the games they are involved in, which may have a family resemblance.

The pawn, used in chess, has meaning, but not because it points to some hidden abstract pawn object.  One may use many symbols for pawn, and they all mean the same thing because they're all used according to the rules of chess.

Likewise, '2' is used in many games.  It may be a symbol used to label bus routes ('2A' might be another route).

To think there's some "Number" that is a kind of object we can never see, which "Numerals" somehow point to, is shared by both Platonism and Nominalism.  Not all philosophies reinforce such thinking.  The one I study the most does not.

Now, if we get technical, there are several mathematical objects that
also represent the number Five. These we can consider as some sort of
"abstract structure numerals", each one being akin to a computer-
language object from a different computer-language class. In that
sense we could say there are different "types of number Five" because
you can technically distinguish between 5 as a cardinal number, 5 as a
natural number, 5 as an integer, 5 as a rational number, 5 as a real
number, and 5 as a complex number.

There are different language games, different namespaces.  '5' used to label a bus route is different from 5 + 0i the complex number, because of the language games involving each one.   We need not believe in mysterious abstract objects in some invisible abstract realm.

The cardinal number 5 is that which is common to all sets with five
objects; or the proper class of all such sets.

This stems from the Bertrand Russell era when the "essence of numbers" was distilled to a language of sets -- one more language game (or many partially overlapping ones).

As far as I'm concerned, we are in a new chapter and this set-based "definition" of Number is but one ethnic group's way of talking.  We don't need to think like those Anglos if we don't want to (I don't want to).

Thinking in terms of tools is helpful.  What is the meaning of a screwdriver?  A hammer?  Not some invisible "object" in the background.  The meaning derives from usage patterns.  Likewise the symbol "5" and the symbol "Five".  There is no need to imagine some kind of pointing to a great Number in the sky.

Kirby

[Christian Baune]

Hi,

I would like to add that the choice of symbol may greatly.change the way you think!
Do :
IIX + II
Quite easy, remove II from both sides of "+" and that's it.
Now do:
8 + 2
Not as easy ! You've to look at your summation table!

Now do these:
9 + 3
And
IX + IIV

Yes, roman litterals became harder!

Now, here is the value 3 , 9 , 8 written in 4 numerals systems:
3 and 9 snd 8
♧♧♧ and ♧♧♧♧♧♧♧♧♧ and ♧♧♧♧♧♧♧♧
IIV and IV and IIX
10 and 1010 and 3

Now try writing 65535 in these 4 numeral systems.
Try to do an addition using each system, find the CGD or wethever a given number is prime!

As with language, the choice of symbols affect your way of thinking wich in turn affect your choice of symbols !
You can't ellaborate on something you can't handle/comprehend/approach.

Kind regards,
Christian

Now try writting

[Christian Baune]

IX, not IV for 9 of course.

[Juan]

Bradford,
Continuing with my reference to your post of Thursday, December 13, 2012 9:31:19 AM, now specifically about your line:

"I have to consider the point symbol as a small circle since I do not see points referred to as squares."

Geometrical points do not have shape, as they do not have any positive perimeter, area, volume or weight. The ink marks on the paper we use to indicate points may have some or all of these properties, because ink is made of physical molecules. However points themselves only have identity, and location but they do not have any physical property. They do not exist as material objects. Points have zero length, zero width, zero height. We cannot actually see them with our biological eyes, nor even with microscopes, or electronic microscopes. Points are smaller than atoms, smaller than any physical particle, smaller even that those vibrating strings out of String Theory.
We could say a point is a circle of radius zero; as well as it is a square with sides of length zero, at the same time; or an equilateral triangle with sides of length zero. In this particular case the circle, the square and the triangle would all be congruent because of having zero area. However, when a general shape becomes a single point, those are not really circles, squares, or triangles. They are so only technically. They are called degenerate circles, squares, or triangles.
As far as physical representations go, if we think about pencil or pen drawings on paper, the point symbol varies widely because if you magnify, and closely examine the ink marks people make to represent points, they are all different in size and shape. Even when people intend to draw small circles they most likely produce some sort of irregular, oval shape. Now, in computer screens the point symbol is usually a set of pixels determined by the browser (or some other program) while interpreting the graphic output of the particular program it may be running to render the drawing. In that case, the shape of the point symbol is a combined effect of graphic design and computer graphic algorithms.
In any case, geometrical points are not the same as their representations. Geometrical points are abstract concepts. We usually make connections between these abstract ideas, on one hand, and things in the "real" world, on the other hand. However, they are not the same.

Juan

[kirby urner]

In any case, geometrical points are not the same as their representations. Geometrical points are abstract concepts. We usually make connections between these abstract ideas, on one hand, and things in the "real" world, on the other hand. However, they are not the same.

Juan

On the other hand, defining a point as explained above is definitional and other definitions might as well serve.

For example, Karl Menger, a dimension theorist, proposed "a geometry of lumps" in which points, lines and planes are not considered to differ in dimension and all are considered to be lump-like.  A plane is flat but not infinitely flat.  A point is relatively small, but not infinitely small, and so on.  Such a mathematics is useful in purging infinity from considerations, either infinitely big or infinitely small.

We pick a namespace first, and then develop our mathematics within it.  Euclidean points, the the metaphysical baggage they come with, need not be the only points we care about.

Kirby

[Juan]

Kirby,
I know pretty much nothing to speak of about philosophy. I have not studied Wittgenstein or any other philosopher. One reason I haven't is because of my belief that the number Two was prime before Wittgenstein was born, and it still continues to be a prime number today.
I am under the impression that there are many philosophical theories, coming from different philosophers and philosophical schools. Many of these philosophical theories have parts, or specific doctrines, or beliefs, that contradict some other parts or doctrines of other philosophical theories. So, I believe philosophers are pretty much always arguing among themselves. That is not my cup of tea. I have no patience whatsoever for that ongoing argumentation. That intellectual activity I see as part of what I call "the maddening cacophony of the real world."
On the other hand, as long as Two plus Two equals Four, I do not care whether the real numbers "really are" sets of Cauchy sequences, or "really are" Dedekind Cuts, or that they "really are" anything specific. I definitely, very much prefer to imagine them as Big Numbers in the Sky. That just works for me like a charm.
I agree with you in that you "need not believe in mysterious abstract objects in some invisible abstract realm" but I want to believe in them so I choose to do so. That just makes a whole lot more sense to me, and it makes my mental life a lot easier, and happier.
I also agree with you in that "we don't need to think like those Anglos if we don't want to" but I do. I very much like to think along those lines. Are you kidding me? I love set theory. And I love the constructions of mathematics based on set theory as their foundation. I have been fascinated by the ideas of Cantor, Dedekind, Weierstrass, and other early developers of set theory since I first came in contact with them. Set theory, and set theoretical ideas, are among my favorite mathematical topics.
I don't care about the meaning of a screwdriver. I use screwdrivers when I need them but I spend a lot of time thinking about abstract, invisible, Platonic numbers whether I need them or not, just because I enjoy it. And I enjoy them the more because they are abstract, and invisible.

Juan

[Juan]

Christian,
Thank you for the comment. this is the first time I have ever seen the number 3 written with Roman numerals as IIV. Before, I had only seen it written as III.
The same goes for 8, before, I had only seen it written as VIII but never as IIX, until now. It is still consistent. Quite interesting.

Juan

[kirby urner]

On Fri, Dec 14, 2012 at 4:15 PM, Juan <here...@gmail.com> wrote:

Kirby,
I know pretty much nothing to speak of about philosophy. I have not studied Wittgenstein or any other philosopher. One reason I haven't is because of my belief that the number Two was prime before Wittgenstein was born, and it still continues to be a prime number today.

Also 17 and 23.  However -1 is usually not considered prime, though some have proposed it be added.
 

I am under the impression that there are many philosophical theories, coming from different philosophers and philosophical schools. Many of these philosophical theories have parts, or specific doctrines, or beliefs, that contradict some other parts or doctrines of other philosophical theories. So, I believe philosophers are pretty much always arguing among themselves. That is not my cup of tea. I have no patience whatsoever for that ongoing argumentation.

Yes, mathematics tends to attract those who are temperamentally averse to the "many ways to do it" philosophy.

For this reason, a math class often feels like one where indoctrination occurs i.e. "this is how it is".

However, what I think is valuable, especially for people new to maths, is to remind them it's a beach of many sand castles, not just one.   The way people talk about points, lines and planes, or numerals versus numbers, is characteristic of a specific subculture.  When it comes to definitions and axioms, there are many ways to go.
 

That intellectual activity I see as part of what I call "the maddening cacophony of the real world."
On the other hand, as long as Two plus Two equals Four, I do not care whether the real numbers "really are" sets of Cauchy sequences, or "really are" Dedekind Cuts, or that they "really are" anything specific. I definitely, very much prefer to imagine them as Big Numbers in the Sky. That just works for me like a charm.

Adding social security numbers or zip codes has no real meaning in the language game in which they occur.  They're more like character strings.

I like maths that don't necessarily use numbers e.g. A + A == AA.  2 + 2 = 22.  That's treating numbers as string type objects.  Sometimes useful as well.
 

I agree with you in that you "need not believe in mysterious abstract objects in some invisible abstract realm" but I want to believe in them so I choose to do so. That just makes a whole lot more sense to me, and it makes my mental life a lot easier, and happier.

Yes, it should be put that way to kids:  here's something you might choose to believe in, that might add to your sense of satisfaction.  It's an elective act, this believing.
 

I also agree with you in that "we don't need to think like those Anglos if we don't want to" but I do. I very much like to think along those lines. Are you kidding me? I love set theory. And I love the constructions of mathematics based on set theory as their foundation. I have been fascinated by the ideas of Cantor, Dedekind, Weierstrass, and other early developers of set theory since I first came in contact with them. Set theory, and set theoretical ideas, are among my favorite mathematical topics.

When I came across the unit volume tetrahedron and the hierarchy of volumes based on that:
Cube == 3, Octahedron == 4, Rhombic dodecahedron == 6... and when I saw how this paradigm was easily dismissed simply because it's different, my faith in the Anglos and their way of thinking somewhat deteriorated. 

Thinking mathematics has to have "foundations" and that these will look like set theory and/or Principia Mathematica is not an attractive idea to me.

I go with Wittgenstein's dictum:  set theory / logic provides a foundation for mathematics in the same sense a painted foundation supports a painted castle.  (paraphrase).
 

I don't care about the meaning of a screwdriver. I use screwdrivers when I need them but I spend a lot of time thinking about abstract, invisible, Platonic numbers whether I need them or not, just because I enjoy it. And I enjoy them the more because they are abstract, and invisible.

Juan

I think that's fine.  On the other hand, I don't think the fact that you like your cage and want to stay in it means that people new to math should be given the impression that there is this one way that's right and true and all the other cages should be left empty. 

The sense of "shopping around", of "choice" should be preserved.  I like to present alternatives to the standard / traditional / conventional definitions with this aim in mind.  I hope that's OK with you.  I'm not suggesting your abandon your beliefs.  I'm just keeping the beach free and open to other sandcastle builders who might want to start in a different place.


Kirby

[michel paul]

On Fri, Dec 14, 2012 at 4:37 PM, kirby urner <kirby...@gmail.com> wrote:

On Fri, Dec 14, 2012 at 4:15 PM, Juan <here...@gmail.com> wrote: 

 

>>Two was prime before Wittgenstein was born, and it still continues to be a prime number today. 

>Also 17 and 23.  However -1 is usually not considered prime, though some have proposed it be added.

I like the idea of treating -1 as a kind of prime. However, it would behave a little differently from other primes in that it would not terminate a factoring process. I could conceivably factor 1 into (-1)^2 or (-1)^4, etc, so 'prime' factorizations wouldn't be unique unless we agreed to use the smallest power of -1, an agreement we don't need to make for the other prime factors.

However, in the same way we might also think of unit fractions as kinds of primes. In fact, this is even what you see in something like Sage:

sage: factor(-391/14691)

-1 * 3^-1 * 17 * 23 * 59^-1 * 83^-1

Or in Mathematica:

In[6]:= FactorInteger[-391/14691]

Out[6]= {{-1, 1}, {3, -1}, {17, 1}, {23, 1}, {59, -1}, {83, -1}}

Would the character string "17" created from the concatenation '1'+'7' be considered prime? I'd say no more so than the character string "18". Or the character string "seventeen". In digit string math it would probably make more sense to consider single digit characters as 'prime' and any group of two or more as 'composite'.

What is prime or not in our usual number sense is not a symbolic string but a quantity. Quantities occur naturally, prior to human naming, and they have their own properties. Take a flock of birds, for example. Or dinosaurs. (same thing) Quantities of different types interact with each other in various ratios prior to human attempts to represent them. If these ratios did not already exist in nature prior to our attempt to represent them, there would be no reason to do science. There do exist naturally occurring physical constants and relations that our approximations approach asymptotically over time.

There's definitely a difference in the use of '5' to describe the 5 actions of Shiva vs. the use of '5' to refer to the #5 bus. The difference is that I could change the way I name busses from digits to pieces of fruit. So the #5 bus could be changed to the 'apple' bus, and the #3 could be changed to the 'banana'. Everything would work just like it did before, but the schedules and bus displays would look a little funny, like an amusement park. However, I couldn't just as easily refer to the 'apple' actions of Shiva or the 'banana' actions.

I think it actually is possible to talk about what is common to 5 actions, 5 people, or 5 locations without having to postulate an other-worldly realm of abstract objects. What I'd like to say is that abstract objects such as numbers and patterns are just as real as sticks and stones. They have properties, and they are part of the physical world.

"A pattern has an integrity independent of the medium by virtue of which you have received the information that it exists."  - Buckminster Fuller

Even more, ideas can be natural forces in the world in that they motivate people to create physical changes. Ideologies can change landscapes. Ideas are a part of physical reality, not separate from it.

>mathematics tends to attract those who are temperamentally averse to the "many ways to do it" philosophy.

Really? Schoolish math perhaps, but not mathematics itself. Take Gauss for example, as a kid using pairing to creatively find the sum of 1+2+3+...+100. Good mathematical thinking is creative, not confined to a box.

>what I think is valuable, especially for people new to maths, is to remind them it's a beach of many sand castles

I agree. There's all kinds of ways to do math, and we should encourage that. It's something that the calculus funnel in our curriculum shuts down. A colleague used to frequently assert at meetings, "Math is sequential in nature." It always bothered me when he said that, especially with such an air of authority. Finally I said, "Being 'sequential in nature' means that there is always a 'next' term, right?  Well, that is not the case for either the reals or the complex numbers!"

Mathematics also did not evolve 'sequentially' in history.  It was more like various centers of mathematical activity gradually reached out and communicated with each other.  And that's also how it happens in our brains.

-- Michel

[Christian Baune]

Juan,

I did "automatically" optimize the number of digits !
VIII has 4 digits, IIX only 3.
For IIV, I found it easier to understand than III :-p

I did really don't know the exact rules of roman number so did I a search and found this : http://www.factmonster.com/ipka/A0769547.html

Giving those rules to pupils and asking them to write some number would probably make them understand the bases of a numeral system !

Kind regards,

Christian

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[kirby urner]

On Sat, Dec 15, 2012 at 8:39 AM, michel paul <python...@gmail.com> wrote:

There's definitely a difference in the use of '5' to describe the 5 actions of Shiva vs. the use of '5' to refer to the #5 bus. The difference is that I could change the way I name busses from digits to pieces of fruit. So the #5 bus could be changed to the 'apple' bus, and the #3 could be changed to the 'banana'. Everything would work just like it did before, but the schedules and bus displays would look a little funny, like an amusement park. However, I couldn't just as easily refer to the 'apple' actions of Shiva or the 'banana' actions.

Funny you should mention this, as I have a web page called Vegetable Group Soup which uses vegetables (and a mushroom as I recall) to stand for low order integers.  I also raise vegetables to integer powers.  This isn't to contradict anything you've said, as using elaborate icons in place of the usual numerals is a pain.

http://www.4dsolutions.net/ocn/flash/group.html  (should turn down sound or switch it off as I used some annoying sound effects in the Flash demo of multiplication happening).

In base64 encoding the need is for a lot more numerals.  The logical solution was to use all upper and lowercase ASCII letters (26 + 26), plus the digits (+ 10), plus a couple more, like =.  Base64 is useful for sending email attachments as the SMTP (simple mail transfer protocol) is only comfortable with 7-bit ASCII i.e. the first bit should be 0.  The base64 algorithm takes 3 8-bit bytes and converts them to 4 6-bit chunks with two leading zeros, then maps those permutations to the symbols described.  A picture attachment to an email is run through this transformation such that it goes over the wire in readable/printable ASCII gobbledygook and then reconverts on the other end.

http://en.wikipedia.org/wiki/Base64

Yet another language game, yet another use of mathematics in STEM.

I think it actually is possible to talk about what is common to 5 actions, 5 people, or 5 locations without having to postulate an other-worldly realm of abstract objects. What I'd like to say is that abstract objects such as numbers and patterns are just as real as sticks and stones. They have properties, and they are part of the physical world.

Books like Cirlot's Dictionary of Symbols will take a number like Five and build out associational webs, drawing on subcultures which encode ideas and concepts in various pithy ways. 

Five fold symmetry is important for breaking us out of a lattice i.e. regular pentagons won't tile and the 5-ish Platonics won't space-fill while the others will in combination with one another (e.g. tetrahedron + octahedron, and of course cube).  The virus, schematically an icosahedron, is a most primitive "living machine" that breaks free of The Matrix (i.e. the lattice) to become an individual, an agent.  Hargittai & Hargittai wrote Symmetry:  A Unifying Principle, and more recently Istvan edited Five Fold Symmetry in particular, playing up these webs of association.

http://books.google.com/books/about/Symmetry.html?id=ITsy4v5DuYwC

http://books.google.com/books?id=-Tt37ajV5ZgC

Bradford's way for writing about Zero puts me in a more Jungian head space, where I'm weaving a lot of cosmic significance into low order integers.  I'm not against doing this, but in the Anglo compartmentalization schema, talking about the psychological and/or archetypal aspects of numbers is not in the same ballpark as talking about the set of all sets of Five members. 

Putting a wall in here is not bad or wrong, but we should admit it's one ethnic group's way of keeping discourse more orderly.  Psychology over here, logic and set theory over there.

Looking for commonalities is natural.  The concepts of "translation" (as in moving laterally), "rotation" (which may be done in place) and "scaling" (change in size without changing any angles, either surface or central) my be generalized as well of course (not just number-nouns but actions-verbs may be generalized). 

I like the fact that we use "translation" in English to mean re-expressing in a different language, as if we were taking some "meaning body" and simply sliding it ("translating it") from one language to another (language = coordinate system).

 

"A pattern has an integrity independent of the medium by virtue of which you have received the information that it exists."  - Buckminster Fuller

Even more, ideas can be natural forces in the world in that they motivate people to create physical changes. Ideologies can change landscapes. Ideas are a part of physical reality, not separate from it.

>mathematics tends to attract those who are temperamentally averse to the "many ways to do it" philosophy.

Really? Schoolish math perhaps, but not mathematics itself. Take Gauss for example, as a kid using pairing to creatively find the sum of 1+2+3+...+100. Good mathematical thinking is creative, not confined to a box.

Yes, schoolish math. 

You will be able to collect personal testimonials from many a math teacher that they didn't like disciplines that seemed relatively undisciplined, especially when it came to grading.  Those who take grades very seriously tend to find math in school a "fairer" subject whereas other subjects seemed more, well, subjective, and the grades might seem to depend more on who the teacher thought was smarter in some nebulous hard-to-prove way. 

Mathematics was "objective" in the sense that every question on the test had a single right answer, and the steps to getting that answer tended to be unambiguous.

In math, a student might raise objections ("but but....") e.g. it seems hard to grasp how infinitely small points of no dimension may somehow "add up" to give an infinitely long line of only one dimension, lines which may "raft together" to make a plane and so on. 

With actual objects, like marbles, it's easy to make a line of marbles, and pencils make a nice raft or plane, but then we're always told actual objects are inferior stand-ins for the "real things", which turn out to be fantasies and little plays in the mind's eye -- also pretty subjective when you get down to it. 

But a math teacher can easily dismiss such objections and say "sorry, that's just how we think, this is the way math defines them" and that's the end of the matter.  A schoolish math teacher effectively squelches dissent / debate, whereas in other subjects it can seem as if nothing ever gets settled.

 

>what I think is valuable, especially for people new to maths, is to remind them it's a beach of many sand castles

I agree. There's all kinds of ways to do math, and we should encourage that. It's something that the calculus funnel in our curriculum shuts down. A colleague used to frequently assert at meetings, "Math is sequential in nature." It always bothered me when he said that, especially with such an air of authority. Finally I said, "Being 'sequential in nature' means that there is always a 'next' term, right?  Well, that is not the case for either the reals or the complex numbers!"

Yeah, I much prefer to think of maths as a network wherein one bounces around among related topics.  In a network or graph, of edges and nodes, some nodes have many more edges coming into them.  I call these "grand central stations" in that they relate to so many other ideas.  "Pascal's Triangle" is an example of a GCS node.

An esoteric pass time in ages past was to organize polyhedrons in concentric arrangements in memorable / re-constructable ways. 

Unfortunately, this pass time got left on the "wrong side" of the wall in the Anglo system of compartmentalization i.e. the cosmic / Jungian / psychological side.  Polyhedrons are "back of the bus" in today's schoolish math, despite their relevance to STEM (chemistry, architecture, generic spatial geometry). 

I consider an arrangement of nested polyhedrons to be a Grand Central Station node, but mostly missing from Anglo-American schoolish maths. 

I like the wholesome whole numbers that fall out when the tetrahedron is used as the unit, versus the cube.  Finding a way to work this in has been an uphill battle, but I believe I've found a working solution in the form of Martian Math, one of four components in a Digital Math that covers a lot of ground.

http://wikieducator.org/Martian_Math
 

Mathematics also did not evolve 'sequentially' in history.  It was more like various centers of mathematical activity gradually reached out and communicated with each other.  And that's also how it happens in our brains.

-- Michel

History is massively parallel / concurrent.  Given global telecommunications and the Web, it has become even more so.

Kirby

[Juan]

Christian,

That is a good reference. Thank you for the link. This Wikipedia article
http://en.wikipedia.org/wiki/Roman_numerals

has interesting explanations about the usage, and history of Roman numerals. They also state that "There has never been a universally accepted set of rules for Roman numerals." So Roman numerals can be a bit confusing at times but if you avoid confusing your students by choosing one clear way of writing them, and sticking to it, then students can benefit by increasing the resources they have to do mental math.
One benefit of familiarizing yourself with Roman numerals is that at some point it becomes automatic to think this way:

6 = 5+1;  7 = 5+2;  8 = 5+3;  and  9 = 10-1,

without any extra effort, nor being prompted to. It becomes a habit after writing VI, VII, VIII, IX  enough times.
Having these facts available all the time for conscious thinking, helps mental calculations like this:

7+8 = (5+2) + (5+3) = (5+5) + (3+2) = 10 + 5 = 15

or,

8+3 = (5+ 3) + 3 = 5 + (3+3) = 5 + 6 = 5 + (5+1) = (5+5) + 1 = 10 + 1 = 11
 
Juan

[Juan]

Kirby,

If we are ever going to be able to communicate at all, I must tell you I cannot possibly read all you write, nor follow all the directions of your thinking. My mind uses processors a lot slower than those that seem to produce your writing output. This is not a critique but just an observation.
From my perspective, your posts jump all over the place, cover way too many topics, go off in too many different directions, and make way too many connections to other topics. I have to ignore about 95% of what you write because your posts feel to me like fractal, ever branching, never ending monologues.
For example, I mentioned earlier that Two is a prime number before and after Wittgenstein just to point out the fact that Wittgenstein's philosophy (or any other philosophy for that matter) does not change, affect, or alter any mathematical fact.
From there, you inserted that also 17 and 23 are primes. Yes, the number of prime numbers is infinite. The number Two is not the only prime. Yes, yes, true enough but what is the point of saying it at this moment? What relevance does that have?
Then you start on the number One not being a prime. Also true but again, what is the need for bringing that up into the discussion? Just because One is not a prime should I start studying Wittgenstein's philosophy? What is the connection?

Why would you add social security numbers?

Yes, string concatenation can be written with the "+" sign as in computer languages. That does not make it addition even if it "looks like" addition just because we are using the "+" sign for it.
Addition is addition and string concatenation is string concatenation. Again, so what? Why does that matter? What does it relate to?

Scrolling down to the end of your post I think I start to see your points:

"I don't think the fact that you like your cage and want to stay in it means that people new to math should be given the impression that there is this one way that's right and true and all the other cages should be left empty." 

and

"The sense of "shopping around", of "choice" should be preserved.  I like to present alternatives to the standard / traditional / conventional definitions with this aim in mind.  I hope that's OK with you.  I'm not suggesting your abandon your beliefs.  I'm just keeping the beach free and open to other sandcastle builders who might want to start in a different place."

 
OK, so for the sake of "shopping around," could you please give me one alternative definition of something that is not "my cage"? Could you please show me "another cage," so to speak? But just one please, one at a time, don't show me a full menu because multiple choices confuse me. Please focus on one, just one, not more, please. What would be one "alternative  cage"? Maybe I can understand what you are talking about.

Juan

[Juan]

Bradford,

In relation to your phrase:

" The concentric nature of the circle suggest to me that with expanding scale the number lines increase possibilities for realized potential of greater expanded alignment."

I'm not sure what you are referring to. The only thing (math-wise) I think it could be related to it is this:

The equation of a circle with center at the origin (0,0) and radius r is

x^2 + y^2 = r^2

Changing the value of the 'r' parameter gives you a different circle. All these circles have the same center, so they are concentric. Setting the 'r' value as a function of time, for example

r(t) = t

we can "run the movie," and imagine the circle with center at the origin expanding uniformly with an ever increasing radius.
All along, the horizontal and vertical lines given by these equations:

x = t;   x = -t;   y = t; and  y = -t

keep moving away from the origin, all the time being tangents to the circle.
Is that what you meant?

Juan

[kirby urner]

On Sat, Dec 15, 2012 at 1:21 PM, Juan <here...@gmail.com> wrote:

Kirby,

If we are ever going to be able to communicate at all, I must tell you I cannot possibly read all you write, nor follow all the directions of your thinking. My mind uses processors a lot slower than those that seem to produce your writing output. This is not a critique but just an observation.

I don't assume anyone is reading everything I write.  You and I have had some dialog.  I try to keep my comments self-contained enough for the dialog to make sense in isolation.

We were talking about whether points are zero dimensional, whether numerals point to numbers in a different kind of object space.  We were talking about the definitional foundations of various brands / types of mathematical system.

I was pointing out:  hey, we can have a non-Euclidean geometry that differs in how points, lines and planes are defined.  Karl Menger proposed this.  Here's a link:

http://coffeeshopsnet.blogspot.com/2009/03/res-extensa.html
 

From my perspective, your posts jump all over the place, cover way too many topics, go off in too many different directions, and make way too many connections to other topics. I have to ignore about 95% of what you write because your posts feel to me like fractal, ever branching, never ending monologues.
For example, I mentioned earlier that Two is a prime number before and after Wittgenstein just to point out the fact that Wittgenstein's philosophy (or any other philosophy for that matter) does not change, affect, or alter any mathematical fact.

J.H. Conway, a famous mathematician, suggested that -1 be added to the list of primes, to encompass negative primes in that way.  I haven't followed all the arguments pro or con.

Philosophy and mathematics overlap, especially where "foundations" are concerned.  What philosophers think and do does change the shape of mathematics, Bertrand Russell a case in point.  Do we call Lewis Carroll a philosopher?  Issues of taxonomy obtrude (how to pigeon-hole people).
 

From there, you inserted that also 17 and 23 are primes. Yes, the number of prime numbers is infinite. The number Two is not the only prime. Yes, yes, true enough but what is the point of saying it at this moment? What relevance does that have?

A was agreeing there are an infinite number of mathematical facts one might cite.  I didn't want "two" (2) to seem too special.  Yes, it's the only even prime.

2 + 2 = 4.  Is that always true?  One may define the symbol '2' in a framework where addition is modulo 3.  Or maybe this is the base 3 system.  So no, not always true.  Depends on the language game.

Does that seem relevant to you?  I'm saying context matters.  Not every mathematics needs to include the concept of "real numbers" (historically speaking, none of them did before a certain era, when they emerged in that shape (and continue to evolve e.g. in co-dependent relationship with "infinitesimals").

Then you start on the number One not being a prime. Also true but again, what is the need for bringing that up into the discussion? Just because One is not a prime should I start studying Wittgenstein's philosophy? What is the connection?

No, I brought up -1 as *maybe* being prime. You were probably reading quickly at that point.  There's been some real debate on whether it should be included.

 

Why would you add social security numbers?

I'm pointing out that symbols like 97214 can't be assumed to point to some number if they're involved in a different game.  In the idea that numerals point to numbers, which are in some invisible realm, as a type of Platonic object, there's the assumption that we know what 97214 means by just looking at it.  But what if it's a zip code?

It takes the rules of chess to define what a "pawn" means and, beyond that, it takes the rules of a civilization to define what "chess" means.  It takes the whole to define the parts.  It's not like the parts (e.g. 97214, 2 or +, or =) have independent meaning, minus the games they're involved in.  There's no "pawn" outside of chess.

Math fragments into many language games, not necessarily all logically consistent because when it comes to axioms and definitions, over-arching consistency is not the goal or challenge.  There's no requirement that maths "unify" or be "just one thing".  Philosophies help them divide and thereby multiply sometimes.
 

Yes, string concatenation can be written with the "+" sign as in computer languages. That does not make it addition even if it "looks like" addition just because we are using the "+" sign for it.

The word "addition" takes on different meanings depending on the game.

"Be fruitful and multiply".  What does "multiply" mean?  No one particular thing.
 

Addition is addition and string concatenation is string concatenation. Again, so what? Why does that matter? What does it relate to?

Scrolling down to the end of your post I think I start to see your points:

"I don't think the fact that you like your cage and want to stay in it means that people new to math should be given the impression that there is this one way that's right and true and all the other cages should be left empty." 

and

"The sense of "shopping around", of "choice" should be preserved.  I like to present alternatives to the standard / traditional / conventional definitions with this aim in mind.  I hope that's OK with you.  I'm not suggesting your abandon your beliefs.  I'm just keeping the beach free and open to other sandcastle builders who might want to start in a different place."

 
OK, so for the sake of "shopping around," could you please give me one alternative definition of something that is not "my cage"? Could you please show me "another cage," so to speak? But just one please, one at a time, don't show me a full menu because multiple choices confuse me. Please focus on one, just one, not more, please. What would be one "alternative  cage"? Maybe I can understand what you are talking about.

Juan

I will refer you to the idea that 2nd powering may be effectively modeled with a triangle, 3rd powering with a tetrahedron.

This is something I make sure they know before age 10 if possible.

Kirby

[Bradford Hansen-Smith]

" The concentric nature of the circle suggest to me that with expanding scale the number lines increase possibilities for realized potential of greater expanded alignment."

I'm not sure what you are referring to. The only thing (math-wise) I think it could be related to it is this:

The equation of a circle with center at the origin (0,0) and radius r is

x^2 + y^2 = r^2

Changing the value of the 'r' parameter gives you a different circle. All these circles have the same center, so they are concentric. Setting the 'r' value as a function of time, for example

r(t) = t

we can "run the movie," and imagine the circle with center at the origin expanding uniformly with an ever increasing radius.

All along, the horizontal and vertical lines given by these equations:

x = t;   x = -t;   y = t; and  y = -t

keep moving away from the origin, all the time being tangents to the circle.

Is that what you meant?

Juan

Jaun, that is not quite what I mean, but it works. First I do not see the origin of the circle as the center point; the circle is the center. Concentric circles more away from the center as well as into the center; self-alignment. Movement goes minimum in two directions. In that regard the diameter is the measure of a circle and radii are subsets. Origin is different than beginning. We begin drawing the image from center point out.  Origin is where the pattern of circle originated which we have no clue since we have a beginning and were not around to know. Concentric in both directions as far as anyone knows.

Is folding circles in your cage? I think not, but then what you do is not in mine. Otherwise there would be little discussion and no progress.

Kirby, I support your focus on the tetrahedron as foundational. It is the first movement of folding the circle in half. The other four regular polyhedra are easily modeled by joining multiple regular tetrahedra folded from the same diameter circles. This is a very different understanding of relationships between polyhedra then when edge lengths are changed to show nesting. Each give us a greater understanding about the primacy of the tetrahedron pattern and the many possibilities for formed expressions. This modeling is not beyond 6 and 7 year old students and can start as early as when a child can fold a circle in half.

Brad

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[Juan]

Bradford,

By origin I mean the point (0,0) in the XY-plane, or the point (0,0,0) in 3-D space. I do not mean the origin of the circle in history, or in human culture.
The diameter of the circle is one possible measure of the circle. We can also measure it by its circumference, it area, or its radius. Knowing any one of these four: radius, diameter, circumference, or area of a circle, we can calculate its other three numbers.
The word "concentric" only means "having the same center." So, a circle can be concentric with an ellipse, or with a square, or a hyperbola, since all of them, each has a single, well-defined center point.

Juan

[Juan]

Kirby,

Thank you for the reference to Karl Menger. I will take a look at it later.
Do you actually play any games? I used to play Chess but I stopped playing it once I leaned how to play Go.
I am under the impression you always seem to be saying:
"Well, you can play that game but that is not the only game, there is this other game, you know, and this other one, too. There is a whole bazaar full of different games over there."
My point is that shopping for games is not the same as playing any particular game. Yes, context matters. Yes, sometimes you need to explicitly clarify your assumptions. That does not mean you need to change them. Constantly switching to other contexts is like rushing from country to country taking snapshots of churches and temples, without ever taking the time to stay in one place, and getting to know it in detail. It's like if you said: "But this is also France, we don't have to stay in France, we can go to Germany, they have different architectural styles over there; or we can go to Italy, they have different food. Or, we could even go to Jakarta, we don't need to stay in Europe." And my reaction is: "Can we please stay put in a single city for at least one week at a time? We don't need to go around the globe every single day!"

As for this thread, what do you say? Is Zero even? Yes or no? In what contexts is Zero even? How could Zero not be even? I mean in math, within the confines of well defined, well constructed mathematical "cages," or frameworks, or axiomatic developments. Is Zero even?

Just saying: "Well, depending on the context, Zero may not even be a number, so it wouldn't make sense to consider it even or not."
Or saying: "You can make a sculpture in the shape of the numeral for zero, and you can make it very even, so yes it can be even."

Those kind of answers to me are unacceptable. They are rather, non-answers, they are just a way to keep talking without addressing the question but just questioning the question's context.

Juan

[kirby urner]

On Sun, Dec 16, 2012 at 12:46 PM, Juan <here...@gmail.com> wrote:

Kirby,

Thank you for the reference to Karl Menger. I will take a look at it later.
Do you actually play any games? I used to play Chess but I stopped playing it once I leaned how to play Go.
I am under the impression you always seem to be saying:
"Well, you can play that game but that is not the only game, there is this other game, you know, and this other one, too. There is a whole bazaar full of different games over there."

Hi Juan --

I do play some games.  We should remember also the word "game" as appropriated by Game Theory is applied in economics and world affairs.  In a "war game" you might have a flotilla of Russian warships (real ones) heading towards the Mediterranean (true recent story).

In the Bucky Fuller namespace we have "World Game" which refers both to a workshop (no longer offered) but also to whatever you're doing with your time on "Spaceship Earth" (another term coined by Dr. Fuller, who also contributed mathematical content to the literature) i.e. we're all playing "the world game" or "the game of the world". 

Then we had the "Glass Bead Game" of Herman Hesse (in his book Magister Ludi), which served as a metaphor for any uber-academic undertaking i.e. whatever "mind game" one wishes to imagine.

My point is that shopping for games is not the same as playing any particular game. Yes, context matters. Yes, sometimes you need to explicitly clarify your assumptions. That does not mean you need to change them. Constantly switching to other contexts is like rushing from country to country taking snapshots of churches and temples, without ever taking the time to stay in one place, and getting to know it in detail. It's like if you said: "But this is also France, we don't have to stay in France, we can go to Germany, they have different architectural styles over there; or we can go to Italy, they have different food. Or, we could even go to Jakarta, we don't need to stay in Europe." And my reaction is: "Can we please stay put in a single city for at least one week at a time? We don't need to go around the globe every single day!"

I agree with this analogy and in terms of it, I think we metaphorically journey to ancient Greece and Europe by introducing some classic well-known concepts, and then we just stay put and never question them.  We don't get outside the cage or even realize it's a cage, because no touring of other vistas happens. 

We're stuck with dimensionless points and rectilinear treatments of 2nd and 3rd powering.  This becomes our prison and we're not even told there might be other ways to go. 

Mathematics comes across as monolithic, settled, undeniable (unlike many of the humanities).

In contrast, when dealing with n x m and showing a rectangular model, I like to show a triangle modeling the same concept.  I'll draw an equilateral triangle and label all sides n, and then write n^2 or n-to-the-second power inside.  Am I allowed to do this?  Am I not breaking some law? 

Then I go further.  I draw an equi-edged regular tetrahedron with all sides labeled n, and inside, for the volume, I write n^3 or n-to-the-third power. 

Is this a published model in the literature and is it self-consistent and logical?  Yes.  As a tour guide, I would take me students here, here and here:

http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
http://grunch.net/synergetics/volumes.html
http://en.wikipedia.org/wiki/Quadray_coordinates

Now I look out over the many mathematics curricula that are out there, and I look for those that provide little "field trips" of this nature, into alternative perspectives. 

How many curricula actually discuss a unit-volume tetrahedron with children or young adults?  I find zero, none, zip that are state mandated, i.e. no existing published textbook used in K-12 in any real classroom today makes any mention of this alternative paradigm.   Everyone grows up thinking 2^2 is "2 squared" and 2^3 is "2 cubed" without a second thought or even a token nod to other possibilities. 

Do I blame mathematics for this?  No.  I blame the way it's taught.  It's too narrowing. 

Math teaching in the future will contain more safeguards against this.  The textbooks of today will be held up as models of how math used to be rather poorly communicated.
 

As for this thread, what do you say? Is Zero even? Yes or no? In what contexts is Zero even? How could Zero not be even? I mean in math, within the confines of well defined, well constructed mathematical "cages," or frameworks, or axiomatic developments. Is Zero even?

Yes, Zero is a number, and is even, in the games I play.  Divided by 2 there is no remainder.  That's my criterion.  That's true for any number i.e. (0/n) never has a remainder (n is complex, which includes all the reals), but having that be true for 2 in particular is all we need for 0 to be considered even.

This is true even where the set of integers is finite and "wraps around" i.e. an "even zero" does not require using an infinite set of any particular type of number.

Just saying: "Well, depending on the context, Zero may not even be a number, so it wouldn't make sense to consider it even or not."
Or saying: "You can make a sculpture in the shape of the numeral for zero, and you can make it very even, so yes it can be even."

Those kind of answers to me are unacceptable. They are rather, non-answers, they are just a way to keep talking without addressing the question but just questioning the question's context.

If you give me an apparatus of three mutually orthogonal line segments ala XYZ and ask me about the evenness or oddness of the various points, including (0,0,0), then I will be far less certain about what move to make in this language game.  Should I say "yes" or "no" or "does not apply"? 

I would say the number 0, and the origin of a coordinate system are not synonymous concepts.

One *might* define "even points" as any n-tuple where all elements were even, e.g. (2, 5, 7) is odd whereas (2,2,8) is even.  But that's not a rule we're taught in school and there's no broad consensus we should talk in this way.

To round out with Wittgenstein, what he wanted to show was that imagining "objects" as "the meanings" of terms is rather tempting, but it's quite possible to dismiss specific imagery, including mental imagery as just more of the usage pattern in which a term is deployed, and it's these patterns that give rise to meaning. 

Imagining objects may be part of the game, but the "meaning" of a mathematical term need not be some "thing", nor must even prosaic words like "cow" and "chair" be considered pointers to objects.  They're tools. 

Meanings are not "things that words point to".  That was an old idea of what in means "to mean" (to "point" in some fashion) but at least in philosophy, a lot of us no longer believe in meanings as any kind of object, visible or invisible, physical or imaginary. 

The meaning of "2" is no kind of object of any kind, ditto a "pawn" in chess or a white or black Go chip.  These are symbols without objects because the idea that symbols need objects behind them (perhaps ghostly) in order to mean something turns out to be a superstition we're free to dispense with.

Kirby

[michel paul]

On Sun, Dec 16, 2012 at 3:06 PM, kirby urner <kirby...@gmail.com> wrote:

>I would say the number 0, and the origin of a coordinate system are not synonymous concepts.

Right. 

A good way to see this is to use the x-y coordinate system to model the rational numbers. 

Let the x axis be the denominator axis, and let the y axis be the numerator axis. 

Where is 0? Since 0=0/1, we'd initially locate it at the ordered pair (1,0). 

The rest of the natural numbers would be located at (1,1), (1,2), (1,3), etc. 

The unit fractions would be located at (1,1), (2,1), (3,1), etc.

Each lattice point would determine a line through the origin (0,0). The slope of that line would be the rational number determined by the lattice point.

Lots of interesting consequences follow. Equivalent ratios would lie on the same line. The ordered pair (1,0) would correspond to infinity. 

And what about the origin, (0,0)? What would that be? Not zero, as we already located that at (1,0).

The origin would be, well, just that, the origin of all the numbers, including zero itself.

-- Michel

[Algot Runeman]

On 12/09/2012 01:25 PM, Sue Hellman wrote:
> Algo -- thanks very much for this your metaphor I'm working on a
> workshop in active learning strategies for math educators and would
> like to use it if I may.
Sorry for the delay Sue. You, and anyone else, may certainly use any
material I've posted. Note that the images were not mine and should be
credited.
--Algot

--
-------------------------
Algot Runeman
47 Walnut Street, Natick MA 01760
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algot....@verizon.net
Web Site: http://www.runeman.org
Twitter: http://twitter.com/algotruneman/
sip:algot....@ekiga.net
Open Source Blog: http://mosssig.wordpress.com
MOSS SIG Mailing List: http://groups.google.com/group/mosssig2

[Algot Runeman]

On 12/15/2012 11:39 AM, michel paul wrote:

Mathematics also did not evolve 'sequentially' in history.  It was more like various centers of mathematical activity gradually reached out and communicated with each other.  And that's also how it happens in our brains.

Michel,

This statement reminds me of the distilled nature of an algorithm (or a teaching sequence). Gaining insight from experience does not generally come from a fully formed method. It comes often through stumbles that reveal something unexpected. A plan to test the new idea then begins to emerge. Nonetheless, the plan may fail to produce the expected result because of unrecognized constraints. Gradually our experiments lead to an "efficient" way to get to the goal. That is the experiment which is reported to the world. It isn't common to see a scientific paper which goes through the (often long) list of abandoned trials. Indeed, other working professionals assume the method didn't come all at once, and they want to see a concise experiment which they can try to reproduce.

Teachers are an interesting group of working professionals. We are not often charged with developing the material we "teach." Being handed the approved textbook and a curriculum guide is the common lot of a new teacher.

Teaching a concept is often called "covering" the material. Efficiency is viewed as important. Each year there is more to cover. Efficiency becomes more important. The cycle leads to a "method which works" being taught. Students move along the chain. Some may fall behind, but the sequence continues.

Learning is personal. Teaching is harder. Getting a learner to the "aha moment" is satisfying. The effort to get *every* learner to that moment is humbling. As it turns out, the best teachers, in my opinion, are those who do actually flex the curriculum, rewrite the textbook and do actual experiments to find the dynamics that help the most students learn. Even better, their willingness experiment may actually (re)awaken the natural curiosity of learners. Learners may begin to see that their guide isn't just giving them something to "get through."

Joy of the season.

[Juan]

Dear Kirby,

The "aesthetically pleasantness" of simplexes is subjective, therefore relative. Not everybody likes tetrahedrons more than cubes, or triangles more than rectangles. Now, in everyday practice, simplexes are bound to be run over, trounced (99.9% of the time) by parallelepipeds when it comes to stacking boxes in warehouses, or inside shipping containers; or stacking containers inside big cargo ships, or trains. Not even bees use simplexes, they use hexagonal prisms. Even mineral crystals show a lot of "prismic" shapes, although I think in that realm you may have a better chance of finding naturally occurring tetrahedrons.
No, of course no mandated state curriculum is going to present that (with all due respect) crazy alternative to volume just because it's there and it's logically consistent. No! It would confuse the heck out of a lot of teachers, let alone their poor pupils, having to learn "the alternative way" from teachers who would probably be confused about that notion themselves. If both a cube and a tetrahedron with edges of the same length are going to have the "same" "volume," when they obviously have very different sizes, you'll have a student population quite confused about the physical concept of volume, let alone what formulas to use when solving volume problems.
I believe every engineer, and most physicists, and chemists, would oppose such volume definition to be placed in textbooks.
No, no way, forget about it. I cannot imagine a school district administrator or board member even proposing such inclusion in the curriculum. People who like simplexes go on to write PhD theses on that specific, tiny mini-mini-cage, whatever its wealth of relationships to other parts of math may be.
I don't want to sound pessimistic but good luck trying to get your hobby accepted within the mainstream school curriculum. I don't see that happening any time soon. Of course, I may be wrong, absolutely but I wouldn't jump on any "volume-by-tetrahedrons-to-the-curriculum" kind of movement. I truly believe you have a better chance of stopping global warming, than to see that alternative volume definition in the textbooks.
Even the (again, with all due respect) loony "tau-ist," "anti-Pi" mathematicians who want to replace 3.14 with 6.28 as "the" "circle number," have a better chance of succeeding than you have with your "simplex-volume-definition."
As frustrating, and limiting as it may feel to you, I think it's just common sense the way it is. Happy Holidays!

Juan

[Maria Droujkova]

On Wed, Dec 19, 2012 at 3:49 AM, Juan <here...@gmail.com> wrote:

Dear Kirby,

The "aesthetically pleasantness" of simplexes is subjective, therefore relative. 

This definition from information theory is one of better attempts (ha! judgment call) to make beauty more objective: http://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_and_mathematical_information_theory

I have not tried to apply it to simplexes. It should be a good exercise for Kirby and friends.

 

I don't want to sound pessimistic but good luck trying to get your hobby accepted within the mainstream school curriculum.

The better bet (ha! again) is to do away with the idea of The One Mainstream Curriculum To Rule Them All (TM). I can imagine some kids, somewhere being very interested in this sort of stuff. Just like other kids are interested in robotics, math storytelling, hyperbolic crocheting and other varied geekiness and niche pursuits. Diverse educational ecosystems for the win!

After all, it is the job advice cliche of the last decade that you need to pick a narrow area and to get exceptionally good at it so that people can't ignore you anymore.

Cheers,
Dr. Maria Droujkova

919-388-1721 

[Phillip Kent]

Yet, there is an important element of truth to Kirby's challenge to
'common sense'.

Witness the extraordinary cost in human life for our collective
obsession with 'common sense' parallelipipeds...

http://fallmeeting.agu.org/2012/events/white-lecture-nh22b-defeating-earthquakes-video-on-demand/

- Phillip

On Wed, 2012-12-19 at 00:49 -0800, Juan wrote:
> Dear Kirby,
>
> The "aesthetically pleasantness" of simplexes is subjective, therefore
> relative. Not everybody likes tetrahedrons more than cubes, or
> triangles more than rectangles. Now, in everyday practice, simplexes
> are bound to be run over, trounced (99.9% of the time) by
> parallelepipeds when it comes to stacking boxes in warehouses, or
> inside shipping containers; or stacking containers inside big cargo
> ships, or trains. Not even bees use simplexes, they use hexagonal
> prisms. Even mineral crystals show a lot of "prismic" shapes, although
> I think in that realm you may have a better chance of finding
> naturally occurring tetrahedrons.

++++++
Dr Phillip Kent, London, UK
mathematics education technology research
philli...@gmail.com mobile: 07950 952034
www.phillipkent.net
++++++
"Man's rush to the n'th floor is a neck-and-neck race
between plumbing and abstraction" - Rem Koolhaas

[Bradford Hansen-Smith]

Maths are ideas in the mind. Experiences are registered in physical reality. Where do we experience mathematics?

Brad

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[michel paul]

On Wed, Dec 19, 2012 at 12:49 AM, Juan <here...@gmail.com> wrote:

Even the (again, with all due respect) loony "tau-ist," "anti-Pi" mathematicians who want to replace 3.14 with 6.28 as "the" "circle number," have a better chance of succeeding than you have with your "simplex-volume-definition."

Hi Juan - 

With all due respect as well, I don't find the tau movement that loony. The issue is not simply about replacing '3.14...' with '6.28...', as there wouldn't be much accomplished in doing that. The real issue is starting with the fundamental idea of 'one turn' as the fundamental circle constant. That's it - tau is simply 'one turn'. Its value is the distance around a unit circle. It is a very simple starting point that has lots of interesting consequences. I know that it initially seems silly, like so what?, but when you work with it you find that it actually simplifies many things. For example, take the notion of angle measurement - an angle is simply a fraction of a turn. 1/4 turn is simply 1/4 tau. If you want a decimal approximation, it's 1/4 of 6.28... . It makes immediate sense. If you substitute 2 pi for tau, you will get 1/4 tau = 1/4 (2 pi) = 1/2 pi, so you haven't lost your pi-radian friends. I actually think the use of tau as one turn and angle measure as a fraction of a turn could help many students get a grip on radian measure.

 

In reality, I know that pi isn't going to go away. People are not going to stop using it. A lot of the tau rhetoric is hyperbole and meant with a sense of humor. However, the use of tau does surprisingly make a lot of sense as you dig into it. It simplifies/unifies many ideas. 

-- Michel

[Juan]

Dear Michael,

I did not use the word "loony" with a demeaning intention. I always use the "fraction of a turn" concept when explaining radian angle measurement. It does make a lot of sense. It's just that in my opinion, "one turn equals 2*Pi" is perfectly clear. One turn is 2Pi, that's it. No need for an extra Greek letter. No need for a symbol that represents "one turn." Why? What for? No need. One turn is one turn, and one turn is exactly 2Pi. The quantity 2Pi can be used as but in itself is not a symbol for one turn. The concept of "one turn" does not need a symbol for it.

The name "radian" comes from radius, the distance from the center of the circle to any point in the circumference. When you wind a string around the whole circle, and measure that piece of string, it turns out to measure 2Pi times the length of the radius, or, equivalently, Pi times the length of the diameter.

Pi, and 2Pi originally are "length-wise" numbers, not symbols for turns or half-turns. Their connection to turns and half-turns is via the ratio of the length of the circumference to that of the diameter.
In my opinion, the diameter and the circumference directly and immediately are the two lengths most naturally associated with the circle, because the diameter is "how big across" and the circumference is "how big around." Pretty simple, it doesn't get any simpler than that.

The radius, on the other hand, even though arithmetically is just one-half of the diameter, is a more abstract concept than the diameter. It only arises when you set out to construct a circle but not necessarily when you are presented with an already built circle. If you move round tables between different rooms, you are concerned with how easily they will go through any given door. Will you have to slant the table so it goes through? Or not? It depends on whether the diameter is shorter or longer than the width of the door. That you will find out immediately (yes/no) when you try to move the table across the door without slanting it. No need for measuring the radius, no need for even locating the center of the table.
So, one turn is 6.28.. radians, that's it, and 6.28.. equals 2Pi, that's it. There is no need whatsoever for an extra symbol to represent "one turn."
I say that with the confidence that I can successfully explain trigonometry to about almost any failing high-school student. Without Tau. No need for Tau. Pi is just fine.

If such usage of Tau ever gets adopted in the mainstream curriculum, I can already imagine the overkill drill exercise textbook sections where students will be asked to convert back and forth from degrees to Pi-radian units to Tau-turn units. And, what for? I do not wish such a nightmarish scenario on any student. The vast majority of them will be even more confused, and will hate math more than they already do.

I know teachers need to understand math before teaching it. Many times in practice this means they need to teach math the way they understand it. I only wish teachers would ask students how the student understands math in his or her own individual way, instead of teachers asking themselves why students do not understand math the way teachers explain it, and then trying to find one way to explain it so that everybody will understand it. That is never going to happen.

Juan

[Maria Droujkova]

On Thu, Dec 20, 2012 at 5:05 PM, Juan <here...@gmail.com> wrote:

I did not use the word "loony" with a demeaning intention. I always use the "fraction of a turn" concept when explaining radian angle measurement. It does make a lot of sense. It's just that in my opinion, "one turn equals 2*Pi" is perfectly clear. One turn is 2Pi, that's it. No need for an extra Greek letter. No need for a symbol that represents "one turn." Why? What for? No need. One turn is one turn, and one turn is exactly 2Pi. The quantity 2Pi can be used as but in itself is not a symbol for one turn. The concept of "one turn" does not need a symbol for it.

Repetition is fun, of course (as in the above paragraph, where the same statement is repeated about 11 times, by my count). So, one can argue that Pi repeated twice is simply more fun that Tau used only once. 

The concept of the ratio of a circumference (one turn) to its diameter needs the symbol no more - and no less - than the concept of the ratio of half a turn to its radius. But diameters USED TO be easier to measure than radii. I mean, you need to fold a circle twice (hat tip to Brad) before you obtain its radius. You only need one fold to obtain the diameter. A click saved is 3/4 customers earned, as any web commerce person will tell you. So historically, Pi used to save people a measurement step - or a division. You would not sneeze at a division if you were an Ancient Greek either - their algorithms for it were on the atrocious side.

These days, such considerations don't matter, because hardly anyone measures circles anymore, and we know exactly what Pi is, and we have handy machines for divisions. So we could use Tau if we wished and nobody would lose any measurements, folds, divisions, or customers over it. It would save some symbols in the writing of some formulas.

But Pi is more historical.

[kirby urner]

Apologies if this is repetitious.  I wanted to get a link working on Math Future for the benefit of math-teach:

http://mathforum.org/kb/message.jspa?messageID=7942009

but then I couldn't seem to make it appear here on Math Future in my browser, in the archive.

So as a stop gap, my reply to Juan went to a different public archive here:

http://tech.groups.yahoo.com/group/synergeo/message/70774

... so I could close this circuit. 

The same verbiage may appear elsewhere on this list and for that I apologize. 

I was in too much of a hurry maybe, you know how it gets in this season.

Season's Greetings,

Kirby

[Bradford Hansen-Smith]

Since diameter (2 radii is a subset) is always discussed as the measure to the circumference and is designated by the irrational number of pi; where is the discussion about, do we have a word for, when the diameter and circumference are exactly equal in measure? (A little more nonsense to think about.)  The two most far apart points on the circumference when curved and touched together form the circumference of a circle that is the same measure of the diameter of the circle being used. This can be done before you crease the first fold in the circle.

That is my Holidays gift to all of you that like thinking about what this is that we call math.  

Brad

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[Juan]

On Thursday, December 20, 2012 2:47:54 PM UTC-8, MariaD wrote:

 

The concept of the ratio of a circumference (one turn) to its diameter needs the symbol no more - and no less - than the concept of the ratio of half a turn to its radius.

I couldn't agree more with the above statement. Both ratios equal the number Pi, so they equal each other. Therefore, whatever one needs, the other does too, because they are one and the same.
Merry Christmas, and a proportionally Happy New Year!!

Juan

[Juan]

On Thursday, December 20, 2012 9:19:56 PM UTC-8, Bradford Hansen-Smith wrote:

where is the discussion about, do we have a word for, when the diameter and circumference are exactly equal in measure? (A little more nonsense to think about.)  The two most far apart points on the circumference when curved and touched together form the circumference of a circle that is the same measure of the diameter of the circle being used. This can be done before you crease the first fold in the circle.

Bradford,
 

This is one way to do it:

(x,y,z) = ( cost, (1/pi)*sin(pi*sin(t), (1/pi)-(1/pi)*cos(pi*sin(t) )

[Juan]

On Wednesday, December 19, 2012 5:01:23 AM UTC-8, MariaD wrote:

This definition from information theory is one of better attempts (ha! judgment call) to make beauty more objective: http://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_and_mathematical_information_theory

That approach to a theory of beauty based on algorithmic information theory definitely takes the math of beauty beyond the mere study of the golden ratio.

[Juan]

On Wednesday, December 19, 2012 5:01:23 AM UTC-8, MariaD wrote:

The better bet (ha! again) is to do away with the idea of The One Mainstream Curriculum To Rule Them All (TM). I can imagine some kids, somewhere being very interested in this sort of stuff. Just like other kids are interested in robotics, math storytelling, hyperbolic crocheting and other varied geekiness and niche pursuits. Diverse educational ecosystems for the win!

When I talk about "the mainstream curriculum" I know it's sort of a fuzzy term, far from being absolute, or uniquely defined but there are some topics, and approaches that I see over and over, treated and explained in the same way, textbook after textbook, in many different classes, whether it is Pre-Algebra, or Algebra Readiness, Algebra 1, Algebra 2, AP-Algebra, Honors Algebra, Intermediate Algebra, or College Algebra, or Math-90, or Math-95, there is a whole lot of repetition out there across math courses, not only in the material covered but also in the way it is presented, explained, and tested for. That is what I call "the mainstream curriculum," some sort of observable, statistically aggregated portion of reality, whether I agree with any part of it or not. It's there, it happens. I am not trying to affect it, define it, or change it. I only watch my students undergoing it, struggling with it, and I try to help them with it. It does exist, exceptions notwithstanding.

[Juan]

Kirby,
I read your post in the other archive. I guess I will start studying the "simplex-based volume" when my students start asking me to help them with it.

Juan

[Maria Droujkova]

On Fri, Dec 21, 2012 at 4:45 AM, Juan <here...@gmail.com> wrote:

cost, (1/pi)*sin(pi*sin(t), (1/pi)-(1/pi)*cos(pi*sin(t)

Really neat, Juan! I plotted it using http://www.math.uri.edu/~bkaskosz/flashmo/parcur/

Attached is the picture that came out. I love that even little kids can experiment with those tools. Change a parameter, see what happens.

[Maria Droujkova]

Well, one point I want to make is that the entity you described varies from country to country. And within the country you described, you can observe more variety if you work, say, with private schools - and even more variety if you work with homeschoolers. Like this guy: http://erikdemaine.org/ I bet his curriculum was a tad different.

As for changes, I was mostly addressing Kirby there. Juan's definition above also doubles as a good explanation for why I think getting topics into that entity - "the mainstream curriculum" - is a futile and possibly harmful goal. It would just support the mono-culture of One Right Curriculum. And this mono-culture is an evil in itself, whatever the contents.

Cheers,

MariaD

[michel paul]

On Thu, Dec 20, 2012 at 2:47 PM, Maria Droujkova <drou...@gmail.com> wrote:

 

>diameters USED TO be easier to measure than radii.

Sure, that makes sense. If you need to perform measurements on a circle, it's relatively easy to obtain the circumference and the diameter. Yeah, that's an interesting point. 

But if you want to define what a circle is, a set of points equidistant from a center, then the radius becomes primary, and that's one of the interesting arguments made in favor of tau. When we engage in circular trig, polar coordinates, and stuff like that, we think in terms of radius and seldom need to refer to diameter.

I think using the radius as our unit to measure aspects of the circle yields more insight than using the diameter. We can easily use the radius to inscribe a hexagon, and this nicely shows us that tau is a little bit more than six. I'm not aware of anything as elegant that we can construct using the diameter to show that pi is a little more than three. There are activities to show that the circumference is a little more than 3 times the diameter, but nothing as immediate as that simple diagram.

And then using the radius as a unit is the very definition of radian measure. Therefore, the ratio C/r IS quite clearly radian measure in a way that C/d is not. C/d is pi radians, sure, but that fact requires explanation for a student. It isn't immediately obvious. So in various ways I think it makes sense to say that the radius and C/r are more 'central' in much of our 'circular reasoning' : ) than the diameter and C/d.

 

> Pi is more historical.

Right, and I think the fact that the diameter used to be a little easier to find than the radius does have something to do with that. I accept that pi won't be going away anytime soon, nor does it need to. However, do I find this investigation into tau useful. It's fun to explore, and I've experienced, at least for myself, that contemplating tau has led to some refreshing insights.

[Bradford Hansen-Smith]

After having looked at the parametric curve machine suggested by Maria, as interesting as it might be, I fail to see the connection to reforming the diameter to a circumference where one is equal to the other. I am not qualified to know if Jaun's equation describes touching two opposite points on a circle forming another circle from the same diameter. This is about the geometry of spatial intervals any child can understand. I am still a child in this world of mathematical adults. Can someone explain (x,y,z) = ( cost, (1/pi)*sin(pi*sin(t), (1/pi)-(1/pi)*cos(pi*sin(t) ) in a less abstract way?

Brad

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[kirby urner]

Fair enough.  Wait for the IMAX version from Disney or one of those.

I expect the primary incursion / infusion will continue to be through visual media, such as Youtube (already some there, but more needed).

If you live anywhere near San Antonio, the Octa-Tetra Museum has a lot of this info, and kids to go there on field trips and stuff.

http://nowcastsa.com/content/grand-opening-octa-tetra-museum-0

It's also quite possible a student will first encounter the tetrahedral accounting system in a literature class. 

Much of the writing that reviews this information is either in prose form or poetry.  One could argue this info would be needed by any PhD in recent American literature (1900s onward).

Octa-Tetra space frame:

http://www.rwgrayprojects.com/synergetics/s04/figs/f2001.html  (tetrahedron volume: 1, octahedron: 4)

Tetrahedral accounting applied to Cuboctahedron:

http://www.rwgrayprojects.com/synergetics/s02/figs/f2230.html

Kirby

[David Chandler]

I was just looking up the rule for which side of the street has odd or even house numbers (btw: general rule, odd on N and E and even on S and W, with lots of exceptions) and I ran across this question.  It puts a new twist on the concept of confusion!
--David Chandler

----------------

What side of the road are odd / even numbered addresses on?

Are houses with odd-numbered addresses on the left side of a street, and even-numbered on the right? Does it depend on which way you are facing? Does this rule apply everywhere?

Thanks, Jess

[Juan]

On Friday, December 21, 2012 6:29:02 AM UTC-8, MariaD wrote:

Really neat, Juan! I plotted it using http://www.math.uri.edu/~bkaskosz/flashmo/parcur/

Thank you, Maria. That is a nice 3-D parametric plotter. Wolfram Alpha currently only does 2-D parametric plots, or at least I couldn't find a way to make it plot the curve.

[Juan]

On Friday, December 21, 2012 6:58:33 AM UTC-8, michel paul wrote:

 We can easily use the radius to inscribe a hexagon, and this nicely shows us that tau is a little bit more than six. I'm not aware of anything as elegant that we can construct using the diameter to show that pi is a little more than three. There are activities to show that the circumference is a little more than 3 times the diameter, but nothing as immediate as that simple diagram.

 

I agree that constructing a regular hexagon inscribed in the circle, using the length of the radius as the side of the hexagon is very elegant. Six equals 3x2, so the same construction works for showing that Pi > 3.

Now, the following algorithm is not a geometrical construction but you can run it in your browser. I do not consider the code to be elegant at all but I believe the ideas behind the code are indeed, elegant:

<HTML>
<HEAD>
<Title>Approximating Pi by complex multiplication</Title>
</HEAD>

<BODY>
<script language="Javascript">

var n=0;
for (n=2; n<8; n+=1) {
    document.write("<br>");
    document.write("n value = "+n);
    var h=Math.pow(10,-n);
    document.write("<br>");
    document.write("h value = "+h);
    var stp=1;
    var a=1.0;
    var b=h;
    var c=0.0;
    var d=0.0;
    var working = true;
    while (working) {
        c=a-(b*h);
        d=(a*h)+b;
        a=c;
        b=d;
        stp=stp+1;
        working=!(b < 0);
    }
    stp=(stp-1);
    document.write("<br>");
    document.write("Total number of steps: "+stp);
    document.write("<br>");
    var p=stp*h;
    document.write("Approximate Pi value: "+p);
    document.write("<br>");
}

</script>
</BODY>
</HTML>

Juan

[michel paul]

On Fri, Dec 21, 2012 at 8:55 PM, Juan <here...@gmail.com> wrote:

On Friday, December 21, 2012 6:58:33 AM UTC-8, michel paul wrote:

>>We can easily use the radius to inscribe a hexagon, and this nicely shows us that tau is a little bit more than six. 

 

>I agree that constructing a regular hexagon inscribed in the circle, using the length of the radius as the side of the hexagon is very elegant. Six equals 3x2, so the same construction works for showing that Pi > 3.

 

But notice that the construction uses the radius rather than the diameter. 

What the construction actually shows is that a half turn is slightly more than 3. The fact that a half turn also corresponds to pi is an additional fact we have to introduce to the student. If we already know it, we tend to take it for granted. In fact, the construction explicitly shows neither the diameter nor the ratio of circumference to diameter.

What the construction does explicitly show is the hexagon and a full turn. Half a hexagon corresponds to half a turn. 

Therefore, tau is the whole, and pi is half of tau.

>Now, the following algorithm is not a geometrical construction but you can run it in your browser. I do not consider the code to be elegant at all but I believe the ideas behind the code are indeed, elegant:

Well, one good way to make it more elegant is to re-write it in Python.  : )

In looking at your code, I'm not sure why you would bother with the 'working' variable at all? 

Why not just use 'b >= 0' as the loop control? 

I also wonder why you initialize stp at 1 only to decrement it later. 

Why not just initialize it at 0?

But yes, it is definitely an interesting algorithm. It's intriguing that the number of steps correspond to an approximation of pi. I notice it slows down considerably as n increases, so it's not an efficient way to approximate pi. But that it does approximate pi is cool. 

A few years ago when I began learning Python I found out about 'spigot' algorithms. I found one for pi that will generate as many digits as you like. You want 10,000? It will spit them right out, one at a time, very efficiently. The kids are really amazed the first time they see it. 

-- Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

 

[Juan]

On Saturday, December 22, 2012 1:40:36 AM UTC-8, michel paul wrote:

What the construction does explicitly show is the hexagon and a full turn. Half a hexagon corresponds to half a turn. 

Therefore, tau is the whole, and pi is half of tau.

Yes but one turn is also the modulo for angles, the point where we "reset" them to zero.
So, what is the point of giving a special symbol to "one full turn" if it's going to be equivalent to zero anyway?

With Tau you get 1=0, kind of mysterious but too short for my taste.

Instead, with Pi, you get 1+1=0, just like with integers modulo 2.
 

 

But yes, it is definitely an interesting algorithm. It's intriguing that the number of steps correspond to an approximation of pi.

That is the main idea behind the algorithm, to have Pi approximated by the count of steps, a whole number count of how many multiplications it takes for the imaginary part of the complex number to become negative, and then it stops.

Yes, I would be interested in looking at the spigot algorithm, I will google it.

Juan

[michel paul]

On Sat, Dec 22, 2012 at 12:21 PM, Juan <here...@gmail.com> wrote:

>That is the main idea behind the algorithm, to have Pi approximated by the count of steps, a whole number count of how many multiplications it takes for the imaginary part of the complex number to become negative, and then it stops.

I like that! Very cool. Yes, this is worth looking at some more. Thanks for bringing it up. 

>I would be interested in looking at the spigot algorithm

 Here's one version I found in the form of a Python generator:

def pi_digits(n):

    k, a, b, a1, b1 = 2, 4, 1, 12, 4

    while n>0:

        p, q, k = k*k, 2*k+1, k+1

        a, b, a1, b1 = a1, b1, p*a+q*a1, p*b+q*b1

        d, d1 = a/b, a1/b1

        while d == d1:

            yield int(d)

            n -= 1

            a, a1 = 10*(a%b), 10*(a1%b1)

            d, d1 = a/b, a1/b1

Don't ask me to explain it! I can't.  : )   I just appreciate it. I find it mind blowing. A really bright student I once had edited it to produce the digits in binary.

What I can say is that it is a generator, hence the use of 'yield' rather than 'return'.

Here it is being used in the Python shell to create 10000 digits:

>>> list(pi_digits(10000))

[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, 9, 8, 2, 1, 4, 8, 0, 8, 6, 5, 1, 3, 2, 8, 2, 3, 0, 6, 6, 4, 7, 0, 9, 3, 8, 4, 4, 6, 0, 9, 5, 5, 0, 5, 8, 2, 2, 3, 1, 7, 2, 5, 3, 5, 9, 4, 0, 8, 1, 2, 8, 4, 8, 1, 1, 1, 7, 4, 5, 0, 2, 8, 4, 1, 0, 2, 7, 0, 1, 9, 3, 8, 5, 2, 1, 1, 0, 5, 5, 5, 9, 6, 4, 4, 6, 2, 2, 9, 4, 8, 9, 5, 4, 9, 3, 0, 3, 8, 1, 9, 6, 4, 4, 2, 8, 8, 1, 0, 9, 7, 5, 6, 6, 5, 9, 3, 3, 4, 4, 6, 1, 2, 8, 4, 7, 5, 6, 4, 8, 2, 3, 3, 7, 8, 6, 7, 8, 3, 1, 6, 5, 2, 7, 1, 2, 0, 1, 9, 0, 9, 1, 4, 5, 6, 4, 8, 5, 6, 6, 9, 2, 3, 4, 6, 0, 3, 4, 8, 6, 1, 0, 4, 5, 4, 3, 2, 6, 6, 4, 8, 2, 1, 3, 3, 9, 3, 6, 0, 7, 2, 6, 0, 2, 4, 9, 1, 4, 1, 2, 7, 3, 7, 2, 4, 5, 8, 7, 0, 0, 6, 6, 0, 6, 3, 1, 5, 5, 8, 8, 1, 7, 4, 8, 8, 1, 5, 2, 0, 9, 2, 0, 9, 6, 2, 8, 2, 9, 2, 5, 4, 0, 9, 1, 7, 1, 5, 3, 6, 4, 3, 6, 7, 8, 9, 2, 5, 9, 0, 3, 6, 0, 0, 1, 1, 3, 3, 0, 5, 3, 0, 5, 4, 8, 8, 2, 0, 4, 6, 6, 5, 2, 1, 3, 8, 4, 1, 4, 6, 9, 5, 1, 9, 4, 1, 5, 1, 1, 6, 0, 9, 4, 3, 3, 0, 5, 7, 2, 7, 0, 3, 6, 5, 7, 5, 9, 5, 9, 1, 9, 5, 3, 0, 9, 2, 1, 8, 6, 1, 1, 7, 3, 8, 1, 9, 3, 2, 6, 1, 1, 7, 9, 3, 1, 0, 5, 1, 1, 8, 5, 4, 8, 0, 7, 4, 4, 6, 2, 3, 7, 9, 9, 6, 2, 7, 4, 9, 5, 6, 7, 3, 5, 1, 8, 8, 5, 7, 5, 2, 7, 2, 4, 8, 9, 1, 2, 2, 7, 9, 3, 8, 1, 8, 3, 0, 1, 1, 9, 4, 9, 1, 2, 9, 8, 3, 3, 6, 7, 3, 3, 6, 2, 4, 4, 0, 6, 5, 6, 6, 4, 3, 0, 8, 6, 0, 2, 1, 3, 9, 4, 9, 4, 6, 3, 9, 5, 2, 2, 4, 7, 3, 7, 1, 9, 0, 7, 0, 2, 1, 7, 9, 8, 6, 0, 9, 4, 3, 7, 0, 2, 7, 7, 0, 5, 3, 9, 2, 1, 7, 1, 7, 6, 2, 9, 3, 1, 7, 6, 7, 5, 2, 3, 8, 4, 6, 7, 4, 8, 1, 8, 4, 6, 7, 6, 6, 9, 4, 0, 5, 1, 3, 2, 0, 0, 0, 5, 6, 8, 1, 2, 7, 1, 4, 5, 2, 6, 3, 5, 6, 0, 8, 2, 7, 7, 8, 5, 7, 7, 1, 3, 4, 2, 7, 5, 7, 7, 8, 9, 6, 0, 9, 1, 7, 3, 6, 3, 7, 1, 7, 8, 7, 2, 1, 4, 6, 8, 4, 4, 0, 9, 0, 1, 2, 2, 4, 9, 5, 3, 4, 3, 0, 1, 4, 6, 5, 4, 9, 5, 8, 5, 3, 7, 1, 0, 5, 0, 7, 9, 2, 2, 7, 9, 6, 8, 9, 2, 5, 8, 9, 2, 3, 5, 4, 2, 0, 1, 9, 9, 5, 6, 1, 1, 2, 1, 2, 9, 0, 2, 1, 9, 6, 0, 8, 6, 4, 0, 3, 4, 4, 1, 8, 1, 5, 9, 8, 1, 3, 6, 2, 9, 7, 7, 4, 7, 7, 1, 3, 0, 9, 9, 6, 0, 5, 1, 8, 7, 0, 7, 2, 1, 1, 3, 4, 9, 9, 9, 9, 9, 9, 8, 3, 7, 2, 9, 7, 8, 0, 4, 9, 9, 5, 1, 0, 5, 9, 7, 3, 1, 7, 3, 2, 8, 1, 6, 0, 9, 6, 3, 1, 8, 5, 9, 5, 0, 2, 4, 4, 5, 9, 4, 5, 5, 3, 4, 6, 9, 0, 8, 3, 0, 2, 6, 4, 2, 5, 2, 2, 3, 0, 8, 2, 5, 3, 3, 4, 4, 6, 8, 5, 0, 3, 5, 2, 6, 1, 9, 3, 1, 1, 8, 8, 1, 7, 1, 0, 1, 0, 0, 0, 3, 1, 3, 7, 8, 3, 8, 7, 5, 2, 8, 8, 6, 5, 8, 7, 5, 3, 3, 2, 0, 8, 3, 8, 1, 4, 2, 0, 6, 1, 7, 1, 7, 7, 6, 6, 9, 1, 4, 7, 3, 0, 3, 5, 9, 8, 2, 5, 3, 4, 9, 0, 4, 2, 8, 7, 5, 5, 4, 6, 8, 7, 3, 1, 1, 5, 9, 5, 6, 2, 8, 6, 3, 8, 8, 2, 3, 5, 3, 7, 8, 7, 5, 9, 3, 7, 5, 1, 9, 5, 7, 7, 8, 1, 8, 5, 7, 7, 8, 0, 5, 3, 2, 1, 7, 1, 2, 2, 6, 8, 0, 6, 6, 1, 3, 0, 0, 1, 9, 2, 7, 8, 7, 6, 6, 1, 1, 1, 9, 5, 9, 0, 9, 2, 1, 6, 4, 2, 0, 1, 9, 8, 9, 3, 8, 0, 9, 5, 2, 5, 7, 2, 0, 1, 0, 6, 5, 4, 8, 5, 8, 6, 3, 2, 7, 8, 8, 6, 5, 9, 3, 6, 1, 5, 3, 3, 8, 1, 8, 2, 7, 9, 6, 8, 2, 3, 0, 3, 0, 1, 9, 5, 2, 0, 3, 5, 3, 0, 1, 8, 5, 2, 9, 6, 8, 9, 9, 5, 7, 7, 3, 6, 2, 2, 5, 9, 9, 4, 1, 3, 8, 9, 1, 2, 4, 9, 7, 2, 1, 7, 7, 5, 2, 8, 3, 4, 7, 9, 1, 3, 1, 5, 1, 5, 5, 7, 4, 8, 5, 7, 2, 4, 2, 4, 5, 4, 1, 5, 0, 6, 9, 5, 9, 5, 0, 8, 2, 9, 5, 3, 3, 1, 1, 6, 8, 6, 1, 7, 2, 7, 8, 5, 5, 8, 8, 9, 0, 7, 5, 0, 9, 8, 3, 8, 1, 7, 5, 4, 6, 3, 7, 4, 6, 4, 9, 3, 9, 3, 1, 9, 2, 5, 5, 0, 6, 0, 4, 0, 0, 9, 2, 7, 7, 0, 1, 6, 7, 1, 1, 3, 9, 0, 0, 9, 8, 4, 8, 8, 2, 4, 0, 1, 2, 8, 5, 8, 3, 6, 1, 6, 0, 3, 5, 6, 3, 7, 0, 7, 6, 6, 0, 1, 0, 4, 7, 1, 0, 1, 8, 1, 9, 4, 2, 9, 5, 5, 5, 9, 6, 1, 9, 8, 9, 4, 6, 7, 6, 7, 8, 3, 7, 4, 4, 9, 4, 4, 8, 2, 5, 5, 3, 7, 9, 7, 7, 4, 7, 2, 6, 8, 4, 7, 1, 0, 4, 0, 4, 7, 5, 3, 4, 6, 4, 6, 2, 0, 8, 0, 4, 6, 6, 8, 4, 2, 5, 9, 0, 6, 9, 4, 9, 1, 2, 9, 3, 3, 1, 3, 6, 7, 7, 0, 2, 8, 9, 8, 9, 1, 5, 2, 1, 0, 4, 7, 5, 2, 1, 6, 2, 0, 5, 6, 9, 6, 6, 0, 2, 4, 0, 5, 8, 0, 3, 8, 1, 5, 0, 1, 9, 3, 5, 1, 1, 2, 5, 3, 3, 8, 2, 4, 3, 0, 0, 3, 5, 5, 8, 7, 6, 4, 0, 2, 4, 7, 4, 9, 6, 4, 7, 3, 2, 6, 3, 9, 1, 4, 1, 9, 9, 2, 7, 2, 6, 0, 4, 2, 6, 9, 9, 2, 2, 7, 9, 6, 7, 8, 2, 3, 5, 4, 7, 8, 1, 6, 3, 6, 0, 0, 9, 3, 4, 1, 7, 2, 1, 6, 4, 1, 2, 1, 9, 9, 2, 4, 5, 8, 6, 3, 1, 5, 0, 3, 0, 2, 8, 6, 1, 8, 2, 9, 7, 4, 5, 5, 5, 7, 0, 6, 7, 4, 9, 8, 3, 8, 5, 0, 5, 4, 9, 4, 5, 8, 8, 5, 8, 6, 9, 2, 6, 9, 9, 5, 6, 9, 0, 9, 2, 7, 2, 1, 0, 7, 9, 7, 5, 0, 9, 3, 0, 2, 9, 5, 5, 3, 2, 1, 1, 6, 5, 3, 4, 4, 9, 8, 7, 2, 0, 2, 7, 5, 5, 9, 6, 0, 2, 3, 6, 4, 8, 0, 6, 6, 5, 4, 9, 9, 1, 1, 9, 8, 8, 1, 8, 3, 4, 7, 9, 7, 7, 5, 3, 5, 6, 6, 3, 6, 9, 8, 0, 7, 4, 2, 6, 5, 4, 2, 5, 2, 7, 8, 6, 2, 5, 5, 1, 8, 1, 8, 4, 1, 7, 5, 7, 4, 6, 7, 2, 8, 9, 0, 9, 7, 7, 7, 7, 2, 7, 9, 3, 8, 0, 0, 0, 8, 1, 6, 4, 7, 0, 6, 0, 0, 1, 6, 1, 4, 5, 2, 4, 9, 1, 9, 2, 1, 7, 3, 2, 1, 7, 2, 1, 4, 7, 7, 2, 3, 5, 0, 1, 4, 1, 4, 4, 1, 9, 7, 3, 5, 6, 8, 5, 4, 8, 1, 6, 1, 3, 6, 1, 1, 5, 7, 3, 5, 2, 5, 5, 2, 1, 3, 3, 4, 7, 5, 7, 4, 1, 8, 4, 9, 4, 6, 8, 4, 3, 8, 5, 2, 3, 3, 2, 3, 9, 0, 7, 3, 9, 4, 1, 4, 3, 3, 3, 4, 5, 4, 7, 7, 6, 2, 4, 1, 6, 8, 6, 2, 5, 1, 8, 9, 8, 3, 5, 6, 9, 4, 8, 5, 5, 6, 2, 0, 9, 9, 2, 1, 9, 2, 2, 2, 1, 8, 4, 2, 7, 2, 5, 5, 0, 2, 5, 4, 2, 5, 6, 8, 8, 7, 6, 7, 1, 7, 9, 0, 4, 9, 4, 6, 0, 1, 6, 5, 3, 4, 6, 6, 8, 0, 4, 9, 8, 8, 6, 2, 7, 2, 3, 2, 7, 9, 1, 7, 8, 6, 0, 8, 5, 7, 8, 4, 3, 8, 3, 8, 2, 7, 9, 6, 7, 9, 7, 6, 6, 8, 1, 4, 5, 4, 1, 0, 0, 9, 5, 3, 8, 8, 3, 7, 8, 6, 3, 6, 0, 9, 5, 0, 6, 8, 0, 0, 6, 4, 2, 2, 5, 1, 2, 5, 2, 0, 5, 1, 1, 7, 3, 9, 2, 9, 8, 4, 8, 9, 6, 0, 8, 4, 1, 2, 8, 4, 8, 8, 6, 2, 6, 9, 4, 5, 6, 0, 4, 2, 4, 1, 9, 6, 5, 2, 8, 5, 0, 2, 2, 2, 1, 0, 6, 6, 1, 1, 8, 6, 3, 0, 6, 7, 4, 4, 2, 7, 8, 6, 2, 2, 0, 3, 9, 1, 9, 4, 9, 4, 5, 0, 4, 7, 1, 2, 3, 7, 1, 3, 7, 8, 6, 9, 6, 0, 9, 5, 6, 3, 6, 4, 3, 7, 1, 9, 1, 7, 2, 8, 7, 4, 6, 7, 7, 6, 4, 6, 5, 7, 5, 7, 3, 9, 6, 2, 4, 1, 3, 8, 9, 0, 8, 6, 5, 8, 3, 2, 6, 4, 5, 9, 9, 5, 8, 1, 3, 3, 9, 0, 4, 7, 8, 0, 2, 7, 5, 9, 0, 0, 9, 9, 4, 6, 5, 7, 6, 4, 0, 7, 8, 9, 5, 1, 2, 6, 9, 4, 6, 8, 3, 9, 8, 3, 5, 2, 5, 9, 5, 7, 0, 9, 8, 2, 5, 8, 2, 2, 6, 2, 0, 5, 2, 2, 4, 8, 9, 4, 0, 7, 7, 2, 6, 7, 1, 9, 4, 7, 8, 2, 6, 8, 4, 8, 2, 6, 0, 1, 4, 7, 6, 9, 9, 0, 9, 0, 2, 6, 4, 0, 1, 3, 6, 3, 9, 4, 4, 3, 7, 4, 5, 5, 3, 0, 5, 0, 6, 8, 2, 0, 3, 4, 9, 6, 2, 5, 2, 4, 5, 1, 7, 4, 9, 3, 9, 9, 6, 5, 1, 4, 3, 1, 4, 2, 9, 8, 0, 9, 1, 9, 0, 6, 5, 9, 2, 5, 0, 9, 3, 7, 2, 2, 1, 6, 9, 6, 4, 6, 1, 5, 1, 5, 7, 0, 9, 8, 5, 8, 3, 8, 7, 4, 1, 0, 5, 9, 7, 8, 8, 5, 9, 5, 9, 7, 7, 2, 9, 7, 5, 4, 9, 8, 9, 3, 0, 1, 6, 1, 7, 5, 3, 9, 2, 8, 4, 6, 8, 1, 3, 8, 2, 6, 8, 6, 8, 3, 8, 6, 8, 9, 4, 2, 7, 7, 4, 1, 5, 5, 9, 9, 1, 8, 5, 5, 9, 2, 5, 2, 4, 5, 9, 5, 3, 9, 5, 9, 4, 3, 1, 0, 4, 9, 9, 7, 2, 5, 2, 4, 6, 8, 0, 8, 4, 5, 9, 8, 7, 2, 7, 3, 6, 4, 4, 6, 9, 5, 8, 4, 8, 6, 5, 3, 8, 3, 6, 7, 3, 6, 2, 2, 2, 6, 2, 6, 0, 9, 9, 1, 2, 4, 6, 0, 8, 0, 5, 1, 2, 4, 3, 8, 8, 4, 3, 9, 0, 4, 5, 1, 2, 4, 4, 1, 3, 6, 5, 4, 9, 7, 6, 2, 7, 8, 0, 7, 9, 7, 7, 1, 5, 6, 9, 1, 4, 3, 5, 9, 9, 7, 7, 0, 0, 1, 2, 9, 6, 1, 6, 0, 8, 9, 4, 4, 1, 6, 9, 4, 8, 6, 8, 5, 5, 5, 8, 4, 8, 4, 0, 6, 3, 5, 3, 4, 2, 2, 0, 7, 2, 2, 2, 5, 8, 2, 8, 4, 8, 8, 6, 4, 8, 1, 5, 8, 4, 5, 6, 0, 2, 8, 5, 0, 6, 0, 1, 6, 8, 4, 2, 7, 3, 9, 4, 5, 2, 2, 6, 7, 4, 6, 7, 6, 7, 8, 8, 9, 5, 2, 5, 2, 1, 3, 8, 5, 2, 2, 5, 4, 9, 9, 5, 4, 6, 6, 6, 7, 2, 7, 8, 2, 3, 9, 8, 6, 4, 5, 6, 5, 9, 6, 1, 1, 6, 3, 5, 4, 8, 8, 6, 2, 3, 0, 5, 7, 7, 4, 5, 6, 4, 9, 8, 0, 3, 5, 5, 9, 3, 6, 3, 4, 5, 6, 8, 1, 7, 4, 3, 2, 4, 1, 1, 2, 5, 1, 5, 0, 7, 6, 0, 6, 9, 4, 7, 9, 4, 5, 1, 0, 9, 6, 5, 9, 6, 0, 9, 4, 0, 2, 5, 2, 2, 8, 8, 7, 9, 7, 1, 0, 8, 9, 3, 1, 4, 5, 6, 6, 9, 1, 3, 6, 8, 6, 7, 2, 2, 8, 7, 4, 8, 9, 4, 0, 5, 6, 0, 1, 0, 1, 5, 0, 3, 3, 0, 8, 6, 1, 7, 9, 2, 8, 6, 8, 0, 9, 2, 0, 8, 7, 4, 7, 6, 0, 9, 1, 7, 8, 2, 4, 9, 3, 8, 5, 8, 9, 0, 0, 9, 7, 1, 4, 9, 0, 9, 6, 7, 5, 9, 8, 5, 2, 6, 1, 3, 6, 5, 5, 4, 9, 7, 8, 1, 8, 9, 3, 1, 2, 9, 7, 8, 4, 8, 2, 1, 6, 8, 2, 9, 9, 8, 9, 4, 8, 7, 2, 2, 6, 5, 8, 8, 0, 4, 8, 5, 7, 5, 6, 4, 0, 1, 4, 2, 7, 0, 4, 7, 7, 5, 5, 5, 1, 3, 2, 3, 7, 9, 6, 4, 1, 4, 5, 1, 5, 2, 3, 7, 4, 6, 2, 3, 4, 3, 6, 4, 5, 4, 2, 8, 5, 8, 4, 4, 4, 7, 9, 5, 2, 6, 5, 8, 6, 7, 8, 2, 1, 0, 5, 1, 1, 4, 1, 3, 5, 4, 7, 3, 5, 7, 3, 9, 5, 2, 3, 1, 1, 3, 4, 2, 7, 1, 6, 6, 1, 0, 2, 1, 3, 5, 9, 6, 9, 5, 3, 6, 2, 3, 1, 4, 4, 2, 9, 5, 2, 4, 8, 4, 9, 3, 7, 1, 8, 7, 1, 1, 0, 1, 4, 5, 7, 6, 5, 4, 0, 3, 5, 9, 0, 2, 7, 9, 9, 3, 4, 4, 0, 3, 7, 4, 2, 0, 0, 7, 3, 1, 0, 5, 7, 8, 5, 3, 9, 0, 6, 2, 1, 9, 8, 3, 8, 7, 4, 4, 7, 8, 0, 8, 4, 7, 8, 4, 8, 9, 6, 8, 3, 3, 2, 1, 4, 4, 5, 7, 1, 3, 8, 6, 8, 7, 5, 1, 9, 4, 3, 5, 0, 6, 4, 3, 0, 2, 1, 8, 4, 5, 3, 1, 9, 1, 0, 4, 8, 4, 8, 1, 0, 0, 5, 3, 7, 0, 6, 1, 4, 6, 8, 0, 6, 7, 4, 9, 1, 9, 2, 7, 8, 1, 9, 1, 1, 9, 7, 9, 3, 9, 9, 5, 2, 0, 6, 1, 4, 1, 9, 6, 6, 3, 4, 2, 8, 7, 5, 4, 4, 4, 0, 6, 4, 3, 7, 4, 5, 1, 2, 3, 7, 1, 8, 1, 9, 2, 1, 7, 9, 9, 9, 8, 3, 9, 1, 0, 1, 5, 9, 1, 9, 5, 6, 1, 8, 1, 4, 6, 7, 5, 1, 4, 2, 6, 9, 1, 2, 3, 9, 7, 4, 8, 9, 4, 0, 9, 0, 7, 1, 8, 6, 4, 9, 4, 2, 3, 1, 9, 6, 1, 5, 6, 7, 9, 4, 5, 2, 0, 8, 0, 9, 5, 1, 4, 6, 5, 5, 0, 2, 2, 5, 2, 3, 1, 6, 0, 3, 8, 8, 1, 9, 3, 0, 1, 4, 2, 0, 9, 3, 7, 6, 2, 1, 3, 7, 8, 5, 5, 9, 5, 6, 6, 3, 8, 9, 3, 7, 7, 8, 7, 0, 8, 3, 0, 3, 9, 0, 6, 9, 7, 9, 2, 0, 7, 7, 3, 4, 6, 7, 2, 2, 1, 8, 2, 5, 6, 2, 5, 9, 9, 6, 6, 1, 5, 0, 1, 4, 2, 1, 5, 0, 3, 0, 6, 8, 0, 3, 8, 4, 4, 7, 7, 3, 4, 5, 4, 9, 2, 0, 2, 6, 0, 5, 4, 1, 4, 6, 6, 5, 9, 2, 5, 2, 0, 1, 4, 9, 7, 4, 4, 2, 8, 5, 0, 7, 3, 2, 5, 1, 8, 6, 6, 6, 0, 0, 2, 1, 3, 2, 4, 3, 4, 0, 8, 8, 1, 9, 0, 7, 1, 0, 4, 8, 6, 3, 3, 1, 7, 3, 4, 6, 4, 9, 6, 5, 1, 4, 5, 3, 9, 0, 5, 7, 9, 6, 2, 6, 8, 5, 6, 1, 0, 0, 5, 5, 0, 8, 1, 0, 6, 6, 5, 8, 7, 9, 6, 9, 9, 8, 1, 6, 3, 5, 7, 4, 7, 3, 6, 3, 8, 4, 0, 5, 2, 5, 7, 1, 4, 5, 9, 1, 0, 2, 8, 9, 7, 0, 6, 4, 1, 4, 0, 1, 1, 0, 9, 7, 1, 2, 0, 6, 2, 8, 0, 4, 3, 9, 0, 3, 9, 7, 5, 9, 5, 1, 5, 6, 7, 7, 1, 5, 7, 7, 0, 0, 4, 2, 0, 3, 3, 7, 8, 6, 9, 9, 3, 6, 0, 0, 7, 2, 3, 0, 5, 5, 8, 7, 6, 3, 1, 7, 6, 3, 5, 9, 4, 2, 1, 8, 7, 3, 1, 2, 5, 1, 4, 7, 1, 2, 0, 5, 3, 2, 9, 2, 8, 1, 9, 1, 8, 2, 6, 1, 8, 6, 1, 2, 5, 8, 6, 7, 3, 2, 1, 5, 7, 9, 1, 9, 8, 4, 1, 4, 8, 4, 8, 8, 2, 9, 1, 6, 4, 4, 7, 0, 6, 0, 9, 5, 7, 5, 2, 7, 0, 6, 9, 5, 7, 2, 2, 0, 9, 1, 7, 5, 6, 7, 1, 1, 6, 7, 2, 2, 9, 1, 0, 9, 8, 1, 6, 9, 0, 9, 1, 5, 2, 8, 0, 1, 7, 3, 5, 0, 6, 7, 1, 2, 7, 4, 8, 5, 8, 3, 2, 2, 2, 8, 7, 1, 8, 3, 5, 2, 0, 9, 3, 5, 3, 9, 6, 5, 7, 2, 5, 1, 2, 1, 0, 8, 3, 5, 7, 9, 1, 5, 1, 3, 6, 9, 8, 8, 2, 0, 9, 1, 4, 4, 4, 2, 1, 0, 0, 6, 7, 5, 1, 0, 3, 3, 4, 6, 7, 1, 1, 0, 3, 1, 4, 1, 2, 6, 7, 1, 1, 1, 3, 6, 9, 9, 0, 8, 6, 5, 8, 5, 1, 6, 3, 9, 8, 3, 1, 5, 0, 1, 9, 7, 0, 1, 6, 5, 1, 5, 1, 1, 6, 8, 5, 1, 7, 1, 4, 3, 7, 6, 5, 7, 6, 1, 8, 3, 5, 1, 5, 5, 6, 5, 0, 8, 8, 4, 9, 0, 9, 9, 8, 9, 8, 5, 9, 9, 8, 2, 3, 8, 7, 3, 4, 5, 5, 2, 8, 3, 3, 1, 6, 3, 5, 5, 0, 7, 6, 4, 7, 9, 1, 8, 5, 3, 5, 8, 9, 3, 2, 2, 6, 1, 8, 5, 4, 8, 9, 6, 3, 2, 1, 3, 2, 9, 3, 3, 0, 8, 9, 8, 5, 7, 0, 6, 4, 2, 0, 4, 6, 7, 5, 2, 5, 9, 0, 7, 0, 9, 1, 5, 4, 8, 1, 4, 1, 6, 5, 4, 9, 8, 5, 9, 4, 6, 1, 6, 3, 7, 1, 8, 0, 2, 7, 0, 9, 8, 1, 9, 9, 4, 3, 0, 9, 9, 2, 4, 4, 8, 8, 9, 5, 7, 5, 7, 1, 2, 8, 2, 8, 9, 0, 5, 9, 2, 3, 2, 3, 3, 2, 6, 0, 9, 7, 2, 9, 9, 7, 1, 2, 0, 8, 4, 4, 3, 3, 5, 7, 3, 2, 6, 5, 4, 8, 9, 3, 8, 2, 3, 9, 1, 1, 9, 3, 2, 5, 9, 7, 4, 6, 3, 6, 6, 7, 3, 0, 5, 8, 3, 6, 0, 4, 1, 4, 2, 8, 1, 3, 8, 8, 3, 0, 3, 2, 0, 3, 8, 2, 4, 9, 0, 3, 7, 5, 8, 9, 8, 5, 2, 4, 3, 7, 4, 4, 1, 7, 0, 2, 9, 1, 3, 2, 7, 6, 5, 6, 1, 8, 0, 9, 3, 7, 7, 3, 4, 4, 4, 0, 3, 0, 7, 0, 7, 4, 6, 9, 2, 1, 1, 2, 0, 1, 9, 1, 3, 0, 2, 0, 3, 3, 0, 3, 8, 0, 1, 9, 7, 6, 2, 1, 1, 0, 1, 1, 0, 0, 4, 4, 9, 2, 9, 3, 2, 1, 5, 1, 6, 0, 8, 4, 2, 4, 4, 4, 8, 5, 9, 6, 3, 7, 6, 6, 9, 8, 3, 8, 9, 5, 2, 2, 8, 6, 8, 4, 7, 8, 3, 1, 2, 3, 5, 5, 2, 6, 5, 8, 2, 1, 3, 1, 4, 4, 9, 5, 7, 6, 8, 5, 7, 2, 6, 2, 4, 3, 3, 4, 4, 1, 8, 9, 3, 0, 3, 9, 6, 8, 6, 4, 2, 6, 2, 4, 3, 4, 1, 0, 7, 7, 3, 2, 2, 6, 9, 7, 8, 0, 2, 8, 0, 7, 3, 1, 8, 9, 1, 5, 4, 4, 1, 1, 0, 1, 0, 4, 4, 6, 8, 2, 3, 2, 5, 2, 7, 1, 6, 2, 0, 1, 0, 5, 2, 6, 5, 2, 2, 7, 2, 1, 1, 1, 6, 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3, 1, 9, 3, 9, 5, 0, 9, 7, 9, 0, 1, 9, 0, 6, 9, 9, 6, 3, 9, 5, 5, 2, 4, 5, 3, 0, 0, 5, 4, 5, 0, 5, 8, 0, 6, 8, 5, 5, 0, 1, 9, 5, 6, 7, 3, 0, 2, 2, 9, 2, 1, 9, 1, 3, 9, 3, 3, 9, 1, 8, 5, 6, 8, 0, 3, 4, 4, 9, 0, 3, 9, 8, 2, 0, 5, 9, 5, 5, 1, 0, 0, 2, 2, 6, 3, 5, 3, 5, 3, 6, 1, 9, 2, 0, 4, 1, 9, 9, 4, 7, 4, 5, 5, 3, 8, 5, 9, 3, 8, 1, 0, 2, 3, 4, 3, 9, 5, 5, 4, 4, 9, 5, 9, 7, 7, 8, 3, 7, 7, 9, 0, 2, 3, 7, 4, 2, 1, 6, 1, 7, 2, 7, 1, 1, 1, 7, 2, 3, 6, 4, 3, 4, 3, 5, 4, 3, 9, 4, 7, 8, 2, 2, 1, 8, 1, 8, 5, 2, 8, 6, 2, 4, 0, 8, 5, 1, 4, 0, 0, 6, 6, 6, 0, 4, 4, 3, 3, 2, 5, 8, 8, 8, 5, 6, 9, 8, 6, 7, 0, 5, 4, 3, 1, 5, 4, 7, 0, 6, 9, 6, 5, 7, 4, 7, 4, 5, 8, 5, 5, 0, 3, 3, 2, 3, 2, 3, 3, 4, 2, 1, 0, 7, 3, 0, 1, 5, 4, 5, 9, 4, 0, 5, 1, 6, 5, 5, 3, 7, 9, 0, 6, 8, 6, 6, 2, 7, 3, 3, 3, 7, 9, 9, 5, 8, 5, 1, 1, 5, 6, 2, 5, 7, 8, 4, 3, 2, 2, 9, 8, 8, 2, 7, 3, 7, 2, 3, 1, 9, 8, 9, 8, 7, 5, 7, 1, 4, 1, 5, 9, 5, 7, 8, 1, 1, 1, 9, 6, 3, 5, 8, 3, 3, 0, 0, 5, 9, 4, 0, 8, 7, 3, 0, 6, 8, 1, 2, 1, 6, 0, 2, 8, 7, 6, 4, 9, 6, 2, 8, 6, 7, 4, 4, 6, 0, 4, 7, 7, 4, 6, 4, 9, 1, 5, 9, 9, 5, 0, 5, 4, 9, 7, 3, 7, 4, 2, 5, 6, 2, 6, 9, 0, 1, 0, 4, 9, 0, 3, 7, 7, 8, 1, 9, 8, 6, 8, 3, 5, 9, 3, 8, 1, 4, 6, 5, 7, 4, 1, 2, 6, 8, 0, 4, 9, 2, 5, 6, 4, 8, 7, 9, 8, 5, 5, 6, 1, 4, 5, 3, 7, 2, 3, 4, 7, 8, 6, 7, 3, 3, 0, 3, 9, 0, 4, 6, 8, 8, 3, 8, 3, 4, 3, 6, 3, 4, 6, 5, 5, 3, 7, 9, 4, 9, 8, 6, 4, 1, 9, 2, 7, 0, 5, 6, 3, 8, 7, 2, 9, 3, 1, 7, 4, 8, 7, 2, 3, 3, 2, 0, 8, 3, 7, 6, 0, 1, 1, 2, 3, 0, 2, 9, 9, 1, 1, 3, 6, 7, 9, 3, 8, 6, 2, 7, 0, 8, 9, 4, 3, 8, 7, 9, 9, 3, 6, 2, 0, 1, 6, 2, 9, 5, 1, 5, 4, 1, 3, 3, 7, 1, 4, 2, 4, 8, 9, 2, 8, 3, 0, 7, 2, 2, 0, 1, 2, 6, 9, 0, 1, 4, 7, 5, 4, 6, 6, 8, 4, 7, 6, 5, 3, 5, 7, 6, 1, 6, 4, 7, 7, 3, 7, 9, 4, 6, 7, 5, 2, 0, 0, 4, 9, 0, 7, 5, 7, 1, 5, 5, 5, 2, 7, 8, 1, 9, 6, 5, 3, 6, 2, 1, 3, 2, 3, 9, 2, 6, 4, 0, 6, 1, 6, 0, 1, 3, 6, 3, 5, 8, 1, 5, 5, 9, 0, 7, 4, 2, 2, 0, 2, 0, 2, 0, 3, 1, 8, 7, 2, 7, 7, 6, 0, 5, 2, 7, 7, 2, 1, 9, 0, 0, 5, 5, 6, 1, 4, 8, 4, 2, 5, 5, 5, 1, 8, 7, 9, 2, 5, 3, 0, 3, 4, 3, 5, 1, 3, 9, 8, 4, 4, 2, 5, 3, 2, 2, 3, 4, 1, 5, 7, 6, 2, 3, 3, 6, 1, 0, 6, 4, 2, 5, 0, 6, 3, 9, 0, 4, 9, 7, 5, 0, 0, 8, 6, 5, 6, 2, 7, 1, 0, 9, 5, 3, 5, 9, 1, 9, 4, 6, 5, 8, 9, 7, 5, 1, 4, 1, 3, 1, 0, 3, 4, 8, 2, 2, 7, 6, 9, 3, 0, 6, 2, 4, 7, 4, 3, 5, 3, 6, 3, 2, 5, 6, 9, 1, 6, 0, 7, 8, 1, 5, 4, 7, 8, 1, 8, 1, 1, 5, 2, 8, 4, 3, 6, 6, 7, 9, 5, 7, 0, 6, 1, 1, 0, 8, 6, 1, 5, 3, 3, 1, 5, 0, 4, 4, 5, 2, 1, 2, 7, 4, 7, 3, 9, 2, 4, 5, 4, 4, 9, 4, 5, 4, 2, 3, 6, 8, 2, 8, 8, 6, 0, 6, 1, 3, 4, 0, 8, 4, 1, 4, 8, 6, 3, 7, 7, 6, 7, 0, 0, 9, 6, 1, 2, 0, 7, 1, 5, 1, 2, 4, 9, 1, 4, 0, 4, 3, 0, 2, 7, 2, 5, 3, 8, 6, 0, 7, 6, 4, 8, 2, 3, 6, 3, 4, 1, 4, 3, 3, 4, 6, 2, 3, 5, 1, 8, 9, 7, 5, 7, 6, 6, 4, 5, 2, 1, 6, 4, 1, 3, 7, 6, 7, 9, 6, 9, 0, 3, 1, 4, 9, 5, 0, 1, 9, 1, 0, 8, 5, 7, 5, 9, 8, 4, 4, 2, 3, 9, 1, 9, 8, 6, 2, 9, 1, 6, 4, 2, 1, 9, 3, 9, 9, 4, 9, 0, 7, 2, 3, 6, 2, 3, 4, 6, 4, 6, 8, 4, 4, 1, 1, 7, 3, 9, 4, 0, 3, 2, 6, 5, 9, 1, 8, 4, 0, 4, 4, 3, 7, 8, 0, 5, 1, 3, 3, 3, 8, 9, 4, 5, 2, 5, 7, 4, 2, 3, 9, 9, 5, 0, 8, 2, 9, 6, 5, 9, 1, 2, 2, 8, 5, 0, 8, 5, 5, 5, 8, 2, 1, 5, 7, 2, 5, 0, 3, 1, 0, 7, 1, 2, 5, 7, 0, 1, 2, 6, 6, 8, 3, 0, 2, 4, 0, 2, 9, 2, 9, 5, 2, 5, 2, 2, 0, 1, 1, 8, 7, 2, 6, 7, 6, 7, 5, 6, 2, 2, 0, 4, 1, 5, 4, 2, 0, 5, 1, 6, 1, 8, 4, 1, 6, 3, 4, 8, 4, 7, 5, 6, 5, 1, 6, 9, 9, 9, 8, 1, 1, 6, 1, 4, 1, 0, 1, 0, 0, 2, 9, 9, 6, 0, 7, 8, 3, 8, 6, 9, 0, 9, 2, 9, 1, 6, 0, 3, 0, 2, 8, 8, 4, 0, 0, 2, 6, 9, 1, 0, 4, 1, 4, 0, 7, 9, 2, 8, 8, 6, 2, 1, 5, 0, 7, 8, 4, 2, 4, 5, 1, 6, 7, 0, 9, 0, 8, 7, 0, 0, 0, 6, 9, 9, 2, 8, 2, 1, 2, 0, 6, 6, 0, 4, 1, 8, 3, 7, 1, 8, 0, 6, 5, 3, 5, 5, 6, 7, 2, 5, 2, 5, 3, 2, 5, 6, 7, 5, 3, 2, 8, 6, 1, 2, 9, 1, 0, 4, 2, 4, 8, 7, 7, 6, 1, 8, 2, 5, 8, 2, 9, 7, 6, 5, 1, 5, 7, 9, 5, 9, 8, 4, 7, 0, 3, 5, 6, 2, 2, 2, 6, 2, 9, 3, 4, 8, 6, 0, 0, 3, 4, 1, 5, 8, 7, 2, 2, 9, 8, 0, 5, 3, 4, 9, 8, 9, 6, 5, 0, 2, 2, 6, 2, 9, 1, 7, 4, 8, 7, 8, 8, 2, 0, 2, 7, 3, 4, 2, 0, 9, 2, 2, 2, 2, 4, 5, 3, 3, 9, 8, 5, 6, 2, 6, 4, 7, 6, 6, 9, 1, 4, 9, 0, 5, 5, 6, 2, 8, 4, 2, 5, 0, 3, 9, 1, 2, 7, 5, 7, 7, 1, 0, 2, 8, 4, 0, 2, 7, 9, 9, 8, 0, 6, 6, 3, 6, 5, 8, 2, 5, 4, 8, 8, 9, 2, 6, 4, 8, 8, 0, 2, 5, 4, 5, 6, 6, 1, 0, 1, 7, 2, 9, 6, 7, 0, 2, 6, 6, 4, 0, 7, 6, 5, 5, 9, 0, 4, 2, 9, 0, 9, 9, 4, 5, 6, 8, 1, 5, 0, 6, 5, 2, 6, 5, 3, 0, 5, 3, 7, 1, 8, 2, 9, 4, 1, 2, 7, 0, 3, 3, 6, 9, 3, 1, 3, 7, 8, 5, 1, 7, 8, 6, 0, 9, 0, 4, 0, 7, 0, 8, 6, 6, 7, 1, 1, 4, 9, 6, 5, 5, 8, 3, 4, 3, 4, 3, 4, 7, 6, 9, 3, 3, 8, 5, 7, 8, 1, 7, 1, 1, 3, 8, 6, 4, 5, 5, 8, 7, 3, 6, 7, 8, 1, 2, 3, 0, 1, 4, 5, 8, 7, 6, 8, 7, 1, 2, 6, 6, 0, 3, 4, 8, 9, 1, 3, 9, 0, 9, 5, 6, 2, 0, 0, 9, 9, 3, 9, 3, 6, 1, 0, 3, 1, 0, 2, 9, 1, 6, 1, 6, 1, 5, 2, 8, 8, 1, 3, 8, 4, 3, 7, 9, 0, 9, 9, 0, 4, 2, 3, 1, 7, 4, 7, 3, 3, 6, 3, 9, 4, 8, 0, 4, 5, 7, 5, 9, 3, 1, 4, 9, 3, 1, 4, 0, 5, 2, 9, 7, 6, 3, 4, 7, 5, 7, 4, 8, 1, 1, 9, 3, 5, 6, 7, 0, 9, 1, 1, 0, 1, 3, 7, 7, 5, 1, 7, 2, 1, 0, 0, 8, 0, 3, 1, 5, 5, 9, 0, 2, 4, 8, 5, 3, 0, 9, 0, 6, 6, 9, 2, 0, 3, 7, 6, 7, 1, 9, 2, 2, 0, 3, 3, 2, 2, 9, 0, 9, 4, 3, 3, 4, 6, 7, 6, 8, 5, 1, 4, 2, 2, 1, 4, 4, 7, 7, 3, 7, 9, 3, 9, 3, 7, 5, 1, 7, 0, 3, 4, 4, 3, 6, 6, 1, 9, 9, 1, 0, 4, 0, 3, 3, 7, 5, 1, 1, 1, 7, 3, 5, 4, 7, 1, 9, 1, 8, 5, 5, 0, 4, 6, 4, 4, 9, 0, 2, 6, 3, 6, 5, 5, 1, 2, 8, 1, 6, 2, 2, 8, 8, 2, 4, 4, 6, 2, 5, 7, 5, 9, 1, 6, 3, 3, 3, 0, 3, 9, 1, 0, 7, 2, 2, 5, 3, 8, 3, 7, 4, 2, 1, 8, 2, 1, 4, 0, 8, 8, 3, 5, 0, 8, 6, 5, 7, 3, 9, 1, 7, 7, 1, 5, 0, 9, 6, 8, 2, 8, 8, 7, 4, 7, 8, 2, 6, 5, 6, 9, 9, 5, 9, 9, 5, 7, 4, 4, 9, 0, 6, 6, 1, 7, 5, 8, 3, 4, 4, 1, 3, 7, 5, 2, 2, 3, 9, 7, 0, 9, 6, 8, 3, 4, 0, 8, 0, 0, 5, 3, 5, 5, 9, 8, 4, 9, 1, 7, 5, 4, 1, 7, 3, 8, 1, 8, 8, 3, 9, 9, 9, 4, 4, 6, 9, 7, 4, 8, 6, 7, 6, 2, 6, 5, 5, 1, 6, 5, 8, 2, 7, 6, 5, 8, 4, 8, 3, 5, 8, 8, 4, 5, 3, 1, 4, 2, 7, 7, 5, 6, 8, 7, 9, 0, 0, 2, 9, 0, 9, 5, 1, 7, 0, 2, 8, 3, 5, 2, 9, 7, 1, 6, 3, 4, 4, 5, 6, 2, 1, 2, 9, 6, 4, 0, 4, 3, 5, 2, 3, 1, 1, 7, 6, 0, 0, 6, 6, 5, 1, 0, 1, 2, 4, 1, 2, 0, 0, 6, 5, 9, 7, 5, 5, 8, 5, 1, 2, 7, 6, 1, 7, 8, 5, 8, 3, 8, 2, 9, 2, 0, 4, 1, 9, 7, 4, 8, 4, 4, 2, 3, 6, 0, 8, 0, 0, 7, 1, 9, 3, 0, 4, 5, 7, 6, 1, 8, 9, 3, 2, 3, 4, 9, 2, 2, 9, 2, 7, 9, 6, 5, 0, 1, 9, 8, 7, 5, 1, 8, 7, 2, 1, 2, 7, 2, 6, 7, 5, 0, 7, 9, 8, 1, 2, 5, 5, 4, 7, 0, 9, 5, 8, 9, 0, 4, 5, 5, 6, 3, 5, 7, 9, 2, 1, 2, 2, 1, 0, 3, 3, 3, 4, 6, 6, 9, 7, 4, 9, 9, 2, 3, 5, 6, 3, 0, 2, 5, 4, 9, 4, 7, 8, 0, 2, 4, 9, 0, 1, 1, 4, 1, 9, 5, 2, 1, 2, 3, 8, 2, 8, 1, 5, 3, 0, 9, 1, 1, 4, 0, 7, 9, 0, 7, 3, 8, 6, 0, 2, 5, 1, 5, 2, 2, 7, 4, 2, 9, 9, 5, 8, 1, 8, 0, 7, 2, 4, 7, 1, 6, 2, 5, 9, 1, 6, 6, 8, 5, 4, 5, 1, 3, 3, 3, 1, 2, 3, 9, 4, 8, 0, 4, 9, 4, 7, 0, 7, 9, 1, 1, 9, 1, 5, 3, 2, 6, 7, 3, 4, 3, 0, 2, 8, 2, 4, 4, 1, 8, 6, 0, 4, 1, 4, 2, 6, 3, 6, 3, 9, 5, 4, 8, 0, 0, 0, 4, 4, 8, 0, 0, 2, 6, 7, 0, 4, 9, 6, 2, 4, 8, 2, 0, 1, 7, 9, 2, 8, 9, 6, 4, 7, 6, 6, 9, 7, 5, 8, 3, 1, 8, 3, 2, 7, 1, 3, 1, 4, 2, 5, 1, 7, 0, 2, 9, 6, 9, 2, 3, 4, 8, 8, 9, 6, 2, 7, 6, 6, 8, 4, 4, 0, 3, 2, 3, 2, 6, 0, 9, 2, 7, 5, 2, 4, 9, 6, 0, 3, 5, 7, 9, 9, 6, 4, 6, 9, 2, 5, 6, 5, 0, 4, 9, 3, 6, 8, 1, 8, 3, 6, 0, 9, 0, 0, 3, 2, 3, 8, 0, 9, 2, 9, 3, 4, 5, 9, 5, 8, 8, 9, 7, 0, 6, 9, 5, 3, 6, 5, 3, 4, 9, 4, 0, 6, 0, 3, 4, 0, 2, 1, 6, 6, 5, 4, 4, 3, 7, 5, 5, 8, 9, 0, 0, 4, 5, 6, 3, 2, 8, 8, 2, 2, 5, 0, 5, 4, 5, 2, 5, 5, 6, 4, 0, 5, 6, 4, 4, 8, 2, 4, 6, 5, 1, 5, 1, 8, 7, 5, 4, 7, 1, 1, 9, 6, 2, 1, 8, 4, 4, 3, 9, 6, 5, 8, 2, 5, 3, 3, 7, 5, 4, 3, 8, 8, 5, 6, 9, 0, 9, 4, 1, 1, 3, 0, 3, 1, 5, 0, 9, 5, 2, 6, 1, 7, 9, 3, 7, 8, 0, 0, 2, 9, 7, 4, 1, 2, 0, 7, 6, 6, 5, 1, 4, 7, 9, 3, 9, 4, 2, 5, 9, 0, 2, 9, 8, 9, 6, 9, 5, 9, 4, 6, 9, 9, 5, 5, 6, 5, 7, 6, 1, 2, 1, 8, 6, 5, 6, 1, 9, 6, 7, 3, 3, 7, 8, 6, 2, 3, 6, 2, 5, 6, 1, 2, 5, 2, 1, 6, 3, 2, 0, 8, 6, 2, 8, 6, 9, 2, 2, 2, 1, 0, 3, 2, 7, 4, 8, 8, 9, 2, 1, 8, 6, 5, 4, 3, 6, 4, 8, 0, 2, 2, 9, 6, 7, 8, 0, 7, 0, 5, 7, 6, 5, 6, 1, 5, 1, 4, 4, 6, 3, 2, 0, 4, 6, 9, 2, 7, 9, 0, 6, 8, 2, 1, 2, 0, 7, 3, 8, 8, 3, 7, 7, 8, 1, 4, 2, 3, 3, 5, 6, 2, 8, 2, 3, 6, 0, 8, 9, 6, 3, 2, 0, 8, 0, 6, 8, 2, 2, 2, 4, 6, 8, 0, 1, 2, 2, 4, 8, 2, 6, 1, 1, 7, 7, 1, 8, 5, 8, 9, 6, 3, 8, 1, 4, 0, 9, 1, 8, 3, 9, 0, 3, 6, 7, 3, 6, 7, 2, 2, 2, 0, 8, 8, 8, 3, 2, 1, 5, 1, 3, 7, 5, 5, 6, 0, 0, 3, 7, 2, 7, 9, 8, 3, 9, 4, 0, 0, 4, 1, 5, 2, 9, 7, 0, 0, 2, 8, 7, 8, 3, 0, 7, 6, 6, 7, 0, 9, 4, 4, 4, 7, 4, 5, 6, 0, 1, 3, 4, 5, 5, 6, 4, 1, 7, 2, 5, 4, 3, 7, 0, 9, 0, 6, 9, 7, 9, 3, 9, 6, 1, 2, 2, 5, 7, 1, 4, 2, 9, 8, 9, 4, 6, 7, 1, 5, 4, 3, 5, 7, 8, 4, 6, 8, 7, 8, 8, 6, 1, 4, 4, 4, 5, 8, 1, 2, 3, 1, 4, 5, 9, 3, 5, 7, 1, 9, 8, 4, 9, 2, 2, 5, 2, 8, 4, 7, 1, 6, 0, 5, 0, 4, 9, 2, 2, 1, 2, 4, 2, 4, 7, 0, 1, 4, 1, 2, 1, 4, 7, 8, 0, 5, 7, 3, 4, 5, 5, 1, 0, 5, 0, 0, 8, 0, 1, 9, 0, 8, 6, 9, 9, 6, 0, 3, 3, 0, 2, 7, 6, 3, 4, 7, 8, 7, 0, 8, 1, 0, 8, 1, 7, 5, 4, 5, 0, 1, 1, 9, 3, 0, 7, 1, 4, 1, 2, 2, 3, 3, 9, 0, 8, 6, 6, 3, 9, 3, 8, 3, 3, 9, 5, 2, 9, 4, 2, 5, 7, 8, 6, 9, 0, 5, 0, 7, 6, 4, 3, 1, 0, 0, 6, 3, 8, 3, 5, 1, 9, 8, 3, 4, 3, 8, 9, 3, 4, 1, 5, 9, 6, 1, 3, 1, 8, 5, 4, 3, 4, 7, 5, 4, 6, 4, 9, 5, 5, 6, 9, 7, 8, 1, 0, 3, 8, 2, 9, 3, 0, 9, 7, 1, 6, 4, 6, 5, 1, 4, 3, 8, 4, 0, 7, 0, 0, 7, 0, 7, 3, 6, 0, 4, 1, 1, 2, 3, 7, 3, 5, 9, 9, 8, 4, 3, 4, 5, 2, 2, 5, 1, 6, 1, 0, 5, 0, 7, 0, 2, 7, 0, 5, 6, 2, 3, 5, 2, 6, 6, 0, 1, 2, 7, 6, 4, 8, 4, 8, 3, 0, 8, 4, 0, 7, 6, 1, 1, 8, 3, 0, 1, 3, 0, 5, 2, 7, 9, 3, 2, 0, 5, 4, 2, 7, 4, 6, 2, 8, 6, 5, 4, 0, 3, 6, 0, 3, 6, 7, 4, 5, 3, 2, 8, 6, 5, 1, 0, 5, 7, 0, 6, 5, 8, 7, 4, 8, 8, 2, 2, 5, 6, 9, 8, 1, 5, 7, 9, 3, 6, 7, 8, 9, 7, 6, 6, 9, 7, 4, 2, 2, 0, 5, 7, 5, 0, 5, 9, 6, 8, 3, 4, 4, 0, 8, 6, 9, 7, 3, 5, 0, 2, 0, 1, 4, 1, 0, 2, 0, 6, 7, 2, 3, 5, 8, 5, 0, 2, 0, 0, 7, 2, 4, 5, 2, 2, 5, 6, 3, 2, 6, 5, 1, 3, 4, 1, 0, 5, 5, 9, 2, 4, 0, 1, 9, 0, 2, 7, 4, 2, 1, 6, 2, 4, 8, 4, 3, 9, 1, 4, 0, 3, 5, 9, 9, 8, 9, 5, 3, 5, 3, 9, 4, 5, 9, 0, 9, 4, 4, 0, 7, 0, 4, 6, 9, 1, 2, 0, 9, 1, 4, 0, 9, 3, 8, 7, 0, 0, 1, 2, 6, 4, 5, 6, 0, 0, 1, 6, 2, 3, 7, 4, 2, 8, 8, 0, 2, 1, 0, 9, 2, 7, 6, 4, 5, 7, 9, 3, 1, 0, 6, 5, 7, 9, 2, 2, 9, 5, 5, 2, 4, 9, 8, 8, 7, 2, 7, 5, 8, 4, 6, 1, 0, 1, 2, 6, 4, 8, 3, 6, 9, 9, 9, 8, 9, 2, 2, 5, 6, 9, 5, 9, 6, 8, 8, 1, 5, 9, 2, 0, 5, 6, 0, 0, 1, 0, 1, 6, 5, 5, 2, 5, 6, 3, 7, 5, 6, 7] 

Of course, these days you can just type '10000 digits of pi' into Alpha, and it will give you the same thing. But it's fun having the code there to experiment with.

[kirby urner]

What I can say is that it is a generator, hence the use of 'yield' rather than 'return'.

Wow, that is an amazing little gem.

Ramanujan's uber-bizarre generator is one I've used for 1000 digits.

http://planetmath.org/RamanujansFormulaForPi.html  (good test of your browser's ability to render math formulas).

Speaking of Ramanujan, I've always like this epic ballad version of his life story:

http://archive.org/details/Ramanujan


 

Here it is being used in the Python shell to create 10000 digits:

>>> list(pi_digits(10000))

[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6,...]

Wow.  To get it digit by digit is appealing.

Kirby

[n.bar...@gmail.com]

Hi:

I want to know the derivation of this algorithm of Pi. Can you point me to resource or person?

Nikhil

nbarthwal [at] gmail [dot] come