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Dec 3, 2012, 4:45:51 PM12/3/12

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

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

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Algot Runeman

47 Walnut Street, Natick MA 01760

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

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

Algot Runeman

47 Walnut Street, Natick MA 01760

508-655-8399

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

Dec 3, 2012, 5:04:51 PM12/3/12

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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]

Sent: Monday, December 03, 2012 4:45 PM

To: mathf...@googlegroups.com

Subject: [Math 2.0] Zero -- Even or Not

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- J

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From: mathf...@googlegroups.com [mathf...@googlegroups.com] On Behalf Of Algot Runeman [algot....@verizon.net]

Sent: Monday, December 03, 2012 4:45 PM

To: mathf...@googlegroups.com

Subject: [Math 2.0] Zero -- Even or Not

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Dec 3, 2012, 8:07:14 PM12/3/12

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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--

Bradford Hansen-Smith

www.wholemovement.com

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

Bradford Hansen-Smith

www.wholemovement.com

Dec 6, 2012, 3:13:01 PM12/6/12

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

Enjoy,

--Joshua

Dec 6, 2012, 7:54:20 PM12/6/12

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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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Bradford Hansen-Smith

www.wholemovement.com

Brad

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Bradford Hansen-Smith

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Dec 7, 2012, 1:32:29 AM12/7/12

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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

Dec 7, 2012, 1:35:33 AM12/7/12

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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

--

Dec 7, 2012, 2:03:25 AM12/7/12

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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?

Dec 7, 2012, 2:06:20 AM12/7/12

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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

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

Dec 7, 2012, 2:26:22 AM12/7/12

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Hiya Sue –

I too love the “fruit” denominator. Linda

Dec 7, 2012, 2:38:51 AM12/7/12

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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

Dec 7, 2012, 10:19:44 AM12/7/12

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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

--

Bradford Hansen-Smith

www.wholemovement.com

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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Bradford Hansen-Smith

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Dec 7, 2012, 10:25:05 PM12/7/12

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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

--David Chandler

--

Dec 8, 2012, 12:28:25 AM12/8/12

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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--

Bradford Hansen-Smith

www.wholemovement.com

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

Brad

Bradford Hansen-Smith

www.wholemovement.com

Dec 7, 2012, 11:59:16 PM12/7/12

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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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Dec 8, 2012, 11:47:25 AM12/8/12

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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.

===================================

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

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

- Richard Feynman

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

- Dr. Christos Papadimitriou

===================================

===================================

Dec 9, 2012, 2:06:45 AM12/9/12

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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

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

Dec 9, 2012, 11:54:30 AM12/9/12

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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

(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

Dec 9, 2012, 1:25:14 PM12/9/12

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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

BTW: Here's my variation on your 2 apples problem:

plus =

(My additions are adapted from images in wikimedia commons.)

Sue

--

Dec 9, 2012, 1:54:34 PM12/9/12

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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
<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.

>

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.

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

Dec 9, 2012, 2:18:41 PM12/9/12

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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.

Dec 9, 2012, 3:28:26 PM12/9/12

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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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Dec 9, 2012, 3:45:28 PM12/9/12

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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.
> 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

>

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

Dec 10, 2012, 11:09:21 AM12/10/12

to mathf...@googlegroups.com

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

--

Bradford Hansen-Smith

www.wholemovement.com

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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Bradford Hansen-Smith

www.wholemovement.com

Dec 10, 2012, 1:18:36 PM12/10/12

to mathf...@googlegroups.com

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

<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.

>

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

Dec 10, 2012, 8:51:40 PM12/10/12

to mathf...@googlegroups.com

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.

No virus found in this message.

Checked by AVG - www.avg.com

Version: 2012.0.2221 / Virus Database: 2634/5449 - Release Date: 12/10/12

Dec 12, 2012, 6:16:08 AM12/12/12

to mathf...@googlegroups.com

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.

Dec 12, 2012, 7:27:08 AM12/12/12

to mathf...@googlegroups.com, wholem...@gmail.com

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

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

Dec 13, 2012, 12:31:19 PM12/13/12

to Juan, mathf...@googlegroups.com

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

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

Dec 13, 2012, 4:26:48 PM12/13/12

to MathFuture

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:

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:

Dec 13, 2012, 8:04:36 PM12/13/12

to mathf...@googlegroups.com

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

Dec 14, 2012, 1:42:20 AM12/14/12

to mathf...@googlegroups.com

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

Dec 14, 2012, 1:49:58 AM12/14/12

to mathf...@googlegroups.com

IX, not IV for 9 of course.

Dec 14, 2012, 6:33:14 PM12/14/12