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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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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?
Hiya Sue –
I too love the “fruit” denominator. Linda
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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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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.
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
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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.
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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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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.
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.
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
IX, not IV for 9 of course.
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,
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
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.
sage: factor(-391/14691)-1 * 3^-1 * 17 * 23 * 59^-1 * 83^-1
In[6]:= FactorInteger[-391/14691]
Out[6]= {{-1, 1}, {3, -1}, {17, 1}, {23, 1}, {59, -1}, {83, -1}}
>mathematics tends to attract those who are temperamentally averse to the "many ways to do it" philosophy.
>what I think is valuable, especially for people new to maths, is to remind them it's a beach of many sand castles
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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 FullerEven 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 castlesI 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!"
-- Michel
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.
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:and
"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."
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
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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.
>I would say the number 0, and the origin of a coordinate system are not synonymous concepts.
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.
Dear Kirby,
The "aesthetically pleasantness" of simplexes is subjective, therefore relative.
I don't want to sound pessimistic but good luck trying to get your hobby accepted within the mainstream school curriculum.
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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."
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.
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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.
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.
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
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!
Juan
cost, (1/pi)*sin(pi*sin(t), (1/pi)-(1/pi)*cos(pi*sin(t)
>diameters USED TO be easier to measure than radii.
> Pi is more historical.
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Really neat, Juan! I plotted it using http://www.math.uri.edu/~bkaskosz/flashmo/parcur/
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.
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.
>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:
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.
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.
>I would be interested in looking at the spigot algorithm
def pi_digits(n):k, a, b, a1, b1 = 2, 4, 1, 12, 4while n>0:p, q, k = k*k, 2*k+1, k+1a, b, a1, b1 = a1, b1, p*a+q*a1, p*b+q*b1d, d1 = a/b, a1/b1while d == d1:yield int(d)n -= 1a, a1 = 10*(a%b), 10*(a1%b1)d, d1 = a/b, a1/b1
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,...]