My nine most vital maths questions

[magic_castle32]

Pi is a creation, rather - and that it popping up everywhere in the world of mathematics, and in physical descriptions, is as mysterious as it is deeply philosophical. Reasons offered depend on which school of thought you belong to, but in practice this doesn't really affect how mathematical research is conducted.

I don't think you understand where the beauty of mathematics lies. I do have a favourite number (for non-mathematical reasons), but I certainly don't get excited over numbers or debate which between 7 and 19 is the sexier integer. The beauty of mathematics lies in the ideas, the concepts, the creativity involved - all of which reveal the brilliance of the human intellect, and the mysterious unity between the different fields of mathematics, and with nature.

Here are some links which might help:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

http://www.sciam.com/article.cfm?articleID=5B8E1AAE-E7F2-99DF-31FF9E4F79068FBE&sc=I100322

http://video.google.com.au/videoplay?docid=7691494040933085582&q=terence+tao&total=5&start=0&num=10&so=0&type=search&plindex=0

And the prologue and prefaces to the updated version of Courant and Robbins' What is Mathematics.

As I've insisted, you need to do mathematics in order to appreciate its aesthetic appeal - perhaps an arcane form of beauty which only a small, initiated brotherhood could cherish, but the beauty of mathematics (and that of nature) makes much of the creative arts trivial i.m.o. :P

 

[matt grime]

pi as a decimal cannot be imagined,

You're definition of imagine seems very odd. In fact no number can be 'imagined' by it since every number has an infinite decimal expansion, although quite possibly all of the digits after some point are 0, or some repeating pattern. I would say it was a very open question as to whether there was some closed formula that let you determine what the r'th digit of the decimal expansion of pi was 'easily'. There is one way to do it that is not easy since we know many ways of expressing pi. You are now making more subjective statements about what is an 'easy' to remember expression. Why don't you just stick with the simple statements where things are what they are rather than trying to force them to be what you want? (That is purely rhetorical.)

There is a number

\sum_{n=1}^{\infty} \frac{1}{10^{n!}}

that is not rational, in fact not only is it irrational, but it is transcendental (just like pi) but I know precisely what all of its decimal digits are. Does that make it 'imaginable'?

 

[CRGreathouse]

(CRGreathouse didn't quite get the right answer, though: 6 has gcd(10,6)=2, yet 1/6 is not terminating).

No, I wasn't even close; I don't know what I was thinking. I'm not even sure how to express this in terms of standard functions, unless by rad(b) = rad(n) where rad(k) is the greatest squarefree number dividing k.

 

[Diffy]

Ignoring the discreteness arising out of atomic sizes, the above argument is flawed in the same manner as Xeno's.

Exactly what I thought when I read 1111111's response to my arguement. Link to Zeno's Paradox here:
http://en.wikipedia.org/wiki/Zeno's_paradoxes

 

[matt grime]

No, I wasn't even close; I don't know what I was thinking. I'm not even sure how to express this in terms of standard functions, unless by rad(b) = rad(n) where rad(k) is the greatest squarefree number dividing k.

Suppose we wish to work in base N, and let p_1, p_r be the distict primes dividing N, then 1/X has terminating expansion base N if and only if the only prime factors of X are in the collection p_1,..,p_r.

Thus 1/2,1/5,1/10, 1/250, 1/500 etc have terminating decimal expansions.

 

[CRGreathouse]

1. pi as a fraction can be imagined (held in the head in all its completeness), pi as a little table symbol can be imagined, pi as a decimal cannot be imagined, therefore in one way it is, in itself, unimaginable. The number 4 as a fraction can be imagined, as a little three-lines crossing symbol it can be imagined, and as a fraction it can be imagined. Therefore in one way pi is unimaginable and in no ways is 4 unimaginable - unless there are ways of representing these numbers that I am unaware of. I am talking about just the numbers here, with no reference to reality. It seems to me that pi is more useful than 4 when mapping reality (it being "one of the five fundamental constants"? dunno, read that somewhere) which seems to be full of circles, curves, arcs and so on.

Let me try to guess a definition here. In each case the imaginable is finite and the unimaginable is infinite, so I'm going to assume that you mean "finite" and "infinite" when you say "imaginable" and "unimaginable".

Any constructible number has a finite symbolic representation -- in fact, that's what it means to be a constructible number. All rational numbers are constructible (and thus "imaginable" in that sense) because they can be written as one integer divided by another. Any number with a terminating decimal expansion (finite number of symbols in the decimal expansion) is rational. As a result, to see if numbers are "imaginable" in your sense we need only look at their decimal representations and see if they're finite (terminating) or infinite (repeating).

4 has a finite decimal representation ("imaginable")
22/7 does not have a finite decimal representation ("unimaginable")
pi does not have a finite decimal representation ("unimaginable")
1/3 does not have a finite decimal representation ("unimaginable")
A googolplex has a finite decimal representation ("imaginable")
2 to the power of 2 to the power of ... to the power of 2, where there are a googolplex "2"s, has a finite decimal representation ("imaginable")

Now I'm not sure how much sense these make to me, a priori. I think that almost all people can imagine 1/3 'in its totality', and plenty of people can imagine pi in the same way, and yet I suspect that most people cannot imagine the last one, even though it's just a whole number.

2. Which numbers are more beautiful than others. CRGreathouse you tease! Tell me! At least "in your opinion". For me "entirely subjective" is far from "meaningless," even, I am very keen to explore, in mathematics. If anyone has anything to say about subjectivity, beauty or aesthetics, please let me know. If anyone has any insights, links, ideas about how our appreciation of infinity and recurrence in nature corresponds in any way to mathematicians appreciation of infinity and recurrence in numerical abstraction, it would be gold to me.

Well, there's a good list at http://home.earthlink.net/~mrob/pub/math/numbers.html

For me, e^{-e}=0.06598803\ldots and e^{1/e}1.44466786\ldots are beautiful. They are the lower and upper bounds for the equation
x^{x^{x^{x^\ldots}}}
For any x between these two numbers, this infinite exponential actually has a finite value. Cool, eh?

A hypothetical beautiful number would be an odd perfect number -- a number not divisible by 2 where the sum of the proper divisors of the number are equal to the number itself. The only perfect numbers known are even, for example 28 = 1 + 2 + 4 + 7 + 14.

 

[arildno]

Who are you, dron, to assign to yourself the grandiose authority of determining for all others the bounds of their imagination?

Perhaps you should go into yourself a bit and reflect upon that the main reason why you struggle with mathematical concepts is that YOU have incorrect preconceived notions about maths.

 

[Hurkyl]

pi as a decimal cannot be imagined, therefore in one way it is, in itself, unimaginable.

It depends on what you mean by imagined. If by "imagined" you mean "can be written as a finite sequence of digits", then it cannot be imagined. But, IMHO, that's an incredibly limited imagination!

The actual, mathematical content of an infinite decimal is that it's simply some function that tells you what digit is in what position. For finite decimals, the usual "string of digits" method is an easy way to visualize that function -- but there is no reason to stop there.

Not only does there exist such a function for pi, but there are effective algorithms for computing it -- in other words, not only can we imagine the infinite decimal expansion of pi, but we can actually compute with it. (Such numbers are sometimes called "constructible")

 

[111111]

Exactly what I thought when I read 1111111's response to my arguement. Link to Zeno's Paradox here:
http://en.wikipedia.org/wiki/Zeno's_paradoxes[/QUOTE]

Oh thanks, it seemed logical, but apparently it isn't true.

 

[CRGreathouse]

Suppose we wish to work in base N, and let p_1, p_r be the distict primes dividing N, then 1/X has terminating expansion base N if and only if the only prime factors of X are in the collection p_1,..,p_r.

Thus 1/2,1/5,1/10, 1/250, 1/500 etc have terminating decimal expansions.

Yes. Is that easier to understand than what I posted (n has a terminating base-b expansion iff rad(b) = rad(n))?

 

[CRGreathouse]

Even worse, your post #1 reads like a parody of views concerning mathematics which (to judge from popular literature and newspaper stories of the time) were held by many persons at the beginning of the last century

I'd love it if you would expand on what you think those views were in the early 1900s, either in this thread or a new one. I'm curious, largely because I don't have a feel for this aspect of math history.

I suggest that this thread be locked, but perhaps someone will care to start a new threads on "What are the current top ten popular myths about mathematics?", "What is mathematics, that thou are beauteous?", or even "Numbers: is math propaganda in the national interest?"

I may just start that first thread.

 

[matt grime]

Yes. Is that easier to understand than what I posted (n has a terminating base-b expansion iff rad(b) = rad(n))?

As I don't know what rad of a number is, it is a damn sight easier for me to understand. Though of course I should have read your post more thoroughly.

 

[dron]

Not sure what happened to Chris Hillman's post, but would like to know what he or anyone else thinks of this...

When mathematicians describe a proof as "beautiful" they can mean one of three things:

One, because it is succinct, aerodynamic, and efficient, like a golfer’s stroke.
Two, because it links unexpected lines of thought, like a poet’s metaphor.
Three, because it somehow vanishes into infinity, like the light of the ribs of the branches of the trees of the forest of the planet of the space of the light…

Feel free to be as contemptuously dismissive as you like chaps.

 

[matt grime]

The first two would be reasonable, and are almost the descriptions that occur in the book by Gowers that you initially said didn't contain any answers to your questions. I don't remember him putting in the similes. If you want to get a better understanding of mathematics/mathematicians, then I would say that a mathematician wouldn't have inserted those similes since they don't help to convey anything, and are just as open to interpretation. I can't make any sense out of the last one.

 

[Chris Hillman]

Not sure what happened to Chris Hillman's post

Reportedly someone died laughing while reading it so out of concern for public safety...

Feel free to be as contemptuously dismissive as you like chaps.

That's not funny. Try again

 

[dron]

Do you really want an honest response?

Yes, give me an honest response. I didn't read your deleted post, just saw it quoted - see if you can find a less hilarious way of putting it perhaps?

 

[mongoose]

look up the prime number theorem. it draws a connection between the natural log and the distribution of prime numbers.


also, mathetical beauty doesn't necessarily have to be succinct...at least not to me. i think a lot of beauty can come from the results, even if the process that leads to them is complicated and messy.