My nine most vital maths questions

[dron]

First post here. Would dearly like to have some answers for book I am writing on aesthetics and psychology. Can't find FAQ section, or anywhere else these questions are answered succinctly, so here they are. Feel free to give yes/no answers, links or even to contemptuously brush aside (although I'd love to know, if so, why these questions are stupid). Needless to say I am not a mathematician and would appreciate as much simplification as possible without distortion.

1. Is it true that the number that pi represents, being infinitely "long", cannot be imagined / accurately represented / exactly known; only the symbol or approximation can be?
2. Does it follow that it is impossible to imagine / accurately represent / exactly know the area of a circle?
3. Is this also true for numbers like e and i? - that all we can know for sure about the calculations based on them are very good approximations?
4. What calculations are based on these numbers? What branches of modern mathematics are based on pi, e and i? Could you give me a simple list (logarithms, calculus, set theory, etc) or does it pretty much amount to "all"?
5. Is there some way that pi, e, i are connected to either prime numbers or chaos theory?
6. Why are prime numbers useful or beautiful? I read an Oliver Sachs book where two autistic twins were thinking up and then "enjoying" (as if drinking fine wine) massive prime numbers. Any idea why would this be so?
7. When a proof of theorum is said to be "beautiful" what criteria does it satisfy? Efficiency is obviously one of them - succinctness. Use of startling analogy another (i.e. reasoning from quite a remote perspective to the one at hand)? What about harmonic arrangement (the formula in some way "feels" nice, like a Fibonacci series looks nice)?
8. Quantum mechanics is, in some ways, the science of the impossible to imagine - what elements of maths are used to deal with wavicles and time traveling particles and whatnot? Does pi, e, i, primes or chaos figure in any way?
9. Would you say that interesting maths tends to come from interesting mathematicians? Knee jerk response I suppose is "of course not". But do the lives of the most "creative" mathematicians look different to the lives of the less brilliant? I know Fermat, for example, led a dull life, but is there any indication that there was "something about him" that set him aside from others in some other way than mathematical skill?

That's all. Thanks for reading.

Dron.

 

[magic_castle32]

I refer you to Timothy Gower's "Mathematics: A Very Short Introduction".

 

[dron]

Thank you. I have read this book. It does not answer any of these questions. Not that it should or anything; just that it doesn't.

 

[magic_castle32]

In which case you need to read it again. And meditate upon it. Perhaps doing some real math would help, too.

 

[arildno]

First post here. Would dearly like to have some answers for book I am writing on aesthetics and psychology. Can't find FAQ section, or anywhere else these questions are answered succinctly, so here they are. Feel free to give yes/no answers, links or even to contemptuously brush aside (although I'd love to know, if so, why these questions are stupid). Needless to say I am not a mathematician and would appreciate as much simplification as possible without distortion.

1. Is it true that the number that pi represents, being infinitely "long", cannot be imagined / accurately represented / exactly known; only the symbol or approximation can be?
2. Does it follow that it is impossible to imagine / accurately represent / exactly know the area of a circle?

Depends on what you mean by "known".
Sure, the decimal representation of pi is infinitely long, and hence, cannot be written down on a piece of paper.
The same is true, however, of the decimal representation of 1/3.

In both cases however, algorithms are well known that allows you to calculate however many correct decimals of either number's decimal representation.

What more do you really want?

Besides, what's so special about decimal representations of numbers anyway?

Neither of this has much to do with approximations in general; pi is a very well defined number whose properties can be relied upon "exactly" in any proof it occurs; it is only its decimal representation which is infinite and not finite. That's really all worth mentioning.

 

[dron]

Thank you Magic Castle - perhaps you can guide me to the precise pages of Gower's book (which I have in front of me) where my questions are answered (I also mailed my questions to professor Gowers - he seems quite friendly). Unfortunately I do not have the ability or inclination to "do the math" required to answer my own questions in great depth; this is why I have asked others, more knowledgeable, to give me, or guide me towards, something I can comprehend.

Arildno. Thank you. I realize this question enters a realm outside of maths proper; e.g. the meaning of numbers, none of which can be "known" - in that they do no represent anything that can be accurately imagined. Nevertheless the precise number pi, as far as I understand it, is in some way different (like other irrational numbers?) in that not only does the thing this number "represent" elude the brain, but the number itself does. Is this so? Why does this not mean that when I use pi to measure a circle the answer I get is not quite precise? I suppose by "algorithms" you mean there are other non-decimal ways of representing pi which somehow eliminate its infinitely complex nature?

Also, is there not - when it comes to imaginability - some fundamental difference between 1/3, which, although "infinite" repeats itself, and pi, which does not?

The reason I ask these questions is not in order to calculate anything, but to understand better the relationship between mathematics and the capacity to imagine. Numbers like pi, e, zero, infinity, i and phi seem to form the base of all mathematics. Although I am not a mathematician I believe I am capable of grasping what these numbers have in common with each other, and with other realms of human experience and thought.

 

[arildno]

Pi can be adequately "imagined" as the ratio between a circle's circumference and its diameter. End of imagination, since we may prove there is a unique number having that particular property.

 

[matt grime]

Thank you. I have read this book [a VSI b Gowers]. It does not answer any of these questions.

Really? He discusses question 7 in some detail, doesn't he?

Few of your questions seem well posed, nor to have mathematical answers. You'd be far better off posting them in philosophy (of mathematics/science).

1,2,3 are all about numerical accuracy, which is not the same thing as accuracy in mathematics. You're mixing 'real life' (where mathematics is used) with mathematics itself. $\pi$ is a perfectly good and accurate symbol.

 

[dodo]

My two cents. Being much of an ignorant myself, I can volunteer to state some of the obvious, to no-one disgrace. : )

1) (The first will be longer, serving as an introduction.) Imagine you start with the integer numbers only. The moment you want to divide 3 by 4, you can't; you need to extend your numbers with a new concept, where the old is still possible but the new can also be done. A similar thing has been done, when starting from fractions and finding out can you can't extract the square root of 2.

The new constructions (in this case you can google for "Cauchy sequences", or "Dedekind cuts") are provided with mechanisms to determine when two of them are equivalent (and thus represent the same number), or how to add or multiply them, or how to put the old numbers in the new representation. Once this is done, they are just as numbers as the fractions are.

Keep in mind that mathematics is a realm of symbols: "53" is no less a symbol than pi, sparing you the hassle of bringing 53 stones and putting them in a line. For a mathematician, pi or e are as exact and accurate as 53. The integers themselves are symbols constructed from an starting point (1) and a succesor operation. As for their "reality", this is a vague subject (reality is best left to your imagination :). Try to produce a metal bar of exactly 1 meter, and you'll see that the problems with reality have nothing to do with the length being integer, fractional or irrational. Pi or e are only approximations when you compare them to fractions, which are an abstraction in themselves.

The imaginary i is a different kind of animal. When you can't extract the square root of -1, you extend the numbers by a different method than above, by representing them as pairs; the old numbers become (n, 0), and i is (0, 1), the base of the new (second) dimension of numbers of the form (0, n). As such, i is a Gaussian integer: it is just a pair of integers, in many senses much like the fraction 3/4 is also a pair of integers.

2) Thus the area of a circle is exactly pi times the radius squared, provided you have an exact expression for the radius itself.

3) Ditto.

4) This is best left to someone knowing better, but a neat example is Euler's formula for complex numbers, e^ix = cos x + i sin x, or if you prefer, the pair (e, 0) raised to the power of (0, x) gives you the pair (cos x, sin x); a special case of which says that e to the power "i times pi" equals exactly 1.

5) No idea. An answer from an specialist would be in order.

6) They are (or can be) a basement for the construction of numbers. Each integer has a unique set of prime factors; it's like all numbers spawn from the primes. There is an inherent beauty to order (I'm probably quoting Jung) and to simple, elegant relationships. Primes usually share that rank. The relationships mentioned in 4) does too.

7) My gut guess is that it provides a bridge where none existed before. And note that this has a relation to the shortness or succintness of it: if the length of the proof were a measure of the bridged distance, a short bridge makes the two subjects even closer. Also, an "elegance" component might be one of surprisingly bringing to relation a subject which seemed unrelated.

8) Peak ignorance here. My guess is that statistics play a major role.

9) Define "interesting". If an interesting mathematician is one that produces interesting math, then the statement is a tautology. If it's meant that the mathematician is an interesting person, then I'd say the two conditions are mostly unrelated, except perhaps when it comes to divulgation. Claims about mathematicians being dull, introverted or neurotic are probably of a projective nature. :) , and certainly not specific to mathematics, but common to any intense mental activity. Meaning, chess players are worse.

. . .

It is good to note also why these kind of threads usually raise little enthousiasm. Too often they are posed by people who would rather been given the understanding, instead of looking for it by themselves. Not that it is your case; but it hits a tiresome spot. Nothing wrong with asking for references to study, as long as you go and do your homework. (And come back after a few months.)

My apologies for any squeech of my non-native English. I hope this can be of help.

 

[arildno]

As for "occurrence" of the numbers e and pi in practice, the interesting relationship
$$\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}$$
along with that integrand's close intimacy with the probability density function in the normal distribution case makes e and pi occur quite frequently in statistical calculations.

 

[dron]

Pi can be adequately "imagined" as the ratio between a circle's circumference and its diameter. End of imagination, since we may prove there is a unique number having that particular property.

Thank you Arildno. I'm afraid I don't understand "since we may" in this sentence.

Really? He discusses question 7 in some detail, doesn't he?

Question 7 is in the only one. I asked it because I was hoping for some greater insight or other observations.

Few of your questions seem well posed, nor to have mathematical answers. You'd be far better off posting them in philosophy (of mathematics/science).

Thank you Mr Grime. I'm not quite sure what you mean by "well posed". I take it by "well" you are not referring to my hazy understanding of the concepts I am seeking to understand - of this, clearly, I am guilty. If, by "well" you mean a standard used by mathematicians, I am ignorant of it I'm afraid. I'd like to know what you mean though. As for posting in philosophy, you may be right. I posted here because I was hoping from answers by mathematicians and not philosophers.

1,2,3 are all about numerical accuracy, which is not the same thing as accuracy in mathematics. You're mixing 'real life' (where mathematics is used) with mathematics itself. Pi is a perfectly good and accurate symbol.

Yes, I am deliberately mixing "real life" (I think) with maths. I know that pi is a good and accurate in maths. I just can't see if or how it applies to "real life".

Turning to Dodo, for whose long patient reply I am particularly grateful, I'm afraid I still do not understand your answer to my first or second questions. I'm sorry.

2) Thus the area of a circle is exactly pi times the radius squared, provided you have an exact expression for the radius itself.

I realize that the area is exactly pi times the radius squared. But if the radius is, for example, one hundred kilometres; what does that make the area? Isn't the answer imprecise?

They are (or can be) a basement for the construction of numbers. Each integer has a unique set of prime factors; it's like all numbers spawn from the primes. There is an inherent beauty to order (I'm probably quoting Jung) and to simple, elegant relationships. Primes usually share that rank. The relationships mentioned in 4) does too.

How are primes "ordered"? Are they ordered in the same way as a cube, or as a logarithmic spiral - the latter being less beautiful than the former to most people's eyes.

My gut guess is that it provides a bridge where none existed before. And note that this has a relation to the shortness or succintness of it: if the length of the proof were a measure of the bridged distance, a short bridge makes the two subjects even closer. Also, an "elegance" component might be one of surprisingly bringing to relation a subject which seemed unrelated.

Yes, I see. That's clear. I wonder about the "nice feel" element though. Are some proofs elegant like a flower is, or somehow giving a feeling of balance or some such?

Define "interesting". If an interesting mathematician is one that produces interesting math, then the statement is a tautology. If it's meant that the mathematician is an interesting person, then I'd say the two conditions are mostly unrelated, except perhaps when it comes to divulgation. Claims about mathematicians being dull, introverted or neurotic are probably of a projective nature. :) , and certainly not specific to mathematics, but common to any intense mental activity. Meaning, chess players are worse.

Yes, I mean an interesting person, having character, flame, originality about him.

It is good to note also why these kind of threads usually raise little enthousiasm. Too often they are posed by people who would rather been given the understanding, instead of looking for it by themselves. Not that it is your case; but it hits a tiresome spot. Nothing wrong with asking for references to study, as long as you go and do your homework. (And come back after a few months.)

Quite. I would like references. But I am not seeking to work on mathematics; I have my hands full. I know from experience that nothing, no matter how complex, cannot be explained to a laymen so that he grasps it, and this is what I ask for. Specialists are prone to look down on those who do not share their vocabulary, and to hide behind the power that their knowledge gives them - in their world. Often they spend so long in this world that they even forget what it is like to live outside it.

An example might be when Arildno tells me

$$\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}$$

I have pointed out that I am not a mathematician, and so, while grateful for the time taken to post an answer, I cannot understand it. What is the long thing on the right, like a treble clef? What is d? What is x? What is an intrgrand? What is probability density function? What is the normal distribution case?

 

[matt grime]

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

 

[Gokul43201]

Yes, I am deliberately mixing "real life" (I think) with maths.

This is generally not a good thing to do.

I know that pi is a good and accurate in maths. I just can't see if or how it applies to "real life".

Mathematics is not reponsible for how its members appear in real life. Math deals with abstract quantities and structures. Some of these things may map very well onto real-life situations, some not so well, and some not at all. But this no fault of the mathematics.

I realize that the area is exactly pi times the radius squared. But if the radius is, for example, one hundred kilometres; what does that make the area? Isn't the answer imprecise?

No, the area is (half) as precisely $\pi*10000$ sq. kilometers as the radius is 100 km. The imprecision comes from how well we know the radius, not from how well we know pi.

How are primes "ordered"? Are they ordered in the same way as a cube, or as a logarithmic spiral - the latter being less beautiful than the former to most people's eyes.

No, they are not ordered as either. In fact, they are not known to be ordered in any way that can be written down using only previously familiar functions.

 

[christianjb]

If integers are molecules, then primes are atoms. As each molecule can be reduced to its component atoms, so each integer can be expressed as a unique product of primes.

 

[dron]

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

Thank you Mr Grime. I don't think I've made "what I wish to know" clear for you. I am not seeking unambiguous answers for unambiguous questions, but interesting answers for (to me) interesting questions, which would seem to be something else entirely (although perhaps alien to some people).

Mathematics is not reponsible for how its members appear in real life. Math deals with abstract quantities and structures. Some of these things may map very well onto real-life situations, some not so well, and some not at all. But this no fault of the mathematics.

Thank you Goku. You use the word "fault" - although, of course, I am not blaming mathematics or mathematicians. I am simply enquiring as to the relationship between real life and maths.

No, the area is (half) as precisely pi x 10000 sq. kilometers as the radius is 100 km. The imprecision comes from how well we know the radius, not from how well we know pi.

I see. But what is the answer? Forgive my ignorance, please, but if I multiply 10000 by pi to three decimal places won't I get a different answer to multiplying 10000 by pi to three million decimal places?

No, they are not ordered as either. In fact, they are not known to be ordered in any way that can be written down using only previously familiar functions.

I see, thank you. But I asked how primes were ordered because Dodo told me that prime numbers were beautiful because "beauty comes from order". In other words I was referring to the order in primes, rather than the order of them. I am aware that no-one has yet found a way to predict their distribution. It is their apparent "beauty" that interests me.

If integers are molecules, then primes are atoms. As each molecule can be reduced to its component atoms, so each integer can be expressed as a unique product of primes.

This is an interesting way of looking at them Christian. Thank you. Are integers in some way defined as being able to be expressed by primes, or is it just a happy coincidence that they can be? How, do you think, is either fact related to their beauty, if at all?

 

[Quantumduck]

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

Matt, off topic, but here is a different version of your quote in the sig, again, with no attribution.

http://cs.union.edu/~postowb/cookie.html

Search for 'waste' in the page and you will find it.

Here is the same version in a different context, about three jokes down,

http://lithops.as.arizona.edu/~jill/humor.text

And the closest thing to a real quote I could find was this:

Wiener, Norbert (1894-1964)
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.
Ex-Prodigy: My Childhood and Youth.

 

[Diffy]

I see. But what is the answer? Forgive my ignorance, please, but if I multiply 10000 by pi to three decimal places won't I get a different answer to multiplying 10000 by pi to three million decimal places?

You will, because you are multiplying by two different numbers.

pi to three decimal places does not equal pi to three million decimal places does not equal pi.

There is only one answer when you multiply pi by 10000, but there are infinite representations. If you want to see what that looks like, I would ask you how? I can show it to you in terms of square millimeters. Just give me a compass that can measure out 100 millimters, and I will draw you a circle. The space that that cicle takes up is one representation of pi*10000 (note that pi*10000 is yet another representation). I can't show it to you in terms of decimal representation.

But this makes it no less real (or I guess in your terms "imaginable") than the intetegers.

I can represent pi as a symbol $$\pi$$ one two and three don't have greek letters that can represent them. Symply because mathematicians have not defined them. Had we difined our whole universe and number system differently perhaps we would be able to represent the ratio of a circumference and a diameter a bit easier and the intergers might be impossible.

It is simply a matter of how you look at things.

 

[dron]

Thank you Diffy. So what is the most exact or imaginable answer to the question pi x 10000?

(By the way, I apologise to mathematicians reading this, to you, nonsense. It must be like Kasparov being asked why can't the building jump like the horsey. I should be asking these questions elsewhere, I know - but I've started so I'll plug on. All I can say, in begging your indulgence, is that I know there are some great footballers who enjoy kicking a ball around with a five year old).

 

[Extropy]

Thank you Diffy. So what is the most exact or imaginable answer to the question pi x 10000?

I do not quite know what you mean. pi x 10000 is not a question, so it does not have an answer.The number pi x 10000 is pi x 10000, and nothing else, and therefore is most exact.

If you are asking about what an irrational (or real number in general) is, it is merely a partition of the rational numbers into two disjoint sets, such that all elements of one set is greater than all elements of the other, and such that the one with lesser elements has no greatest element. So essentially pi is one such partition of rationals. A decimal representation of a real number is a convenient way of representing real numbers, but it is merely a representation of the real number (i. e. it is a defined symbol and not the actual symbols of the real numbers themselves, and not the real number itself), and is by no means the only representation.

 

[PowerIso]

Well I suppose the exact answer would be pi * 10000 as stated before, but if you wanted a numerical value, then you would have to pick your sig fig. For the purpose of simplicity, I picked 10.

31415.92654

 

[Extropy]

But, of course, this "numeric representation" is not the actual number. A real number is a Dedekind cut of rationals, and these Dedekind cuts are the same if the partitions are exactly the same. In other words, if the respective sets of the Dedekind cuts are subsets of each other. But, of course, they aren't.

 

[111111]

Here is a kind of a relationship between prime numbers and irrational numbers: prime numbers are numbers that cannot be written as the product of two integers (except 1 and itself), and irrational numbers are numbers that cannot be written as the quotient of two integers, in other words they are both types of numbers that cannot be written shorter; you could say there is no redundancy.

 

[dron]

So pi x 10000 has no exact numerical answer? This is news to me. Does this mean that pi r squared has no exact numerical answer? Would pi x 10000 have an exact numerical answer if pi happened to = 10? Does this mean, as I asked, that (putting aside for a moment that nothing can really be precisely measured) there is no exact numerical answer to the question "what is the area of this circle?" (like there is for what is the area of this square). And is that because of pi's irrationality?

111111, thank you. This seems elegant. I believe that phi is an irrational number; one that appears often in nature because of its lack of redundancy and consequent efficiency (in spirals, phyllotactic patterns, etc). I wonder if irrationals and primes have other relationships with themselves, and with natural forms. This is why I threw chaos theory in the mix earlier - if anyone knows how chaos theory uses irrationals, primes or has anything to say about natural forms, please write something.

As I said before, an interesting answer is more valuable to me than an unambiguous one. And even though what I ask is of moronic simplicity it might, just might, lead somewhere new for you chaps too. This is doubtful but given my long experience in other disciplines who knows what might pop up?

 

[matt grime]

An irrational number (and pi is one) is one that is not a rational (a/b for a,b integers) and therefore does not have a decimal expansion that is eventually recurrent, such as 361/9 = 4.01111111111...

When some people say 'no exact numerical answer' they normally mean it is either it is not a terminating decimal expansion, like 7.2, or that it is irrational. This is entirely a matter of personal taste, and nothing particularly mathematical. 10000xpi is a prefectly good number, it just cannot be expressed in a certain fashion. Once more you're mixing up 'real life' where 'imprecise' has a different meaning. These things are not the same.

That there isn't a nice expression of something in one particular format is not the same thing as being 'unknowable'. 1/9 has no 'exact' expression as a terminating decimal but it does if we use base 3 or 9.

 

[J77]

1. Is it true that the number that pi represents, being infinitely "long", cannot be imagined / accurately represented / exactly known; only the symbol or approximation can be?
2. Does it follow that it is impossible to imagine / accurately represent / exactly know the area of a circle?

Didn't Plato wtie about not being able to draw a perfect circle -- and the perfect circle not existing in the physical world?

 

[Gale]

Gron, I think you're confused about the general nature of numbers. For instance, it seems you would be comfortable with the numbers 1, 2, or 3, since you understand what those represent. You were first taught to count with those numbers, and they make sense to you. The Greeks, for instance, used shapes to represent numbers. To them pi was very understandable since its just a ratio of a circle's circumfrence to its diameter. Its a number that is inherent to the properties of a circle. You aren't used to thinking of numbers this way.

To a Greek, the number "two" did not represent 2 sheeps, or 2 fingers, it represented a line on the side of a triangle, or square, of length 2. Using their style of math, you can represent a number by the diameter of a circle. If I want the number 1, I would draw a circle with a diameter 1 length long. Its a simple thing from here to measue the circumfrence using a string. Archimedes then realized that there was a relationship between the diameter and circumfrence. He then proceded to get as precise a measurement as possible using their methods, (you can look it up if you're interested.) Pi is always the same, regardless of the size of the circle. Therefore pi is just as "exact" as any number we count with, except, that when you 'count' with pi, you're counting using circles.

The problem is then our representation of pi. Since we use counting numbers, we cannot represent the number pi exactly in decimal form. That's simply a product of our counting system. Numbers themselves represent exact things, how we represent the numbers depends on us. You ask if pi can be exactly "known", yes it is. You're just uncomfortable with the way we define it since you've only really been introduced to a small understanding of math. Had you been introduced to Greek shaped-base math, you would feel more comfortable with pi than 0. (The Greeks, consequently, couldn't comprehend zero, since no shape could have a side of such length, while in counting zero is more obvious...)

Again, I reiterrate that your issue with pi is in your understanding of numbers, and has nothing to do with pi itself. The number pi represents is exact and can be imagined. How do you imagine "1"? All numbers are just symbols we've created to represent things. Pi is no different; neither are i and e.

 

[dron]

Thank you Gale. Yes I am more familiar with 1,2,3 and so on, and understand them a little better than irrational numbers, but I'm no more comfortable with them than I am with any abstraction. I am much more at home with reality (something which, controversially, for me, is fundamentally the same for everyone).

But I digress.

I can imagine "1" like so... 1. Not in reference to one sheep or one apple. Just that. 1. This is imaginable to me. Whereas when I think of pi I either think of the symbol $$\pi$$ (imaginable) or I think of it as a fraction (imaginable) or in terms of a decimal 3.14 blah blah infinity (unimaginable).

So. Can it not be said that in some way irrational numbers go on forever in a way that rational numbers do not? If so, does it not mean that irrational numbers are more... irrational! It would seem so to me (unless rational numbers can be "rephrased" in such a way to make them irrational?). It would also seem to me that seeing as reality is irrational, irrational numbers are (more) appropriate to describing it. This is what I am trying to discover.

 

[J77]

You know about transcendental numbers ( http://en.wikipedia.org/wiki/Transcendental_number ), right?

 

[Gale]

Rational numbers can be made irrational in a different number base. We count in base 10. If you count in base pi, 10 would represent pi. The number "4" would become irrational in this number system and would be about 10.221... (I think)

Again, the problem is just in your understanding of numbers. "4" is only imaginable because of the way you count and perceive numbers. 4 is a symbol, $$\pi$$ is a symbol. Both represent real, and exact numbers. Pi is no different, except that in our convienient system, its irrational.

 

[dron]

Thank you both.

1. So, given that we use 10 as a base, does this mean that in some way irrational numbers go on forever in a way rational numbers do not?
2. Are irrational numbers in some way more appropriate for describing what we find in natural forms (curves, arcs, circles, fractals and wotnot)?
3. So why do we use 10 as a base and not base pi? Is base 10 somehow easier to think in? Why? Is it because irrational numbers are in some way unthinkable (not merely inconvenient)?
4. And what about beauty? Are some numbers more beautiful than others?

 

[J77]

Sorry about the overuse of wiki but it's good for this kind of thing...

Beauty ---> http://en.wikipedia.org/wiki/Golden_ratio

 

[Gale]

1. I think has been answered.
2. More appropriate than what? Every other sort of number? No. Different numbers represent different things, all of which describe "natural forms".
3. How many fingers do you have?
4. Read the wiki.

 

[Extropy]

2. Are irrational numbers in some way more appropriate for describing what we find in natural forms (curves, arcs, circles, fractals and wotnot)?

The numbers various values are assigned were because of the nature of the object, and not the number.

3. So why do we use 10 as a base and not base pi? Is base 10 somehow easier to think in? Why? Is it because irrational numbers are in some way unthinkable (not merely inconvenient)?

Because we have ten fingers. Irrational numbers are not unthinkable; they merely are an element of a complete ordered field, and nothing more.
Perhaps this may help for an understanding of reals:
http://www.dpmms.cam.ac.uk/~wtg10/reals.html

4. And what about beauty? Are some numbers more beautiful than others?

That depends on what we do with them. It is no inherent property of number itself. As a rule, math is not based upon physics; rather, it is based upon semantics and syntatics.

 

[111111]

4. And what about beauty? Are some numbers more beautiful than others?

You're the aesthetics person, why are you asking us? And secondly what is the deal with the "golden ratio"? Show me some actual evidence that it is somehow beautiful. Like an experiment using children or other unbiased people that shows that they will favor a golden rectangle over another rectangle. Personally I think that a circle is the most beautiful, its the only shape with perfect symmetry, with every part constantly changing at the exact same amount.

 

[matt grime]

1. So, given that we use 10 as a base, does this mean that in some way irrational numbers go on forever in a way rational numbers do not?

No. And yes. Define your terms. I think I already answered this perfectly. The only numbers with terminating decimal expansions are those when the denominators are divisble only by the primes 2 and 5

Any other number will not have a terminating expansion.

Any rational number will have an eventually periodic expansion and that is nothing to do with 10 being the base rather than, say, 11.

I am not going to say that any more. That is at least the 3rd time it's been explained here, and god knows how many other times elsewhere.

2. Are irrational numbers in some way more appropriate for describing what we find in natural forms (curves, arcs, circles, fractals and wotnot)?

No. Yes. Depends on what you mean, obviously.

3.Is it because irrational numbers are in some way unthinkable

What? Define what you mean.

4. And what about beauty? Are some numbers more beautiful than others?

Beauty is in the eye of the beholder, and that is YOU.