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.

 

[arildno]

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?

Decimal* representations of irrationals are non-repeating, but being non-repeating does not mean being more infinite than being repeating infinitely many times..

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

"I would like ONE pint of beer" works adequately for me, so I'd say no to this one.


*Of course, this holds whatever integer base we choose, 10 or some other number.

 

[CRGreathouse]

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?

All irrational numbers are nonterminating ("go on forever") in any integer base, and they never repeat. Some rational numbers terminate -- exactly those which have denominators d (in lowest form) with gcd (d, b) = 1 where b is the base. So in base 10, terminating decimals are of the form a / (2^b * 5^c) for an integer a and non-negative integers b, c.

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

Here's the thing. There are 'many more' irrational numbers than rationals -- if you pick a 'random' real number the probability that it is rational is 0. (It's not impossible, just vanishingly unlikely.) Thus it would be sensible if 'most' things were described by irrational numbers.

I can't comment on natural forms, though; no natural forms can be measured precisely enough to see if they are rational or irrational. Between any two unequal real numbers there are an infinite number of rational numbers and an infinite number of irrational numbers.

3. So why do we use 10 as a base and not base pi?

Probably because we have 10 fingers.

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

Yes.

 

[dron]

Ooh ooh. Which ones? Why?

(longer reply coming soon)

 

[dron]

Another thank you for all your time and perseverance. You are quite right in saying that some of these things have been dealt with before. Partly this is because the concepts are hard to grasp, partly because people skilled in analytical competencies are rarely gifted verbally and find it difficult to express the ideas to newcomers. If I have seemed to asked a question twice it is because, I'm afraid, I still do not understand. My understanding of maths ended with quadratic equations (believe it or not I was a high performer until this point, when I switched to arts) and just about all of the links and equations you have posted, while the intent is appreciated, are meaningless to me. If I were to ask clarification on these, as well as my far simpler questions, my every post would be a book. (If inability to grasp something is an unusual situation for any of you, perhaps you could try working on something that does not come at all naturally to you - I'm sure there's something - and then maybe you can fish out an ounce of sympathy. )

So.

1. We use 10 base because we have ten fingers? Is that true? Is it not the case that 10 is used more for its convenience than its ubiquity?

2. In any case, I am still not clear if irrational numbers have some - any - element of cannot be thought-about-ness that rational numbers do not have.

3. Extropy says that

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

Were natural forms the origin of all these nutty irrationals?

4. Arildno's sarcastic

"I would like ONE pint of beer" works adequately for me, so I'd say no to this one

is dismissive, but counting aside, is it not the case that nature is filled with more curves, circles, ellipses, arcs and so on than straight lines and neat 1:1 proportions?

5. 111111, apparently losing patience with me, said, after I asked if some numbers are 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.

It may be that "I am the aesthetics person" - but you are still a "person" - and capable of appreciating beauty. Knowing numbers better than I, I was hoping you could show me something of their aesthetic quality, if they have it; as I might, if you were interested, be able to show you beauty in, say, literature.

(And CRGreathouse says that some numbers are more beautiful - although he may have been pulling my leg)

I cannot show you the evidence you ask for about the golden mean, and note that I am not particularly championing it. Its just that I have heard that it seems to play a role in phyllotactic natural patterns, and is very common in animal symmetry. I have heard that it is an irrational number - and sought to learn some connection between it and pi and e, other irrational numbers I have heard of. And primes. I too find circles beautiful.

6. CRGreathouse, thank you. Very interesting (although the gcdb thing lost me). Although you say that

no natural forms can be measured precisely enough to see if they are rational or irrational. Between any two unequal real numbers there are an infinite number of rational numbers and an infinite number of irrational numbers.

but is it not the case that logarythms and calculus were invented to measure curves and arcs and circular motions and stuff - and that these are based on irrational numbers? and that although they "cannot be measured precisely enough" it tends to be the case that certain numbers - e, i, pi, infinity, zero - are more appropriate to measure nature than others?

7. Finally, Mr Grime. I tire of your tone. Men generally and scientists specifically are often unable to even talk about "tone" - it being, alone with most other genuinely beautiful and valuable things in life, ambiguous, undefinable, amorphous and undetectable by the brain - so you might not be able to understand what I have to say about your inexcusable boorishness. While I am genuinely grateful for the time you have spent answering my very basic and persistent questions, and while I understand your lack of patience, your disgraceful and thinly concealed tone of contempt is insupportable. I don't understand why you answer me if it should be in such a vile way. Look at how you have dealt with my questions. No doubt you will look through and think, "what's he on about?" "whats the big deal" "sissy!" "oversensitive" and so forth. No doubt you think you are being witty (the wit of a certain kind of scientific intellect is always and everywhere the same - where it ventures out of narrow bounds of culture-sex-violence it is only ever arrogantly sarcastic). This is because you have lobotomised the part of your affective apparatus that is even able to perceive such things as warmth, wit and fellow-feeling - using your large brain to try and undestand what I am talking about here, and what I am endeavouring to explore in my questions, is useless. This doesn't mean that discussion is necessarily impossible - for I would like to make use of that large brain (a metaphor of course) - but in this case, without the more basic humanity and subtle pre-brain intelligence that everyone has in common, alas, it is futile. You say you are "not going to say anymore" - this is my hope. Please do not respond to anything else I write. Goodbye.

 

[Gokul43201]

1. We use 10 base because we have ten fingers? Is that true? Is it not the case that 10 is used more for its convenience than its ubiquity?

What makes 10 more convenient than say, 7, 8, 9 or 11 (were it not for the fact that we have 10 fingers)? For convenience, I'd have picked 7 or 11 as a base for unfingered beings.

 

[Gale]

Dron, I feel like you've been given good explanations for everything you've asked. The problem is simply that you don't understand some fundamental concepts of math. If you're honestly interested in knowing more, find some articles, or read more books. Even if you don't understand all of it now, the more you read the more comfortable the maths will become.

I'd say you should look up counting bases (base 10 is arguably not the most convienient base to count in, however, yes, because we have 10 fingers it became convention). Computers for instance use binary, programers use hexadecimal. Laymen use 10 because that's what history dictates.

You should also look up some basic information on irrational numbers. The problem seems to be that you cannot comprehend certain aspects of them. If you do some research, at the very least you'll have more specific questions to be answered. You claim you stopped math around quadratic equations? Well its no wonder irrationals are uncomfortable for you, since that's when they first really pop up. I sincerely suggest you look up the way greeks did math, since they came across irrationals often in their style of math. Look up sqrt(2). In many ways its an easier irrational to comprehend than say, pi or e, since the maths are simpler.

As for natural forms and beauty, it sounds like you're leaning to some sort of platonic logic. Maybe look up some of his philosphy and again, have more specific questions to ask here, so that you get more specific answers.

Also, I'd like to point out that you seem to have this fixation about irrationals being "measured acuractely enough". You're hung up on the decimal representation of irrationals. You would have just as much difficulty cutting a string of exactly length "2" as a string of exactly length pi.

 

[CRGreathouse]

1. We use 10 base because we have ten fingers? Is that true? Is it not the case that 10 is used more for its convenience than its ubiquity?

The true reason is no doubt 'lost to the depths of time', but it seems very likely, especially considering the way that many civilizations (esp. indigenous South American cultures) name their numbers and their fingers with the same words. Other bases have been used, though; I know of at least one Amazonian tribe which used base five (presumably also related to the number of fingers on a hand).

Of course this is a linguistic question not a mathematical one, but I have a side interest in linguistics (and a friend with a degree in the field to recommend books to me).

2. In any case, I am still not clear if irrational numbers have some - any - element of cannot be thought-about-ness that rational numbers do not have.

Even real mathematians dispute how much "thought-about-ness" the real numbers (most of which are irrational) have. http://www.cs.auckland.ac.nz/CDMTCS/chaitin/olympia.pdf covers exactly that topic, though it's probably not accessible to you.

(And CRGreathouse says that some numbers are more beautiful - although he may have been pulling my leg)

Not pulling your leg.

6. CRGreathouse, thank you. Very interesting (although the gcdb thing lost me). Although you say that but is it not the case that logarythms and calculus were invented to measure curves and arcs and circular motions and stuff - and that these are based on irrational numbers? and that although they "cannot be measured precisely enough" it tends to be the case that certain numbers - e, i, pi, infinity, zero - are more appropriate to measure nature than others?

Logarithms don't have to use irrationals; the common log as in log_10 (1000) = 3 was the original sort of logarithm*, designed to simplify calculations by reducing multiplications to lookups and additions. It was later discovered that the use of e was most "natural" and that began to dominate other choices for many purposes.

I'm not sure that measuring nature takes anything more than rational numbers. Maybe we mean different things when we say this, though.

gcd(a, b) is the greatest common factor between a and b. gcd (4, 6) = 2 since 4 and 6 are both divisible by 2 (no remainder) and no larger number has this property.

* Actually Napier used a slight shifted variant on this, but that's just a quibble.

Also, please be kind to matt_grime. It is hard to convey tone in this written medium, and I would advise honest patience here. The repetition needed in a thread like this bothers him -- understandably, most math-minded people are bothered by redundancy -- but his answers are useful, if you spare the time to reconsider them. In his last post:

• He explained what makes a terminating decimal number for you. I tried to explain it differently myself, hoping to be more understandable -- but he tried before I did.
• He asked for your definition of "unthinkable". This probably seemed nickpicky to you, but that's because you're not a mathematician: we are truly lost without coherent definitions. He approached this differently than I did; I instead suggested a correspondence to a known definition, if you will. Either way, we both needed something much more precise that what you gave to answer the question.
• He told you that you were the standard for what makes a number beautiful. Again, this is definitional -- what makes beauty? I was sufficiently uncomfortable with the generality of the statement that I didn't even give any examples.

In any case, even if you don't appreciate their insights, please don't insult the members here. At the very least it's not conducive to discussion.

 

[CRGreathouse]

Also, I'd like to point out that you seem to have this fixation about irrationals being "measured acuractely enough". You're hung up on the decimal representation of irrationals. You would have just as much difficulty cutting a string of exactly length "2" as a string of exactly length pi.

Yes, thank you, I was trying to make that point.

 

[Diffy]

2. In any case, I am still not clear if irrational numbers have some - any - element of cannot be thought-about-ness that rational numbers do not have.

Cannot be thought about ness made me think that these irrationals are "that which should not be named"!

I can offer you this:

Imagine a tree, that has no branches and grows straight from the ground, directly up.

It starts as nothing and gorws to be 5 meter tall let's say.

As it grows from 0 to 5 meters, at some point that tree is 1, 2, 3,... pi meters tall, is it not? As it grows it will at some point be every rational and irratinoal number (between 0 and 5, and in meters) tall. At any point in time is the height of the tree not imaginable? Can you not take a picture of it at any point?

You can imagine the tree being 1 meters tall because you can hold a meter stick to it and see when hit matches up, yes? Well, if you take that meter stick, split it in half, pin the middle to the ground draw out a circle , and measure out enough string to rap around the circle once, then you have something to measure out when that tree reaches exactly pi meters tall.

You may say, well that is not exact, then I would ask you, how do you really know when that tree exactly matches up with the meter stick?

 

[111111]

As it grows from 0 to 5 meters, at some point that tree is 1, 2, 3,... pi meters tall, is it not? As it grows it will at some point be every rational and irratinoal number (between 0 and 5, and in meters) tall.

Well probably not every number, in fact not even close, since there are an infinite number of numbers between 0 and 5 so it would take an infinite amount of time to go through all of them, and since atoms have a certain size.

 

[111111]

I cannot show you the evidence you ask for about the golden mean, and note that I am not particularly championing it. Its just that I have heard that it seems to play a role in phyllotactic natural patterns, and is very common in animal symmetry.

If you go to the bottom of the wikipedia page on golden ratios, it has a bunch of stuff basically disproving most of the sightings of it.

From Wikipedia:
"Some specific proportions in the bodies of many animals (including humans[45][46]) and parts of the shells of mollusks[3] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in a specific individual and the proportion in question is often significantly different from the golden ratio."

 

[Gokul43201]

Well probably not every number, in fact not even close, since there are an infinite number of numbers between 0 and 5 so it would take an infinite amount of time to go through all of them, and since atoms have a certain size.

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

 

[matt grime]

You have asked two types of question, Dron. Those that are mathematical such as which numbers have terminating decimal expansions (CRGreathouse didn't quite get the right answer, though: 6 has gcd(10,6)=2, yet 1/6 is not terminating). These have been answered multiple times and it is up to you to go away and understand the answer. THis is not the 'snootiness of a certaintype of scientist', just a simple fact, and one not necessarily common to other sciences. If you want to understand maths then *do* the maths and understanding it is a lot easier. Unlike other subjects where you learn facts (effect of gravity is proportional to square of separation distance and mass) which you apply, in maths you can actually play with these things and see why 1/5 has a terminating decimal expansion, and 1/3 doesn't. And this maths is a lot more elementary than the quadratic equations you cite.

The second type of question is metamathmatical: what is a beautiful number. There is no definition of beauty that let's one say 3 is and 7 isn't. Any answer you get is entirely subjective and in some sense 'meaningless' - no one person's opinion here ought to count more than anyone else's.

Some metamathical questions do have mathematical content, and have been formalized into (different) mathematical systems*, but the subjective question of which one is 'correct' is still unanswerable and mostly a matter of personal taste.

* axioms of choice and constructability, the law of the excluded middle, constructivism, categories v. sets, constructible real numbers, etc.

 

[arildno]

I'd like to add that on a personal level, the idea of mathematical "beauty" is not insignificant, but may well be one of the reasons why a particular individual chooses to pursue a career in mathematics, rather than doing something else.

Getting a subjective reward, however, like aesthetic experiences, does not mean that these types of rewards have a necessary connection to the subject matter, or indeed to be regarded as part of mathematics as such.

 

[dron]

I have to leave on a long journey and cannot now keep up with this thread as I'd like. Nevertheless I will have to time digest it (printed out) and may be able to return in a few weeks. Thank you all so much. You have cleared up much more than it seems for me. I still feel somehow that many of you have not quite grasped the essence of what I was hoping to discuss, but that's probably because I've not been able to put my questions precisely enough. I'll ask two of them once more. If you think they've already been answered, please just swallow your aggravation, count to ten, and walk away.

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.

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. (Thanks for the golden mean wiki 111111 - happy to have a good debunking.)

Thank all again.

Dron

www.natureculturenothing.co.uk