My nine most vital maths questions

[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

 

[magic_castle32]

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

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

Here are some links which might help:

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

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

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

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

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

 

[matt grime]

pi as a decimal cannot be imagined,

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

There is a number

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

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

 

[CRGreathouse]

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

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

 

[Diffy]

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

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

 

[matt grime]

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

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

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

 

[CRGreathouse]

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

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

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

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

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

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

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

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

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

 

[arildno]

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

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

 

[Hurkyl]

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

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

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

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

 

[111111]

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

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

 

[CRGreathouse]

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

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

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