http://www.maths.ex.ac.uk/~mwatkins/

http://www.maths.ex.ac.uk/~mwatkins/zeta/ss-a.htm

... so what?

If we create a game that says that by these rules of division only certain numbers can follow, then they will follow...

... and? Are they indicative of how works the universe, by design or by default, or are they indicative of how works the numbers game?

Very fine reference, indeed!

A puzzle: Why is it that any number multiplied by its inverse fraction always equals to one? i.e., 1/n * n = 1

Any number will work, prime or not. Would zero * infinity = 1 too?



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I have formally 'resigned' (tactical withdraw) from the Space-Talk boards; mine were many questions, ideas, but no real answers. Thanks. 04/10/04.

Disclaimer: Please note the ideas expressed here by me are cutting edge theory, very speculative in nature, and not physics as it is being currently taught. Caveat lector.

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A puzzle: Why is it that any number multiplied by its inverse fraction always equals to one? i.e., 1/n * n = 1

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Shambolic, since I happen to be around (having survived the worse case of flue ever) I thought it fun to check out tmorten's page.

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Ummm, how about...because that's the definition of an inverse?

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In effect, does zero really exist or not?

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Shambolic, in ur:

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Are you ever going to admit that you know less about something than those who study it?

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Sorry, that last post was very hurried, and did a bad job of making any point at all (except the additive identity part).

You're right that definitions are very important in maths. But that doesn't mean you can define whatever you like. The limit as n -> infinity of n * 1/n = 1, but that doesn't mean you can say 0 * infinity = 1, because that leads to contradictions, as I pointed out earlier.

As for the primes...what I was trying to say earlier is that the properties of the natural numbers don't arise from definitions; they really are fundamental. If you take 12 marbles, you can put 4 into each of 3 bags, or 6 into each of 2 bags. Call these experimental facts, if you like.
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Shambolic - keeping it complex analytic

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You're right that definitions are very important in maths. But that doesn't mean you can define whatever you like. The limit as n -> infinity of n * 1/n = 1, but that doesn't mean you can say 0 * infinity = 1, because that leads to contradictions, as I pointed out earlier.

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Yes, of course, that's a given, that we cannot arbitrarily give definitions to math and expect they will work. As you point out, there has to be experimental evidence to support the structure of how the math is defined, or it's self contradicting nonsense.

The reason I keep bringing up uncomfortable questions is not because I am ornery, but because I have a genuine interest in knowing why on a grander scale of things, in academia and the real world, some things are allowed, while others are not. Primarily, why is it that we are told, universally by absolutely everyone, that we cannot multiply or divide by zero? This is not a fickle question, because the definition of zero itself is not ironclad. What is nothing?

You are aware that in calculus, progressively smaller deltas are theorized down to near zero to calculate, at a zero point, the slope of a curve. Now, is this zero really a point at nothing? Or is it more like the zero I illustrated in 1/n * n = 1, where n => infinity, so that zero * infinity = 1? Do you see why I question this definition of zero? On one hand, the one where rightly we cannot use zero to multiply or divide, zero is nothing. But on the other hand, where we can use zero as a tendency towards nothingness, such as a zero point on a curve, we can use this "zero" though it does not represent nothingness, and that number can be multiplied or divided, because by definition that "zero" has a value to it, the slope of the curve.

This is why I showed these definitions in succession: 1. The inverse function, and 2. inverse taken to some infinite end, where it approximates zero and infinity, 3. zero times infinity is equal to one.

These do not constitute a proof, nor do they dispel the notion that we cannot multiply or divide by zero, or infinity, but it shows that in theory, it can be done. Why is this important? Because in the future, we may need to do the math by multiplying zero and infinity, though we have no way to do so now, except to "normalize" it by removing these math nuissance obstructions.
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As an aside, the Pythagoreans were so violent in their arguments over math that they would even kill each other over obscure points!.. obviously a bit excessive.

__________________
I have formally 'resigned' (tactical withdraw) from the Space-Talk boards; mine were many questions, ideas, but no real answers. Thanks. 04/10/04.

Disclaimer: Please note the ideas expressed here by me are cutting edge theory, very speculative in nature, and not physics as it is being currently taught. Caveat lector.

This discussion shouldn't lead to homicide

Trivial point: we can multiply by zero.

As I said before, the best way to 'define' zero is probably by 0 + x = x + 0 = x for all x. I know that's a very abstract way to do it, but that's how maths is.

The theory of limits is fairly technical. But never do we divide by zero. sin(x)/x is undefined at x=0, but the limit as x -> 0 of sin(x)/x = 1.

One of the problems with what you were saying (1/n * n as n -> infinity) is that your logic is...
1. n * 1/n = 1
2. Now let n -> infinity
So far so good. The next step is the mistake. You say "therefore infinity * 0 = 1" but in doing so, you implicitly assumed that 1/infinity = 0. You can't assume what you're trying to prove!!
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Shambolic - keeping it complex analytic

Coppernicus2, I don't think that 1/n * n = 1 is anything particularly interesting. If you perform an action on something, and then perform the opposite action, it's intuitive that you should end up with what you started with. If you divide 1 by a number, and then do the opposite (multiple that result by the same number) you'll end up with what you started with, 1. However, as Shambolic pointed you, you do need to put restrictions on n. Algebraicly, you should just see at a glance that the n's cancel and you just equal with 1=1. The only exceptions being when n=0 or infinity.

Mathematical proofs are an entirely different beast that you might want to stay away from. Any number divided by 0 is undefined in the real domain (we can annotate these numbers in the complex domain, i.e. imaginary numbers). Unlike physical sciences, laws of math are immutable. Math is not negotiable.