Hatrack River Forum: Infinity Debates - Papa Moose? Hobbes? Math geeks?

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Posted by imogen (Member # 5485) on :

In his very sweet thread on Hobbes' and Annie's 1 year anniversary, Hobbes said

quote:
Contrary to popular opinion, infinity is not an actual number, it is the concept of a number. A true number can never be infinite, because a true number has a defined value, and as soon at is defined it is finite, it does not extend past all other possible numbers. When dealing with infinity the word is often used to mean the concept of forever, but in math, the best way of representing it in an equation, to obtain a meaningful result one must use limits.

lim x-> infinity, and then use x where you want infinity to be. You treat x as a number that approaches infinity; due to not being a number itself, infinity can not actually be reached, but “approaching” means “if it ever did reach that illustrious number, this would be it”. A very handy concept, one used frequently in math.

a definition which Pop described as "limited".

So what is a proper (unlimited [Smile]

) definition of infinity?

As a number I guess you can simply define it as 1/0 (or x/0 for that matter).

Posted by Boris (Member # 6935) on :

I thought that was an imaginary number...Or am I thinking of something else? (I'm an English/Computer geek, not a math geek)

Posted by imogen (Member # 5485) on :

Naw, imaginary numbers are the square roots of negatives and are represented by i.

Posted by MEC (Member # 2968) on :

imaginary number = i = (-1)^1/2

you cannot put 0 in the denominator. so it would be the limit of 1/x as x approches 0.

Posted by Hobbes (Member # 433) on :

Well actually, techinically, infinity can be treated as a number, it's just very difficult to deal with in an equation. An example:

(x^2+2x)/(4x^2-54x+2)

Where x is infinity. What does that equal? The only way to tell is to put a limit on it, as x approaches infinity. See, x^2 is a bigger inifinity than x, so just using the "number" infinity is insufficent, how big is the infinity? In this case, the answer is actually 1/4 if you're curious, but I'm not going to venture to far into limit theory.

Anyways, yes, Papa is right, and I'm sure when he sees this he'll give you a much more detailed explenation, since he's like 90 bajillion times smarter than I am (why do you think we call him Papa? [Wink]

) but for now, that's my input.

Hobbes [Smile]

Posted by Hobbes (Member # 433) on :

Ohh, and something to keep in mind, limits aren't the actual number. So this:

lim y-> infinity lim x->0 x^(y) is very different than 0^inifinty. The latter would be 0, since zero to any number is always zero, but the former isn't 0, since x is only approaching zero, it isn't actually zero.

Hobbes [Smile]

Posted by TomDavidson (Member # 124) on :

The best example I ever came up with to explain this to someone:

The number of natural numbers is infinite.
The number of whole numbers is one larger than the number of natural numbers.
The number of integers is two times the number of natural numbers, plus one.
The number of real numbers, as there are an infinite number of fractions between each integer, is the square of the number of integers.

So "infinity," while useful as a concept, cannot be said to be a value.

Posted by Hobbes (Member # 433) on :

As a side note, all the number sets Tom listed are the same size of infinitey, but all real numbers is a larger infinitey than any of those.

Hobbes [Smile]

Posted by Mike (Member # 55) on :

Tom:

quote:
The number of real numbers, as there are an infinite number of fractions between each integer, is the square of the number of integers.

You mean the rational numbers, yes? those that can be written as p/q, with p and q both integers? There are more real numbers than rational numbers, in the sense that you can't put the reals and rationals in a one-to-one correspondance. Which, incidentally, you can with the rationals and the naturals. One such mapping is:

1 -> 0
2 -> 1
3 -> -1
4 -> 2
5 -> -2
6 -> 1/2
7 -> -1/2
8 -> 3
9 -> -3
10 -> 1/3
11 -> -1/3
12 -> 4
13 -> -4
14 -> 3/2
15 -> -3/2
16 -> 2/3
17 -> -2/3
18 -> 1/4
19 -> -1/4
...

A proof of why this works is left as an exercise for the reader. [Big Grin]

A set is called countable if such a one-to-one correspondance exists. The rationals are countable, as are the gaussian integers. Reals are not, nor are p-adic numbers.

http://en.wikipedia.org/wiki/Countable

As for whether infinity is a number, that is really a matter of definition. As are many things in math. But keep in mind that there are many kinds of infinities, and the kind you mean when talking about limits is only one example.

Posted by saxon75 (Member # 4589) on :

Yes, there are different types of infinities. Integers, natural numbers, and whole numbers are countable infinities. Real numbers and (I think) rational numbers are uncountable infinities.

I think I remember hearing or reading somewhere that there are specific orders of infinity, but I never learned what they are.

[Edit: You all should pay attention to that smart Mike up there. This one down here doesn't know what he's talking about.]

[ October 22, 2004, 11:20 AM: Message edited by: saxon75 ]

Posted by Mike (Member # 55) on :

Thanks, saxon. [Blushing]

If I had gone to the trouble of actually reading the link I posted, I wouldn't have bothered to write out that list. But the link might be a little too dense for some.