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One, Two, ... Infinity, ... Infinities?

Date: 11/04/2010 at 20:33:56
From: David
Subject: Infinity/ies

A recent thread on a social forum reminded me of an incident that occurred
during the later years of my teaching career. While teaching primary
level, I overheard the following conversation between two ten year-olds:

   Q: What is the largest number?
   A: Infinity.
   
   Q: How many integers are divisible by 2?
   A: Mmmmm... infinity.
   
   Q: Divisible by 3?
   A: Infinity.
   
   Q: By any given integer?
   A: Infinity.
   
   Q: Does that mean that there is an infinite number of infinities?
   A: O my god!

They looked at me and I shrugged. Tapping into my ability to enthuse
children of this age, I posed the question, "What do you think are the
factors of infinity if infinity is an integer?"

In the end we decided that we did not know enough mathematics to be able
to answer our questions.

I've often puzzled over these ideas; but not being a mathematician, my
concept of infinity is too simple.



Date: 11/04/2010 at 23:21:51
From: Doctor Vogler
Subject: Re: Infinity/ies

Hi David,

Thanks for writing to Dr. Math.

You are right to question your concept of infinity as too simple: infinity
is not an integer. It's not even an number. So it makes no sense to ask
whether it is "divisible by two," or for that matter divisible by any
other number.

(The integer zero, on the other hand, is divisible by every integer, even
though there aren't infinitely many zeros. There's just one.)

Infinity is a fascinating concept. I can think of at least three different
ways in which infinity is used; some of them are related, but they are all
quite different from one another.

One of them is limits, which belong to the domain of calculus. You can
take limits as some variable grows to infinity (or negative infinity); and
the value of a limit (even if the variable goes to a finite number) can be
infinity or negative infinity. In each case, it refers to the variable or
function getting arbitrarily large (or large negative). For example, as x
grows to infinity, the limit of 1/x is zero, as is the limit of 2^(-x); of
arctan(x) is pi/2; and of log(x) is infinity.

Another context is sets, where the size, or "cardinality," of a set might
be infinity. For finite sets, you can only make a one-to-one pairing of two
sets (matching up each element of one set with its own element of the
other, and missing none) if both sets have the same number of elements.
Applying the same idea to infinite sets, as one of your former students
may have done, leads to the surprising conclusion that the set of even
integers has the same number of elements as the set of all integers. (Just
pair up x in the set of all integers with 2x is the set of even integers.)
It takes a little more work to show that the set of rational numbers has
the same size as the set of integers. You might then conclude that all
infinite sets have the same size. But it can be proved that there is no
way to pair up the set of integers with the set of all real numbers. So
the size of the set of real numbers is a *bigger* infinity than the size
of the set of integers. It's a fascinating subject. See also

    http://mathforum.org/library/drmath/sets/high_sets.html 

The third is poles of functions, which belongs to complex analysis. A
fourth is projective space. But you're less likely to run into that,
unless you study algebraic geometry. I won't go into any more detail about
that.

If you have any questions about this or need more help, please write back
and show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Large Numbers
Elementary Number Sense/About Numbers

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