What is infinity multiplied by infinity?

Answer 1

One normally says that infinity times infinity is infinity.

Most mathematicians would accept this as valid if presented in a proper mathematical context.

Context:

The mathematical concept of infinity depends upon the context.

As far as lower level mathematics that most people are familiar with are concerned, infinity is not a number and thus can not participate in arithmetic in the usual fashion.

In Calculus, you can work with infinity, but only through the language of limits. In this case, if you multiply two functions whose limit approaches infinity, the result's limit will also approach infinity. However, this tells us nothing about how "big" this infinity is.

In an example from higher level mathematics, you can use the Aleph

numbers, which is the set of all cardinal numbers of the infinite sets of finite numbers. Aleph

numbers are transfinite, and are for all intents and purposes considered infinite.

Mathematical operations with infinity, or transfinite numbers, produce a peculiar arithmetic. If "a" is "infinite" then the following

relationships are taken to hold.

a + a =

a
a * a =

a
a^a

=

a