If Infinity is larger than the largest number imaginable, then how do you square infinity? Because if Infinity is the largest number then even...

Hello!

Infinity is not an ordinary number, and its square isn't a number also. Actually, the square of infinity is also infinity.

Consider a model where ordinary numbers and infinity are represented as a limits of sequences.

If a sequence `{a_n}` has a limit `a` (a number), i.e.

`AA` e>0 `EE` N(e) | `AA` n>N(e) `|a_n-a|lte,`

then it is considered as a representative of a number `a.`

If a sequence `{a_n}` has an infinite limit,...

Hello!

Infinity is not an ordinary number, and its square isn't a number also. Actually, the square of infinity is also infinity.

Consider a model where ordinary numbers and infinity are represented as a limits of sequences.

If a sequence `{a_n}` has a limit `a` (a number), i.e.

`AA` e>0 `EE` N(e) | `AA` n>N(e) `|a_n-a|lte,`

then it is considered as a representative of a number `a.`

If a sequence `{a_n}` has an infinite limit, i.e.

`AA` E>0 `EE` N(E) | `AA` n>N(E) `|a_n|gtE,`

then it is considered as a representative of the infinity.

In this model we can add and multiply numbers AND infinity (with some restrictions). In particular, infinity squared is also infinity.

There is another model, where infinity is a cardinality of an infinite set. There are many different infinities in this model, some of them are greater than another:)
But "infinity squared" (the cardinality of the Cartesian product of the corresponding infinite set) is the same infinity.