# If Infinity is larger than the largest number imaginable, then how do you square infinity? Because if Infinity is the largest number then even infinity squared could still be smaller.

Borys Shumyatskiy | Certified Educator

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.

cmsmichael | Student

Infinity is a term used to describe anything that is endless. That can apply to a scientific principle, mathematics, spiritual belief, or a philosphical idea. Something that is endless has no beginning or end and cannot be measured or managed using traditional knowledge. Therefore, attempting to apply known principles or formulas to the concept of endlessness, or infinity in this case, is not possible.

I thought of the line from the song in "The Sound of Music" that goes "How do you hold a moonbeam in your hand?" when I read your question. There are some things that we intuitively know are real but can never fully explain or manage with our limited human resources. That's what keeps life interesting and drives us toward greater understanding of our world and the universe that surrounds us.