Basically, there's three types of answers you could give...

a) "Infinity is not defined"

There are in fact lots of different concepts of infinity, and some things are more infinite than others! In any case, any infinity is a different kind of thing to a normal number.

b) "Square root is not defined for that kind of entity"

There are some mathematical "systems" in which infinities are defined. But the square root function is not defined in them.

With either of the (a) or (b) answers, asking the square root of infinity is like asking the square root of cheese. It doesn't mean anything. In a computer program it would be a type error.

c) "The most sensible answer is that the square root of infinity is infinity"

For a start it's clearly not any normal number x. x*x would always be finite.

Then depending on how you define infinity, there are a couple of arguments that would lead you to conclude sqrt(infinity) = infinity.

One argument is: As x heads towards infinity, what does f(x) head towards?
If f(x) = 1/x, f(x) head towards 0, without ever gettign there. That would suggest if it is meaningful at all f(infinity) = 0. Likewise sqrt(x) could be said to head towards infinity.

Another argument, which I won't spell out, uses something often dubbed "diagonalisation". It's a way of showing that there are exactly as many fractions as whole numbers. The same kind of method could be used to show that if you had a "square" whose sides were infinite, the number of "points" in the square was the same as the number of "points" in the side of the square.

As I mentioned there is more than one kind of infinity. These arguments don't even make sense for all of them!

I'm pretty sure all this is far more than you ever wanted to know, and your question was probably just a bit of fun!
Source(s):
College math


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