## Infinity, part I

In which, not wanting one of his sleepless nights to go to waste, a humble engineering student tries to explain infinity to “normal” people. This is part I of a three-part series, if all goes according to plan.

What is infinity?

Is it a number? Let’s suppose it is, and try to do some arithmetic (basic operations):
infinity+1=…infinity?
infinity+any number = infinity
infinity+infinity = infinity
infinity-infinity=???
Infinity doesn’t behave like a number in the traditional sense – you need crazy stuff like hyperreal numbers to make that work.
Therefore, it isn’t a number, but rather a notion, a property of something.

How do we define infinity?

I’m pretty sure you know there are infinitely many whole numbers; they just go on forever. And I’m equally certain someone had to explain that to you; surely, a child might say: what if there is a largest number? At that, the explainer would point out that if you add one to that supposedly biggest number, you get an even larger one; add one again, and it’s larger still, and so on, “forever”.
But what do mathematicians call infinity?

And that is more or less one of the two main ways in which mathematicians define infinity (more on the second in part II),  believe it or not. Infinity is what can get larger than any number; you could think of a ludicrously large figure, and infinity would still be bigger. More precisely, a sequence of numbers (for example 1,2,4,8,16…) is said to “tend to infinity” if for any number A, you can find a point in the sequence where all the numbers past that point are larger than A. Notice how nothing is said to be “equal” to infinity – it’s not a number, remember.

Infinity is what can be as big as you want; that’s how the ancient Greek philosophers, and the scholars of the medieval world saw it, and it’s the definition most commonly used in calculus even today. But infinity is – fittingly – so much more than that, as we’ll see in the next two parts.