What Is Infinity?

Step 1 of 5

Name the biggest number you can.

A million. A billion. A billion billion. Whatever you say, adding one gives something bigger — so your number was not the biggest.

Step 2 of 5

So the counting numbers never end.

There is no last one. Not because we have not found it, but because the rule for making a new one always works. That endlessness is what infinity names.

Step 3 of 5

Infinity is not a number you can do sums with.

You cannot write infinity minus one and get something sensible. It describes a property of a collection — how it behaves, not how big it is.

Step 4 of 5

Endless things can fit in small spaces.

Between 0 and 1 there are infinitely many fractions: a half, a third, a quarter, and on forever. Infinity does not need infinite room.

Step 5 of 5

And some infinities are bigger than others.

Cantor proved in 1874 that the endless list of fractions is smaller than the endless list of all decimals. Mathematicians thought he had lost his mind.

The ∞ symbol was invented by John Wallis in 1655. Nobody is sure why he chose that shape.