To Infinity And Beyond

Mathematician Eugenia Cheng has written a popular, amusing, and fascinating book on the concept of infinity.

Beyond Infinity: An Expedition to the Outer Limits of Mathematics takes up its paradoxes, mathematical conundrums, and important uses.

For example, one mathematical axiom is that:

infinity X infinity = infinity.

But if you work out this equation by dividing both sides by infinity, you get:

infinity = 1

Since that can’t be, infinity must not be a number, exactly. But what is it?

Read an excerpt from her book, taken from Science Friday.

From Eugenia Cheng, Why Infinity Is No Ordinary Number – Science Friday [an excerpt from her book Beyond Infinity: An Expedition to the Outer Limits of Mathematics:

In the previous chapter I listed some beginning ideas about infinity.

Infinity goes on forever.

Does this mean infinity is a type of time, or space? A length?

Infinity is bigger than the biggest number.
Infinity is bigger than anything we can think of.

Now infinity seems to be a type of size. Or is it something more abstract: a number, which we can then use to measure time, space, length, size, and indeed anything we want? Our next thoughts seem to treat infinity as if it is in fact a number.

If you add one to infinity it’s still infinity.

This is saying

∞ + 1 = ∞

which might seem like a very basic principle about infinity. If infinity is the biggest thing there is, then adding one can’t make it any bigger. Or can it? What if we then subtract infinity from both sides? If we use some familiar rules of cancellation, this will just get rid of the infinity on each side, leaving

1 = 0

which is a disaster. Something has evidently gone wrong. The next thought makes more things go wrong:

If you add infinity to infinity it’s still infinity.

This seems to be saying

∞ + ∞ = ∞

that is,

2∞ = ∞

and now if we divide both sides by infinity this might look like we can just cancel out the infinity on each side, leaving

2 = 1

which is another disaster. Maybe you can now guess that something terrible will happen if we think too hard about the last idea:

If you multiply infinity by infinity it’s still infinity.

If we write this out we get

∞ x ∞ = ∞

and if we divide both sides by infinity, canceling out one infinity on each side, we get

∞ = 1

which is possibly the worst, most wrong outcome of them all. Infinity is supposed to be the biggest thing there is; it is definitely not supposed to be equal to something as small as 1.

What has gone wrong? The problem is that we have manipulated equations as if infinity were an ordinary number, without knowing if it is or not.