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How big is infinity? – Isabelle Greco

How big is infinity? Well… it’s infinitely weird.

I remember as a kid playing games where you have to try and shout out the biggest number you could. I would always be really proud when I remembered a number like googol and even prouder when I remembered an even bigger number like googolplex. But what really annoyed me was when someone shouted out ‘infinity’, because how was I meant to top that!

So, apart from revealing that I was a very competitive child, the point of that story is to introduce us to this idea of infinity, which we tend to use all the time but not really think about very much. Is it a number? The biggest number? How big? Can you get bigger (and win the game!)?

Let’s try and answer that first question – is infinity a number? We’re going to use a mathematical technique called ‘proof by contradiction’. Proof by contradiction works like this:

  1. Assume that something is true.
  2. Do some maths based on this assumption.
  3. Get to something that is very obviously not true (like 0 = 1)
  4. Because of the previous step, we conclude that what we assumed back in step 1 must, in fact, be false.

So let’s try this out! To avoid writing ‘infinity’ infinitely many times, I’m going to sometimes use the symbol ∞.

  1. Let’s assume that infinity is, in fact, a number.
  2. So, if infinity is a number we can add other numbers to it. For example, what is ∞ + 1? Let’s try visualise this. If we have an infinitely large number of cakes, for example, and add an extra cake to our pile of infinite cakes how many cakes do we have? Still infinity! So we have ∞ + 1 = ∞. Now, since infinity is just a number, let’s take infinity away from both sides. That leaves us with 1 = 0!
  3. Uh oh! The equation 1 = 0 cannot be true! In fact, the assumption that one and zero are different numbers actually underlies our whole number system so we know that this one is wrong.
  4. Our conclusion is therefore that what we assumed in step one is wrong, so infinity is not a number!

Awesome! So infinity isn’t a number. What is it then? One way of thinking about infinity is as a destination that numbers can approach but never reach. For example, a number can get bigger and bigger and bigger and go towards infinity but it will never actually reach infinity.

We’ve answered one of our questions, but we still haven’t thought about whether we can get ‘bigger infinities’. For example, let’s think about all the counting numbers. We can even list them:

1, 2, 3, 4, 5, 6, 7, 8, …

So there must be infinitely many counting numbers, because we could keep writing that list and it would never ever end. As soon as we had a ‘biggest’ number, we could just add one and get a new bigger number for ever and ever (that was actually a mini proof by contradiction right there – we’re getting good at these!). What about then all the number between zero and one? We could start looking at decimals like:

0.1000, 0.1001, 0.1002, 0.1003,…

but what’s stopping us from using more decimal places:

0.100000, 0.100001, 0.100002, 0.100003,…

or even more decimal places:

0.10000000, 0.10000001, 0.10000002, 0.10000003,…!

Since we could use as many decimal places as we liked, we have no choice to conclude that there are in fact infinitely many numbers between zero and one. But is this the same infinity as the number of counting numbers. Actually, it’s not – it’s bigger. This one is a little more complicated to explain, but this video does a great job. 

One last dose of weirdness for you: so there are infinitely many numbers between zero and one. How many numbers are between zero and two? Our intuition says twice as many, but it’s actually not – it’s the exact same size of infinity! For us Fault in Our Stars tragics out there, that does mean that although some infinities are bigger than other infinities, but these two infinities are not a good example of that fact. (As a side note, for some mathematically correct logic by John Green, I do however recommend An Abundance of Katherines.)

So, next time someone tries to tell you that they are ‘infinitely better than you’ join me in feeling just a little bit smug as you remember that infinity is not actually a number, and a little bit awestruck when you remember that infinity is far, far weirder than it seems!

About Isabelle: Isabelle is lucky enough to be studying maths and French; two subjects that she absolutely loves. When she’s not thinking about French maths, you can find her rereading Harry Potter, fighting for equality, or soaking up some sunshine.