How big is infinity?
Recently Avengers Infinity War hit cinemas, breaking all the records for a Superhero movie and taking over £29.4m in its opening weekend.. So, of course, Georgia Mills has taken the opportunity to brush up on some maths.
Georgia - Have you ever asked someone under the age of ten what they think the biggest number in the world is? The results are invariably amazing…
One thousand and one million hundred and three
One hundred and twenty nine billion and eighty three.
That’s quite a big number isn’t it?
Georgia - Thank you to Emma, Claire, Sarah and Jim for sending those in. And, to be fair to those kids, it’s kind of a trick question.
Eugenia - There is no biggest number in the world because if there were we could also add one and it would be bigger. So you might think that the biggest number in the world is ‘infinity.’
Georgia - This is Eugenia Cheng. She’s scientist in residence at the School of the Art Institute in Chicago and also the Author of Beyond Infinity.
Eugenia - Infinity is very large, but actually you don’t have to be that dumb not to understand it because it’s very difficult to understand. And although it’s something that you can think about, even when you’re a small child and you are having those arguments about who’s more right - “I’m infinity times right”! Actually, it took mathematicians thousands of years to pin down what infinity is in a way that actually stands up to logic.
One way you can describe infinity is you can simply say it is the quantity of whole numbers that exist. You might think that’s cheating because you haven’t said what it is, but mathematicians kind of like cheating. And then, there’s this funny thing you can do where you can actually make a bigger infinity. So, if someone says to you “oh, I’m right times infinity”, then you can beat them. Not that everything is about beating people. You can say “I’m right times two to the power of infinity”, and that’s actually bigger.
Georgia - But isn’t infinity already the biggest number? How can there be a bigger one?
Eugenia - That’s a good question. The thing is that if you use the fact that infinity is the absolutely biggest thing and then you work with it logically in mathematics, you very quickly find you can prove that everything equals zero, which is a bit of a problem. Because if everything equals zero, then there isn’t really anything in the world, and that’s okay in mathematics, you’re in the zero world, but it’s not a very interesting world to be inside. But, of course, there are other worlds in which everything doesn’t equal zero and that’s why we actually need higher and higher orders of infinity to avoid everything collapsing to zero.
Georgia - Infinity simply refuses to behave itself. Everything goes a little bit weird when you think about it. And a famous example of this involves an infinite hotel…
Eugenia - Oh yes. The Hilbert Hotel is one of the famous thought experiments involving infinity. Hilbert’s Hotel is an infinite hotel and the idea is that there is a room for every number 1, 2,3,4, 5, 6, and so on. Suppose your hotel is full and you’re the manager of the hotel and a new guest arrives, you could either turn the guest away - say “bye, no room at the hotel”. Or you could go ahh, but I could make some more money if I move everyone up one room. So the person in room 1 goes to room 2; the person in room 2 goes to room 3; the person in room 3 goes to room 4, and so on and there’s an infinite number of rooms, so you never run out of rooms. But now room one is empty so the new guest can move into room 1 so, miraculously, the hotel was full and now there’s a spare room and this is something you can’t do in a normal hotel.
In a finite hotel, if it’s full, it’s just full, but infinity is weird like that. It’s sometime called a paradox but there's nothing wrong with it, so it’s the type of paradox which doesn’t involve an actual fallacy, it just contradicts our intuition about the world, and that’s not wrong, it’s just that we’ve discovered something. So sometimes, when we're thinking about infinity, the idea isn’t exactly to understand infinity better, but it’s to shed light on our actual world. We also learn a lot about numbers that do come up in life by imagining what would happen if infinite things were true.
Georgia - This is quite mind blowing the whole idea, and I suppose our experience of the world is finite. I only have ten cakes or whatever, I’m never going to have infinity….
Eugenia - Poor you.
Georgia - Yes, it’s a real shame. I’ll never have infinite cakes. So I suppose it’s not surprising that well, to me at least, the whole idea is a bit like ‘pow’.
Eugenia - I think that’s right. But the funny thing is that although we don’t have infinite cakes, we do sort of have infinite pieces of cake. Because, if you cut your piece of cake in half, and then you take half of the rest, and half of the rest, and half of the rest, and so on. That’s what small children do to make their cake last forever or, at least, that’s what I did.
Georgia - So infinity is useful for me to decide that my cake is going to last forever, but is a useful concept in mathematics? Does it have any applications?
Eugenia - It’s the study of infinity that lead to the whole field of calculus, and calculus is all about things that continuously change. And it leads to things like differential equations, which are probably the most applied, the most applicable and used parts of mathematics and everything that’s around us. If you point at any object around you, there’s going to be tons of calculus that went into the making of it: from the electricity that was used to make it to the tools that were used to make it to the design, to the way it was shipped around the world - it’s everywhere around us.
Georgia - So infinity is all around us, but it still makes my head spin, and yearn for the simple days, when the highest number conceivable was “seven”.