# Approaching Infinity

Last summer, I worked at and gave some lectures at the National Maths Summer School. The students submitted feedback forms, and a surprisingly large number mentioned that they would have liked to have a session about ‘infinity’. I was reminded of this by a post on an interesting blog that I’d seen linked to by, of all people, Stephen Fry. It is easy to forget, a full three years after a first university course on analysis, that the infinity had once seemed so confusing.

The problem is as much one of presentation as of mathematical content. The impression often given is that mathematical statements concerning infinity are not properly defined, or can’t be understood in a ‘real world’ setting. Unqualified and often rather misleading explanations are absolutely rife. And even some well-qualified scientists have put forward theories that are questionable at best. First we talk about some of the usual problems, and why they might not be so significant after all.

• No-one can imagine what infinity is: I’m not sure whether this is true – I personally feel I have a reasonable idea. But even this doesn’t matter. Arguments like this often reference the fact that there are $10^{80}$ atoms in the universe (or something similar) and how this doesn’t even compare to infinity. This is true, but it doesn’t affect our ability to understand and make deductions about a concept. I can’t imagine what 5-dimensional space looks like, but with five co-ordinates (x,y,z,w,v) I can describe it in mathematical terms that are entirely reasonable. This allows me to start working out properties of the object even if I can’t visualise it.
• Infinity is about philosophy: This might well stem from its appearance in popular culture (‘to infinity and beyond’) and the metaphysical (‘the Father of an infinite majesty’ etc). I would suggest that if you are worried about coming to a philosophical understanding of infinity, first you should question whether you have a true philosophical understanding of seven. I can picture seven oranges in my mind, but does that alone really explain all the seven-ness of seven? In any case, we can learn some simple rules to deal with seven (like 3+4=7) in a concrete way, and though the rules aren’t as ‘obvious’, we can do the same for infinity.
• Infinity is not a number: Again, this is in some sense true (see below). But it doesn’t make any difference if you use it correctly. At various points in time 0 has been considered ‘not a number’, as have negative numbers. If you build up the world of complex numbers by defining the square root of -1, is this a number? As with many words, infinity means different things in different contexts. This is actually often really about the following:
• Infinity messes up arithmetic: Because for example $\infty-\infty$ is not necessarily = 0. Or that $\frac{1}{0}=\infty$ (at least informally) but $0\times\infty$ is not 1. By the time we are at secondary school we feel pretty comfortable about arithmetic, and don’t want any surprises cropping up. Well, maybe we will just have to relax the strength of our feelings about how it works when we look at infinity. Just because subtraction fails to behave normally doesn’t mean we can’t do cool things with infinity. We just have to make sure we don’t subtract at any point!

It is surely the case, however, that infinity is hard to teach as part of a school syllabus. Topics which work well as part of a curriculum typically offer a huge number of theoretical examples, plenty of applied or practical examples and, most importantly, don’t rely on technical details for understanding. Take quadratic equations, a focal point of middle school maths. You can come up with examples with integers solutions (like $x^2+6=5x$) in a matter of seconds, and there is an endless range of ‘Alice runs 10m/s faster than Bob but starts 10 minutes later and has to cover twice the distance’ type problems with which to apply the skills. Finally, there is the quadratic formula, which though many could, the students do not need to learn how to derive.

With infinity, we don’t have this luxury. We can come up with lots of infinite sets, or sequences that tend to infinity, but relatively few of them will be easy to study without technical analysis, and hardly any will have ‘real world’ applications. Unfortunately, by its very nature, infinity sits more comfortably in the definition-theorem-proof regime of undergraduate mathematics. However, I hope some of the following exploration should be accessible and perhaps even interesting!

A collection of numbers, or mathematical objects has a size. This might be finite: the set {1,2,3,4,5} has size = 5 for example. But it might also be infinite: the collection of all positive integers {1,2,3,…} is the best example. But there are many many other examples: the set of all even positive integers, the set of all integers, the set of all rational numbers, the set of all real numbers (informally, any number you can think of). These sets all have infinite size, but do they have the same size?

This is definitely a question worth thinking about, but first we need to decide what we mean by ‘same size’. It is clear what I mean if I say that the set {1,2,3,4,5} has the same size as {3,4,5,6,7}, but with infinite sets it is less obvious. For example, we might suspect that the set of even integers is smaller than the set of integers, as the latter contains the even numbers also includes the set of odd numbers. But I can get from one set to the other by multiplying or dividing all the elements by two. So perhaps we should consider them to be the same size?

In fact, this turns out to be the most convenient conclusion. The moral from the discussion about infinity and arithmetic should be that simplicity is a good thing. We would prefer not to have to worry about exactly what it means for a set to have size $\infty-1$ or $\frac{\infty}{2}$ or something similar. It would be best if two infinite sets have the same size unless they are catastrophically different.

The simplest way to phrase this is to say that two sets A and B have the same size if we can pair up elements of A with elements of B. That is, elements of A get paired with precisely one element of B and vice versa. Note that this is consistent with the definition of finite sets. My own inital intuition when I first learn this was that no infinite sets would pair up nicely, then shortly afterwards, that all sets should pair up. In fact predictably the situation is somewhere in between!

If we can pair up a set with the positive integers, we say that the set is countably infinite or countable. It’s then time to start checking what sets of numbers have this property. Because you can pair n with 2n for every positive integer n, we see that the even positive integers are countable. By a slightly more complicated pairing, you can show that the integers (not just the positive integers) are also countable. The set of rational numbers and the set of real numbers are worth considering yourself. The question of whether these are countable deserves a post of its own, but the answer contains some beautiful mathematics.

Having thought of infinity as the size of a large set, it’s now time for a completely different approach. Casually, we can think of $\infty$ as ‘the point at the end of the real line’. This is best thought about using sequences $a_1,a_2,a_3,\ldots$. If you take a sequence like 1, 1/2, 1/3, 1/4,… you see that the terms approach 0. Formally, say that the sequence $a_n\rightarrow a$ or tends to a if whenever we put a small interval around a, eventually the sequence ends up in the interval and stays there forever. Note that this has to happen however small the interval is. So in the reciprocals example, a small interval around 0 might be [-0.01,0.01] or [-0.000001,0.00001] or even smaller, but however you choose the interval, the sequence will eventually be confined to the interval. You might think of it by a cage analogy: however small a cage you put around a, eventually the sequence will be trapped in the cage.

And we can tend tending to infinity in a similar way. We say that a sequence tends to infinity if whenever you specify a finite number N, however large it is, eventually the sequence is greater than N, and remains so forever. By a similar analogy, you could think of this as putting up a barrier on some number N. Then the sequence will eventually be trapped on the ‘far side’ of the barrier. This definition applies equally well to $-\infty$, where instead you need to be confined on the lower side of the barrier.

This may look quite abstract, but in fact you’ve probably already been using it. Think about infinite sums. You get taught how to do finite sums: you add them up one at a time and eventually you are done. But with infinite sums, you never finish. Instead, what you are doing is computing large finite sums, and seeing whether they tend to a limit as described above.

The best example is 1/2+1/4+1/8+…=1. Here’s a nice reason: set 1/2+1/4+1/8+…=S, then multiply both sides by 1/2 to get: 1/4+1/8+1/16+…=S/2. But then we can substitute this back into the original result to get 1/2+S/2=S, and conclude that S=1. The problem is that this method ‘works’ for 1+2+4+8+…=S equally well. You should try it yourself, and you may well be surprised by what answer you get. This definitely doesn’t look right does it? Under any normal reasoning, this sum must surely = infinity?

So what went wrong? Well, in both examples we took an equation and did some arithmetic on it. But in the second example, we suspect that S was not finite. And we’ve seen earlier that infinity messes up arithmetic, hence why we got nonsense. By thinking about cutting pieces of a cake in half repeatedly, we would hope that 1/2+1/4+1/8+…=1, which is finite, and so it is more likely that we are ‘allowed’ to do arithmetic there, hence why that calculation gave an answer that looked right.

Now, let’s look at finite sums in the first example. You can check that $\frac12+\frac14+\ldots \frac{1}{2^n}=1-\frac{1}{2^n}$. Now look at the definition of limits of sequences. For basically the same reason that 1, 1/2, 1/3 tended to 0, $1-\frac{1}{2^n}\rightarrow 1$, and so this formal definition of infinite sums gives the same answer as the original working. Similarly, you should check that the formal definition of tending to infinity shows that 1+2+4+… does this. An infinite sum is often called a series, and one like this adivergent series.

So what are the morals of this story? These two examples of infinity in action are not easy, but hopefully not too scary either. If you approach it from a philosophical or intuitive point of view, it may seem like an erratic wild animal, but infinity can be tamed if you have suitable tools for thinking about it.