Why Dividing by Zero Is a Terrible Idea

Katie Steckles

A little while ago, I was contacted by the producers of BBC Radio 4’s More or Less to help them settle a mathematical argument – specifically, this one:

In the post, a disgruntled parent explained that their third grade (roughly nine-year old) child had been taught that one divided by zero is zero – which they were so outraged by, they felt the need to complain to the school. Given that the purpose of teachers is to guide a child’s education and help them to understand how the world works, it is certainly infuriating to learn that a teacher might be giving them wrong information. But why might someone get this wrong so easily – and what actually is the answer?

Understanding Zero

Firstly, it is worth acknowledging that not all teachers will be trained mathematicians, and it will not be often they are called upon to answer questions like this – but all it takes is a curious student thinking beyond the content of the lesson to put us somewhere unexpected.

The concepts involved in this particular question – “what is one divided by zero?” – all seem like familiar, simple, manageable ideas: the numbers one and zero, and the concept of division, which we usually learn at a younger age than nine. Who could possibly expect that just arranging them in this particular way would result in something that does not have a simple answer? Or indeed, any answer at all!

When learning about division at school, it is often framed in terms of sharing: If I have twelve sweets, and there are four people, we can share the sweets out equally by dividing the number of sweets by the number of people, to get the number of sweets each person gets (in this case, three). But if we change the number of people and share them between three people instead, we get four each; between two people, there are six sweets each. And if we take the greedy option and share the sweets between one person (if such a thing even counts as sharing), that person would get all 12 sweets.

But beyond this point, the question becomes slightly meaningless. If I had zero people to share sweets between, I could give them any number of sweets each (including numbers that are more than the number of sweets I have) – because there are no people to give them to, I can perform the action of giving nobody some sweets as much as I like, and still have all the sweets I started with.

This starts to hint at the idea that dividing by zero does not work in the same way as dividing by other numbers does. This is partly because the process of division is something that we are used to being able to run in both directions: We say we can invert an operation like division, and by knowing how many sweets I ended up with and how many people have shared the sweets, I could calculate how many sweets were being shared out in the first place by multiplying these two numbers together.

But this breaks down entirely when we divide by zero. If I can invert the calculation 12 ÷ 4 = 3 by finding 4 × 3 = 12, I should be able to figure out what was originally divided by zero by multiplying each share by zero. But anything multiplied by zero is zero – as any third-grader would be able to tell you – so no matter what size of share we assign to each of the zero people, we cannot invert the process and find the original total.

In general, we say that dividing by zero is not defined, or that it is not a permitted operation; calculators asked to divide by zero will sometimes return ERR (error) or NAN (not a number), since the value of anything divided by zero is undefined. Mathematicians will avoid dividing by zero, since it creates logical contradictions – many of the enjoyably puzzling ‘false proofs’ you can find, including the classic proof that 1 = 2, rely on a step which is equivalent to dividing by zero (but disguised using algebra, so you may not realise what is happening). 

Reaching My Limit

Following my appearance on the radio to explain this, a number of listeners contacted the show (sadly, the number was not zero) to complain that I was wrong, and that of course it is possible to assign a value to the quantity given when you calculate one divided by zero: obviously, the answer is infinity. Despite having carefully explained that we deliberately do not assign a quantity to the answer, many people insisted that I had just failed to explain it properly.

And I can understand why these people thought they were right – it is well known that the smaller the number you are dividing by, the larger the result. In our twelve sweets scenario, dividing by four people gave 3 each, by three people gave 4 each, by two people 6 each, and dividing by one person meant 12 sweets each – the number increases as the amount you are dividing by decreases.

We can even continue this beyond the point where our sweet analogy fails us – if we divide 12 by 0.5, which is less than one, we get 24: Dividing by a half is the same as multiplying by two. With this logic, dividing by 1/100 would give us 1200, and dividing by 1/1,000,000 – a millionth – would give us an answer of 12 million. It doesn’t matter what we start from: Whether we’re dividing 12 by something, or dividing 1 by something, we can keep playing this game as long as we want, and the smaller we make the number we divide by, the larger the answer we get.

Often in mathematics, we play this kind of game when we have a situation that will theoretically continue forever: if I wanted to show what would happen if I kept adding together all the numbers in an infinite series, as I wrote about here back in 2019, we can show that something will get to infinity, or converge to a particular value, without actually adding things up forever.

As long as you keep asking ‘can you make a number bigger than this?’, and I can keep suggesting something to divide by that satisfies your question, we can keep going: And as the number we divide by gets closer and closer to zero, it can be tempting to conclude that when it reaches zero, the answer will be infinity.

But this does not work as an answer, for multiple reasons; not least because infinity is, as is often stated, ‘not a number’ – the same ‘not a number’ your calculator is not prepared to display. But it is even worse than that: There is more than one way to get closer and closer to zero, and if you use a different approach, you get a different answer.

If you can wrap your head around dividing by a negative number, you should be happy with the idea that 12 divided by -3 is -4. (Again, invertibility helps here, since multiplying together -3 and -4 logically gives a positive answer of 12). And we can also do this for smaller and smaller negative numbers: we could divide by -0.5, -1/100 or -1/1,000,000, and the answers we get would be larger and larger; and, importantly, negative.

In the same way that our positive numbers are somehow approaching infinity, these numbers are approaching minus infinity. And even if you are so convinced of your correctness that you would write in to a radio show to correct a mathematician, you must surely agree that negative infinity is a long way away from being infinity, even though the numbers we are dividing by are also getting closer and closer to zero.

This idea, sometimes called ‘two-sided limits’, is a great way to understand why dividing by zero is a terrible idea: In order for something to be consistently defined as the limit of a process, it needs to work from both directions (in the case of numbers on a one-dimensional line, there are only two directions to check, but in higher-dimensional analysis, things get even more difficult).

While I would not expect the third-grade teacher to explain all of this to a nine-year-old – and I can appreciate how difficult it is to admit you do not know the answer to something, especially when you are in a position of authority – the teacher’s reaction on being queried could most certainly be described as unhelpful. The maths behind zero and division involves a lot of mysterious behaviour and unanswered questions, and understanding it can take you down a real rabbit hole – to minus infinity and beyond. 

The post Why Dividing by Zero Is a Terrible Idea originally appeared on the HLFF SciLogs blog.