What is Zero Divided by Zero?

Описание: Learn what Zero divided by Zero really means.

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The short answer is that division by zero, whatever the dividend, is undefined in regular arithmetic. That is, this particular operation is not allowed by the rules of mathematics. People often take this to mean that the operation makes no sense physically, that the mathematics don’t model any sensical real-world situation. In the case of division by zero, this is a helpful visual. Division represents evenly splitting a certain number of objects (the dividend) into a certain number of groups (the divisor) and counting the number of objects in each resulting group (the quotient). The number of groups can be five, one million, or one, but it cannot be zero. Side note, people may argue that dividing some number of objects into zero groups is the same as not dividing it into groups, but that’s not true. Not dividing it into groups is the same as dividing it into one group, leaving the original group unchanged. This gives a good visual explanation of why dividing by zero makes no sense, but the truth is that a lot of things in math have no direct visual or physical representation and yet are useful in modelling the world around us. It’s better to think of an undefined operation as a banned rule breaker. Math is, at its core, a list of self-consistent rules, and if something breaks that self consistency, it must be thrown away.

The big rule that division by zero breaks is that, for a function to have an inverse, the function must be bijective. Let’s break that down. Normally, a function like “multiplication by 4” is undone by the function “division by 4”, and vice versa. So we say those two functions are inverses of each other. But this only works because multiplication by 4 is bijective, that is, for any number, if we multiply that number by 4, we get one and only one result. We can’t multiply a number by 4 and get two different answers, and we can’t multiply two different numbers by 4 and get the same result. To get the inverse function “division by 4”, we simply reverse these arrows. Multiplication by zero is different though. Any number multiplied by zero is zero, and no number can be multiplied by zero to give us a non-zero result. If we try to define the function “division by zero” by reversing these arrows, we have major problems. 0/0 has an infinite number of solutions, while any other number divided by zero has no solution. This is simply not a useful or sensical function, so it is not allowed. Multiplication by zero actually doesn’t break any mathematical rules for functions, it’s just not bijective. And it has a sensical physical representation to boot, since, for example, having three groups of no objects is a total of no objects. This is why multiplication by zero is allowed, but division by zero is simply not a thing.

We hope that gives you some deeper insight into why division by zero is undefined. If you have any questions or know of something we left out leave a comment below!