Have you ever been told that division by zero is undefined? What do we mean by undefined, anyway? It sounds like mathematicians just never bothered getting around to defining it. Imagine Newton and Leibniz (the fathers of Calculus) sitting down at a café in Paris, sipping on coffee. Newton says, “Should we define this whole division by zero thing?” Leibniz pauses a moment, swirling the last drops in his mug, “No, let’s order another round instead.”
That’s not actually what we mean by undefined. Perhaps a better term would be undefinable, although that isn’t quite right either. We could define what 5 divided by 0 is. It’s just that our definition would generate contradictions. In math a definition that causes contradictions is said to not be well defined. We could actually make any number of (infinitely many) poor definitions, but no perfect one that works every time. So, we leave it undefined.
Consider 5 divided by 0. Suppose we said it was a number. Any number will work equally well here. Actually any number will work equally unwell, I suppose. Let’s say that 5 divided by 0 is 100. This is silly, but it turns out to be no sillier than anything else.
For any definition we make, we want everything else defined to still work. The usual way to define division is in terms of multiplication. We say 8 divided by 4 is 2 because 2 times 4 is 8, and so on.
Let’s try that with our definition for 5 divided by 0. Since 5 divided by 0 is 100, then 100 times 0 is 5. But, 100 times 0 is supposed to be 0. (That is called, unimaginatively enough, the multiplication property of 0.) This means either our definition is wrong or 0 equals 5. Wait, what? Well we just said 0*100 = 0 and 0*100 = 5. Since both left hand sides are the same, that would imply that the right hand sides are also the same. We know that 0=5 is absurd (properly we call it a contradiction), so we conclude that our definition is wrong.
Perhaps 100 was just a bad choice. Let’s say it is some other number. To save time I’m going to use the number x — a variable. We’ll say 5 divided by 0 is some number x, and that we’ll figure out which one it is later. By the definition of division, that means that 0 times x is 5. So, x is the number that gives you 5 when you multiply it by 0. But any number times 0 is 0, and we’ve already ruled out the possibility that 5 and 0 are the same, so that means any number we choose for x gives us a contradiction. 5 divided by 0 is not a number.
Hey, that’s not too bad. Maybe 5 divided by 0 is a non-number like the word Aardvark. That would cause even more problems. We have only defined our operations like add, subtract, multiply, and divide for numbers. So, 0 times Aardvark is undefined. Oh no! Not that again! Yep. We’ve just pushed the problem on to the next statement. That doesn’t work either.
But, maybe you’ve heard on the street that 5 divided by 0 is infinity. That is unfortunately misinformation spread about by people learning Calculus that haven’t quite grasped it yet. I could go into more detail, but instead I’ll appeal to the Rule of Two. Either infinity is a number or it is not a number. In either case it doesn’t work by our previous discussion. (In fact, infinity is not a number, but that is a discussion for another time.)
I’ve avoided the case 0 divided by 0, because this is long enough already. If you are clever, you’ll see that you could define 0 divided by 0 in such a way to not violate what we said about 5. I promise to show you why we still don’t define 0 divided by 0 in a future post (probably after we’ve discussed whether infinity is a number or not).
Hopefully I’ve convinced you that division by zero is not defined because it cannot be defined well. If I haven’t reached you with this, you should know that in the mid 1990s a secret organization of retired math teachers was formed to combat the problem of people dividing by zero. Equipped with black helicopters, night-vision goggles, and stun guns (all grant funded, I assure you) they take turns listening in to classrooms all over the nation.
When their surveillance detects someone dividing by zero, they swing into action. A strike team of 4 expert pedagogues repel from their helicopter. Unfortunately, retired math teachers are not known for their peak physical condition or combat operations training. They have spent long careers sitting in uncomfortable chairs drinking too much coffee and grumbling as they grade page after page of division by zero errors. To date, every attempted intervention has resulted in at least one heart attack, stroke, or loss of consciousness from misjudging the distance to the ground. To conceal their nature the mission is then aborted and the poor retired teachers are shuttled to the nearest hospital where the luckiest are treated for shortness of breath and high blood pressure.
Some are not so lucky. So far, 431 retired math teachers have given their lives in service to the mathematics community. So, the next time you are tempted to give any other answer besides undefined for a division by 0 problem, please, think of the huddled squad of out of shape, caffeine addicted, former educators. Don’t let number 432 be on your hands.