In the world of math, many strange results are possible when we change the rules. But there’s one rule that most of us have been warned not to break: don’t divide by zero.
How can the simple combination of an everyday number and a basic operation cause such problems? Normally, dividing by smaller and smaller numbers gives you bigger and bigger answers.
Ten divided by two is five, by one is ten, by one-millionth is 10 million, and so on. So it seems like if you divide by numbers that keep shrinking all the way down to zero, the answer will grow to the largest thing possible. Then, isn’t the answer to 10 divided by zero actually infinity? That may sound plausible. But all we really know is that if we divide 10 by a number that tends towards zero, the answer tends towards infinity. And that’s not the same thing as saying that 10 divided by zero is equal to infinity.
Why not? Well, let’s take a closer look at what division really means. Ten divided by two could mean, “How many times must we add two together to make 10,” or, “two times what equals 10?” Dividing by a number is essentially the reverse of multiplying by it, in the following way: if we multiply any number by a given number x, we can ask if there’s a new number we can multiply by afterwards to get back to where we started.
If there is, the new number is called the multiplicative inverse of x. For example, if you multiply three by two to get six, you can then multiply by one-half to get back to three. So the multiplicative inverse of two is one-half, and the multiplicative inverse of 10 is one-tenth.
As you might notice, the product of any number and its multiplicative inverse is always one. If we want to divide by zero, we need to find its multiplicative inverse, which should be one over zero. This would have to be such a number that multiplying it by zero would give one.
But because anything multiplied by zero is still zero, such a number is impossible, so zero has no multiplicative inverse. Does that really settle things, though?
After all, mathematicians have broken rules before. For example, for a long time, there was no such thing as taking the square root of negative numbers. But then mathematicians defined the square root of negative one as a new number called i, opening up a whole new mathematical world of complex numbers.
So if they can do that, couldn’t we just make up a new rule, say, that the symbol infinity means one over zero, and see what happens?
Let’s try it, imagining we don’t know anything about infinity already. Based on the definition of a multiplicative inverse, zero times infinity must be equal to one. That means zero times infinity plus zero times infinity should equal two.
Now, by the distributive property, the left side of the equation can be rearranged to zero plus zero times infinity. And since zero plus zero is definitely zero, that reduces down to zero times infinity. Unfortunately, we’ve already defined this as equal to one, while the other side of the equation is still telling us it’s equal to two.
So, one equals two. Oddly enough, that’s not necessarily wrong; it’s just not true in our normal world of numbers. There’s still a way it could be mathematically valid, if one, two, and every other number were equal to zero. But having infinity equal to zero is ultimately not all that useful to mathematicians, or anyone else.
There actually is something called the Riemann sphere that involves dividing by zero by a different method, but that’s a story for another day. In the meantime, dividing by zero in the most obvious way doesn’t work out so great.
But that shouldn’t stop us from living dangerously and experimenting with breaking mathematical rules to see if we can invent fun, new worlds to explore.