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.