TedEd: Why can't you divide by zero?

A TedEd by the Ted team

In the mathematical world, one of the unbreakable rules is that you can't divide by zero. But why? Usually, dividing by increasingly small numbers results in bigger and bigger quotients (try it). So, if we keep getting bigger numbers by dividing with smaller numbers, it seems that dividing by 0 should become ∞, the largest number possible. However, we do not necessarily know this. All we know is that as the divisor gets closer to 0, the quotient gets closer to ∞, which is different from knowing x/0 = ∞. What really is division? If we say x/y, we can say how many times must we add y to get x, or y*z=x. Any time we multiply x by y to get z, we can see if there is a number that we can multiply z by to get back to x. That number is also known as the multiplicative inverse. The product of any number and its inverse is always 1. To divide by 0, we must first find its multiplicative inverse, which is 1/0. 1/0 would have to be 1 when multiplied by 0. However, x*0 is always 0, so 0 has no multiplicative inverse. But what if we broke some rules and defined 1/0 as ∞? Then, 0*∞=1, and (0*∞)+(0*∞)=2. We can rearrange the second equation to 0*∞=2. But we already defined 0*∞=1, so this is impossible. Now, we could define have all real numbers equal to 0, but that is pretty useless.