Dividing by Zero

Reddit has started to annoy me. I am finally at the point in math where I can spot misinformation pretty easily. A lot of people were posting in some thread about how dividing by zero was in fact possible. All sorts of answers were being upvoted. I now wonder how often I believe something on reddit that might be completely false just because I don’t know enough about the subject.

Here are a few things people said.

\frac {1}{0} = \infty

Another person mentioned that we just need to define it to be something, similar to how people define
i = \sqrt{-1} because we just didn’t know what \sqrt{-1} was. This is also wrong.

In our number system 1 divided by something is just another way of saying the inverse.

For example \frac{1}{2} is the inverse of 2, which means that when I multiply them together I will get 1.

\frac{1}{2} (2)=1

So trying to define \frac {1}{0} is trying to define an inverse of 0. So we have something that when we multiply it by 0 we get 1. One reason this is impossible is because 0 multiplied by anything should be 0.

If we let \frac {1}{0} = \infty then we have that \infty (0) = 1
This leads to a ton of contradictions, for example:
\infty (0) = 1
multiply by 2 on both sides
\infty (0)(2)  = 1(2)
but 2 times 0 is 0 so we get
\infty (0) = 2

Then we have that

2 = \infty (0) = 1
2 = 1

Besides 2 not equaling 1, we also have that 0 multiplied by infinity can give us two different answers. In fact it can give you any answer just by replacing 2 with something else, which is contradictory.

Later in math we learn about the extended real numbers. Basically you take the real numbers and throw in this “point at infinity,” but it is certainly not the same as 1 divided by 0.

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