You have probably seen a meme similar to this:

But what happens when you divide by zero? Usually my calculator just spits out “error.” Why is that? To investigate this problem, we need our handy dandy friend, “proof by contradiction.”

Suppose we could divide by zero. That would mean that . In particular, since anything divided by itself is always 1. Also, zero times anything is always zero. That is pretty standard knowledge. Ok, here we go.

Hence:

Now divide both sides by zero:

And we can rearrange this equation:

But we know that . Hence:

Finally:

Every kindergarten student knows that 1≠2. But we showed that 1=2. This is an obvious contradiction. We arrived at this contradiction by assuming that we could divide by zero. Thus, we cannot divide by zero. Done.

This proof tells us something very interesting about dividing by zero. The reason your math teacher always told you cannot divide by zero is interesting. If you allow division by zero, then 1=2, and 1=7, and 1=every other number. And when every number equals every other number, numbers are meaningless.

Don’t divide by zero people. You will break math.

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But we know that \frac{0}{0}=1

Do we???

We are pretty sure. By assuming that we can divide by zero, we have to assign a value to \frac{0}{0}. The most logical value would be 1 since \frac{n}{n} = 1 for all n not equal to zero.

You are technically correct that we could assign a different value to it. In that case, you would have to give a good argument as to why \frac{0}{0} = 7, for example. Even still, you can show that it results in a contradiction. I will leave it up to you to show why \frac{0}{0} = 0 also leads to a contradiction.