Never divide by zero. We were all taught this in school, and in most everyday situations, it's good advice. It rarely makes sense to divide anything by zero, and if you try to ask Siri to do it, she will say you have no friends.
But why exactly is dividing by zero such a bad idea? And are there cases where dividing by zero can be a good thing? If you weren't paying attention back in high school, a quick refresher can open your eyes to one of the many wonders of math.
Dividing by zero doesn't make sense because in arithmetic, dividing by zero can also be interpreted as multiplying by zero. 3/0=X is the same equation as 0*X=3. Obviously, there's no number that can be plugged in for X to make that equation work.
A similar situation results if you try and divide zero by itself. 0/0=X can be rewritten as 0*X=0, and the problem here is that every number works. X can be anything, so this equation isn't very useful.
But there are some cases where dividing by zero actually does work, and in fact is essential for solving a problem. This was Newton's insight when he invented calculus.
Say, for instance, you've got a curve and you're trying to find the slope of the curve below at a specific point. This is the same as trying to find the slope of a line that only touches the curve at that point, called a tangent line. In many cases, it is impossible to find that slope using algebra alone.
But there's a trick that uses calculus and the magic of dividing by zero. Instead of finding the tangent line, it's much easier to find the slope of a line that touches the curve at two points. If you move those two points closer and closer together, you get a line that's close to the one you wanted.
If you move the two points so they're on top of each other—such that there's practically only one—you get the original tangent line you wanted, only this time you can find the slope. You're essentially dividing zero by itself and getting an answer.
The trick to divide by zero is essentially to create some extra context. The initial problem with 0/0 is that every number could potentially be an answer, so introducing some limiting factors can narrow down the possible answers.
And once you can divide by zero, a whole new world of mathematics is opened up. Playing with zeros and infinities and all sorts of impossible equations becomes commonplace. Learn to divide by zero and math will never be the same again.
Source: Mathologer
