Here’s something that seems to confuse many people: \[\frac{1}{0} \text{ is undefined}\\ \frac{0}{0} \text{ is indeterminate}\]
If some number, any number at all, divided by zero is undefined, then why isn’t zero divided by zero likewise undefined? And what does “indeterminate” mean anyway?
Let’s start with a more concrete question: What is division?
Assume we have a pizza. Math teachers seem to love using pizza to explain fractions. Anyway, we have a pizza and four people: Each person will get 1/4 of the pizza. The number on the top (the numerator) says how many things we’re splitting up, and the number on the bottom (the denominator) says how many people we’re sharing with. 8/3 means we have eight things to be shared equally among three people… how much does each get? 7/8 means we have seven things to be shared equally among eight people… how much does each get?
So 1/4 means we have one pizza and four people. Each person will get 1/4 of the pizza. That’s what division means. (Notice that we’re talking about fractions here: Division is intimately related to fractions. If you understand one, you can understand the other!)
What does 0/4 mean, using this explanation? It means we have no pizzas, and four people each want some. How much will each get? Zero, zip, zilch, nada. Sorry, everyone. 0/4 = 0.
What does 1/0 mean, using this explanation? It means that we have nobody at all who wants pizza, and we have one pizza. How do we distribute one pizza among zero people? That doesn’t make any sense. We can’t do it.
Another way this is sometimes explained: If we have a pizza and cut off 1/4, we can give that to someone and then we’ll have 3/4 of the pizza left and three people left. We can repeat that, and eventually everyone will have some pizza and there won’t be any left over. Now imagine that 1/0 meant something: We would have a pizza, cut off some portion, give it to nobody, and repeat this until we’re out of pizza… except that we’d never be out of pizza. For this reason, some people say that 1/0 is really equal to infinity, because we’d have to do this an infinite number of times. That’s not quite true, but it’s useful for some basic math.
In general, though, 1/0 is undefined: It’s impossible to distribute one pizza among zero people evenly, and actually get rid of the pizza.
Now think about 0/0. This says we have no pizzas, and nobody gets any. This is conceptually possible. I have no unicorns in my living room, and nobody owns any unicorns. That makes sense, in a weird way.
However, how many unicorns does each person own? How much pizza does each person get? If there are no people owning unicorns or eating pizza, then any amount makes just as much sense as any other: Each person gets five pizzas. Each person gets twenty pizzas. Each person gets five billion pizzas. Since there are no pizzas and no people getting them, then every value for 0/0 makes just as much sense as every other value.
This is why we say that 0/0 is indeterminate: We don’t know what it equals, because while it makes more sense than 1/0, it doesn’t actually tell us anything.
“Indeterminate” means that a particular expression could logically take several different values, and we don’t know which one. “Undefined” means that a particular expression can’t logically take any value at all. And that’s the difference between 0/0 and 1/0.