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Why dividing by zero is undefined

Sal explains why there cannot be a single number that is the result of dividing by zero. Created by Sal Khan.
Video transcript
Comedian Steven Wright-- and I guess we can credit him with being a bit of a philosopher-- once commented that "Black holes are where God divided by zero." And I won't get in to the physics of it, and obviously the metaphor breaks down in certain ways But it is strangely appropriate, because black holes are where our current understanding of physics seems to break down and dividing by zero, as simple of idea as that seems to be, is where our mathematics also breaks down. This is "undefined." Sometimes when you see "undefined" in math class it seems like a very strange thing. It seems like a very bizarre idea. But it really means exactly what the word means. Mathematicians have never defined what it must mean to divide by zero. What is that value? And the reason they haven't done it is because they couldn't come up with a good answer. There's no good answer here, no good definition. And because of that, any non-zero number, divided by zero, is left just "undefined." 7 divided by 0. 8 divided by 0. Negative 1 divided by 0. We say all of these things are just "undefined." You might say, well if we can just define it, let's at least try to come up with a definition of what it means to take a non-zero number divided by zero. So let's do that right now. We could just take the simplest of all non-zero numbers. We'll just do it with one. But we could have done this with any non-zero number. Let's take the example of one. Since we don't know what it means-- or we're trying to figure out what it means to divide by zero Lets just try out really, really, small positive numbers. Let's divide by really, really small positive numbers and see what happens as we get close to zero. So lets divide by 0.1 Well, this will get us to 10. If we divide 1 by 0.01 that gets us to 100. If I go really, really close to zero. If I divide 1 by 0.000001 this gets us-- so this is not a tenth, hundredth, thousandth, ten thousandth, hundred thousandth. This is a millionth. 1 divided by a millionth, that's going to give us 1 million. So we see a pattern here. As we divide one by smaller and smaller and smaller positive numbers, we get a larger and larger and larger value. Based on just this you might say, well, hey, I've got somewhat of a definition for 1 divided by 0. Maybe we can say that 1 divided by 0 is positive infinity. As we get smaller and smaller positive numbers here, we get super super large numbers right over here. But then, your friend might say, well, that worked when we divided by positive numbers close to zero but what happens when we divide by negative numbers close to zero? So lets try those out. Well, 1 divided by negative 0.1, that's going to be negative 10. 1 divided by negative 0.01, that's going to be negative 100. And, if we go all the way to 1 divided by negative 0.000001-- yup, I drew the same number of zeros-- that gets us to negative 1 million. So you when we keep dividing 1 by negative numbers that are closer and closer and closer and closer to zero, we get a very different answer. We actually start approaching negative infinity. So over here we said maybe it would be positive infinity, but you can make an equally strong argument that it could be a very different number. Negative infinity is going the exact opposite direction. So you could make an equally strong argument that it should be negative infinity. And this is why mathematicians say there's just no good answer here. Especially one that's consistent with the rest of mathematics. They could have just said it's equal to 42 or something like that. But that would make no sense. It's neither one of these values, and it wouldn't be consistent with everything else we know. So they just left the whole thing "undefined."