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## Algebra I (2018 edition)

# Why dividing by zero is undefined

As much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be true, because anything times 0 is 0. 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."