So if we want…
> 0 / sqrt(0) = sqrt(0)
But we would also get:
> 0 * (1/sqrt(0)) = 0
as anything multiplied by 0 is 0.
But also, if sqrt(0) = 0, we get:
> 0 / sqrt(0) = 0 / 0 = 1
as anything divided by itself is 1.
And, of course, we get:
> 0 / sqrt(0) = [something] / 0 = ….?
as you cannot divide by 0.
So which is it? Is this sqrt(0), 0, 1 or undefined?
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So sure, if we “divide both sides by sqrt(0)” we get:
> 0 * sqrt(0) = sqrt(0)
which is a valid statement.
The problem with extrapolating this is that
> 1 * sqrt(0) = sqrt(0)
as well. And also:
> 20 * sqrt(0) = sqrt(0)
so if we try to divide, we get:
> sqrt(0)/sqrt(0) = 1 = 20 = 0
which doesn’t work.
When you are asking “what is *a* divided by *b*”, what you’re really asking is “What number, when multiplied by *b*, equals *a*?”
In this case, what number, when multiplied by 0, equals 0? And the answer is, every number. And that’s why the answer is undefined, because it doesn’t give a concrete answer.
We want things to have a single definitive correct answer, so I can say:
x + 2 = 4
and the only right answer for “x” is “2”. “4 – 2 = 2”, that’s it, that’s the answer. We know because we can plug it back in as “x” and get the right result.
But while 0*0=0, we also have 0*1=0 and 0*2=0 and 0*1234567890=0.
So there’s no way to have a single right answer for 0/0, because when we get to “x * 0 = 0”,
any value can be plugged in for “x” and we get the right result.
Think of multiplication as “taking from a box of apples” and division as “putting into a box of apples”.
Using those definitions, “0 * [any number]” is the same as “taking 0 apples from the box [any number] times”. At the end, how many apples do you have? Zero because you never actually took anything from the box.
For division, “[any number] divided by zero” is the same as “how many times can you put 0 apples into the box if you start with [any number]”. It’s obviously a pointless exercise so they call it “undefined”.
When we talk about dividing, what we are really doing is multiplying by the inverse. By that I mean that 7/9 is shorthand for 7*(1/9). So when you ask why we cannot divide by 0, you are really asking why 1/0 isn’t defined.
The reason 1/0 isn’t defined is because imagine 1/0=x. Then multiple both sides by 0 to get 1=x*0. However anything times 0 is 0, so we get 1=0, which is clearly wrong. Therefore 1/0 must be undefined.
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