Why Dividing by Zero is Undefined

Undefined and indeterminate answers

Why Dividing by Zero is Undefined
Why Zero Divided by Zero is Undefined/Indeterminate
Undefined and Indeterminate

Why Dividing by Zero is Undefined Thinking about why dividing by zero is left undefined

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Why Dividing by Zero is Undefined

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Comedian Steven Wright, and I guess we can credit him with being a bit of a philosoper
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 are current understanding of physics seems to break down
and dividing by zero, as simple an 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 bizarre idea
But it really means exactly what the word means
Mathematicians have never said what it means 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'
Seven divided by zero, eight divided by zero
Negative one divided by zero, we say all of these things are just 'undefined'
You might say "well, if we can just define it we can atleast 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'll take the simplest of all non-zero numbers, we'll just take one
(but we could have done this with any non-zero number)
Since we don't know what it means to divide by zero
Lets just try a really really small positive number
Let's see what happens as we get close to zero
So lets divide by 0.1
Well, this will get us to ten
If we divide one by 0.01 that gets us to 100
If I go really really close to zero, if I divide one by 0.000001
This is one divided by a millionth
That's going to give us 1 million!
So we see a pattern here, if we divide one by smaller and smaller positive numbers
We get a larger and larger value
Based on just this you might say "Well, hey, I've got somewhat of a definition for one divided by zero
Maybe we can say that one divided by zero 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 one divided by negative 0.1, that's going to be negative ten.
One divided by negative 0.01, that's going to be negative one-hundred.
And, if we go all the way to 1 divided by negative 0.000001
That gets us to negative 1 million
So you if we keep dividing one by negative numbers that are closer and closer to zero
We get a very different answer!
We actually start approaching negative infinity!
So over here we said it would be positive infinity, but you can make an equaly strong argument
that it could be a very different number!
Negative infinity is going the exact opposite direction
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 was equal to 42 or something like that
But that would make no sense!
Its neither one of these values, and it wouldn't be consistent with everything else we know

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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...

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At 2:33, Sal says "double bonds" but should say "single bonds."

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