Go easy on me.
In: Mathematics
More ELI10: In math there is something called “limits.” It allows us to guess numbers that we can’t actually calculate just by looking at were they would appear to land.
Quick example: Take the sequence `1 2 X 4 5`. We don’t know what `X` is, but if we approach `X` from the left side (start lower and count up), it looks like we’ll land on 3, and if we approach it from the right side (start higher and count down) it looks like we’ll land on 3, therefore we can say with a decent amount of certainty that `X = 3` even though we don’t actually have a way to calculate it.
When it comes to dividing by 0, things get a little tricky. Take the following sequence of calculations:
`1/3 = 0.333_`
`1/2 = 0.5`
`1/1 = 1`
`1/0.5 = 2`
`1/0.01 = 100`
`1/0 = X`
You can see that as we approach dividing by 0, the numbers seem to go up. It may seem easy then to just say “`1/0` equals infinity,” then and be done with it, but there’s more. We just approached it from the right side, but watch what happens when we approach it from the left:
`1/-3 = -0.333_`
`1/-2 = -0.5`
`1/-1 = -1`
`1/-0.5 = -2`
`1/-0.01 = -100`
`1/0 = X`
Now it seems to approach negative infinity. Since approaching from the right points to infinity, and approaching it from the left points to negative infinity, there’s no “true,” answer and therefore we just say that anything divided by 0 is “undefined.”
Let’s say you can divide by 0. This would mean that there would be some number x such that
1/0=x
Which can also be written as
1=0x
But 0 times any number is 0, so there is no solution to this equation, thus its impossible to divide by 0
How many groups of 0 does it take to make 6?
The answer isn’t 0, or infinity, or a complex imaginary number. You CAN’T answer the question. The best we can do is say the answer is “undefined”.
Ask Siri what zero divided by zero is. The answer is hilarious, and it also does an ELI5 really well.
You can do something zero times, but how much of something can you split between zero people?
Multiply x by 0: If you have x groups, and each group has zero people, how many people do you have? That question makes sense, because a group can easily have zero people, and you could have zero groups.
Divide x by 0: If you have x items, how many times can you take zero items away from it before you have no items left?
Well, let’s assume that you have ten items.
10 – 0 = 10.
10 – 0 = 10.
10 – 0 = 10.
10 – 0 = 10.
10 – 0 = 10.
And so on and so forth. Note that the question doesn’t have an answer. It doesn’t matter how many times you take zero away from ten, you’ll never hit a point where you have zero items left.
In contrast, what is 12 divided by 3?
12 – 3 = 9
9 – 3 = 6
6 – 3 = 3
3 -3 = 0.
You can remove 3 from 12 four times: 12/3 = 4.
There are more elaborate and less patronizing ways to explain why dividing by zero doesn’t make sense or doesn’t have a clear answer, but that’s the simplest one. You could say “well, why isn’t it infinity?”, and that works as long as you don’t do it with negative numbers. If you try to divide negative numbers by zero you’ll approach negative infinity: so which is it? infinity or negative infinity? Getting two answers with one question isn’t very useful, and so we rather just say it doesn’t have an answer.
When you multiply or divide you can think about it like making piles of things. So let’s say I have a bunch of marbles and I asked you if I make 5 piles of 2 marbles each, how many marbles do I have. Well that’s 5 (piles) x 2 (marbles) so 10 marbles. Now if I ask you to make 5 piles of 0 marbles how many do you have? It’s a bit weird, but you can say you have 0 marbles in a pile, so you can still work this out logically, and get to an answer of 0 marbles.
Now I do it in reverse, and I say if I split 10 marbles into 5 piles, how many marbles are in each pile? 10/5 = 2. Makes sense. But if I ask you to split 10 marbles into 0 piles…you just can’t do that. There is no way to distribute marbles to no piles, so the answer is undefined.
I can have 20 groups of zero things for a total of zero things (20×0=0)
I can’t have 20 things and separate it into a total of zero groups (20÷0=??)
When you multiply by zero, you always get zero, but if you do a divisor by zero, you are essentially asking what number multiples by 0 to give you the dividend, but of course no number multiples by zero to give you anything other than 0, so you can’t answer
When you divide 0 by 0 you get the opposite problem, any and all numbers multiplied by zero give you zero, so again, you can’t give one answer
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