Why can you multiply by zero but not divide?

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Go easy on me.

In: Mathematics

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

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 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.

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.

You can do something zero times, but how much of something can you split between zero people?