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?

Ask Siri what zero divided by zero is. The answer is hilarious, and it also does an ELI5 really well.

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”.

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

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.”

There’s a sense in which division doesn’t really exist. There’s also a sense in which subtraction doesn’t exist. This goes a bit beyond ELI5 (I literally did it in 2nd year at uni studying maths), but I’ll try my best.

When we say “take away 2”, what we’re really saying is “add on negative 2” where negative 2 is the number such that 2+(-2)=0. We call 0 the additive identity, because x+0=x for any x. I.e. if we add on 0, we get back to wherever we started. For a number x, the number -x such that x+(-x)=0 is what we call the additive inverse, because it sort of takes us in the opposite direction by the same amount. With it so far?

We can play a same game with multiplication. 1 is called the multiplicative identity, since x×1=x. We multiply by 1 and nothing changes. Now, just as there are additive inverses, there are multiplicative inverses. Just as adding additive inverses gave the additive identity, multiplying multiplicative inverses gives the multiplicative identity. So if x and y are such that x×y=1, x and y are multiplicative inverses. Then, we can say that dividing by x is really a shorthand for multiplying by y. Say we say ÷2, that’s really a shortcut for ×½, because 2×½=1 (and ½×2=1). We can find an inverse for any number. Any number, that is, except 0. There is no y such that 0×y=1. Therefore, we can’t say ÷0 because that really means ×y where y doesn’t exist!

You can have zero groups of something.

What you can’t do is work out how many groups of zero objects you would need to total another number than zero.

Ten million groups of zero objects won’t even get you to 1 object in total, nor will 20 million groups, or 12, or 500 billion.

And though zero is an exception, that doesn’t help either. Zero groups of zero objects will total zero. But so will 1, 2, 3, 50 quintillion. So what’s the ‘answer’ when you divide by zero? Either no answer at all, or every single possible answer imaginable.

Multiplying is easy. Dividing by zero is impossible and gives only nonsense answers.

Multiplication is repeated addition. It says if I add a number (let’s say 3) together some number of times (say 6), what does that give me? 3 x 6 = 18

Division is basically repeated subtraction. If I have some number (let’s say 18), how many times can I subtract this other number (like 6) from it. 18 ÷ 6 = 3

How many times can you subtract 0 from 18? Infinity times. Since infinity is not a number, but an idea you cannot divide by 0.

Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.

(Stolen quote)

Let’s say you had a box of 10 tomatoes.

Multiplication says: if you take many of these boxes, how many tomatoes will you have?

Division says: How many of these boxes do you need to get a 100 tomatoes?

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Now say you had a box of 0 tomatoes.

Multiplication says: if you take many of these boxes, you will have no tomatoes.

Division says: How many of these boxes do you need to get a 100 tomatoes?

Uhh. You have the crashing realization that you can’t get a 100 tomatoes no matter how many boxes you take. And moments after, the computer simulating all of us explodes in a puff of 1s and.. you guessed it, 0s.

You can, but it’s undefined. 3/4 is 3 apples n 4 ppl and everyone gets the same amount. 0/4 is no appl n 4 ppl, no appl for the ppl. 3/0 is 3 apples no people all nonexistent ppl get same portion of 3 apples. 0/0 is no appl no ppl all nonexistent ppl get same amount of nonexistent appl.

If you have any number of things in a row it is n x 1 = n. Now you have any number of things with no row which is n x 0 = 0.

If multiplication is repeated addition then division is repeated subtraction. When you divide by 0 you are subtracting 0 from another number infinitely and you never get anywhere.

Just like multiplication is repeated additions, division is repeated subtractions.

You have 12 apples and want to divide them among 4 people. So the question is, how many times can you take 4 from 12?

1. 12 – 4 = 8

2. 8 – 4 = 4

3. 4 – 4 = 0

3 times.

Now then. You have 10 apples and want to divide them among 0 people. So the question is, how many times can you take 0 from 10?

1. 10 – 0 = 10

2. 10 – 0 = 10.

Uh-oh.

You have two apples on your table. Multiply them by two…. how many do you have.

Now you have two other apples on your table. Sort them into zero piles.

Suppose you were holding a pencil in your hand and I asked you to break it in to two pieces (divide by two) you could do that. But, if I asked you to break it in to zero pieces (divide by zero) you couldn’t do it. The universe would explode.

Because Mathematicians refuses to believe that you can have 8 apples and 0 baskets to put them in.