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.
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.
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?
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.
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.
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.
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!
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