Go easy on me.
In: Mathematics
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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