As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?
So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?
In: 27
It’s been a while so this might not be the best way of going about it. But if we define m=1/0 then we are implicitly implying that m belongs to an algebraic structure in which multiplicative inverses are defined (otherwise / is meaningless), and that’s its inverse is therefore 0/1=0, and one in which the additive identity is defined (0). From that we end up with a contradiction. 1=(1/0)(0/1)=(1*0/0*1)=0/0=(0+0)/0=0/0+0/0=2.
Basically, i ends up useful because you can create an algebraically complete field (a field is basically everything that you can do with real numbers being defined, algebraically complete is why C is useful: every polynomial has an answer) with it as a member but m ends up useless because the moment you start trying to the binary operations to the set that contains you’ve used to define it then you start breaking things so you can’t use it for anything.
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