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
The reason is that the basic operations that we define on real numbers and complex numbers (and fields in general – there are more than just these two) can’t fulfill the basic properties we want them to have if m=1/0 was defined.
It’s not too hard to see this:
What’s 0? 0 is the element that you can add to any number without changing it (the “neutral element”): For all numbers a, a+0=a
From this and our idea of how addition and multiplication should be compatible, it follows that any number multiplied by 0 is 0 again: For all a, a*0 = 0 (you can see this by multiplying the above equation with any number b: ab + a*0 = a*b, so a*0=0).
But now, m=1/0 can’t be defined, because m*0 should be 0, but if m=1/0, m*0= 1/0*0, it should also be 1, and that’s not possible at the same time.
Latest Answers