[eli5] Why is imaginary number root -1? Why not root -n, since no number times each other would result in negative number?

Question

[eli5] Why is imaginary number root -1? Why not root -n, since no number times each other would result in negative number?

In: Mathematics

Answer 1

Something has to be the “standard” unit. For real number, it is “1”, for imaginary number, it is “i”. The value of i is conventionally agreed to be sqrt of (-1). Of course if everyone agree, “i” could be sqrt of (-2) or (-3) or (-n), but why do that, it will look weird. Just make an xy graph and you will see why doing so is weird.

And no, it doesn’t solve the problem of “no number times each other would result in negative number”. Per your proposal, i x i = -n which is still a negative number.

But more importantly, there are numbers that when times each other (square), result in negative number. *They are imaginary numbers.*