Is Sqrt(-1) not unique? I.e. is i not the only one?

[Curl]

Sorry for the dumb question, I'm not good at mathematics, but:

what is 1/i ?

1/i * 1/i = 1*1 / (i*i) = 1/-1 = -1 ?

If so the number, 1/i has the property that (1/i)^2 = -1 which is the definition of i.

BUT, 1/i is not equal to i because if it did:

1/i * i would be i*i= -1

however,

1/i * i = i / i = 1 ?

So can someone explain what's going on? Where did I make the mistake(s)?

Thanks!

 

[hilbert2]

$\frac{1}{i}=-i$, because by multiplying the numerator and denominator with $i$ we get $\frac{1}{i}=\frac{i}{i^{2}}=\frac{i}{-1}=-i$. The numbers $i$ and $-i$ both solve the equation $x^{2}=-1$. The square root or higher roots can't be defined uniquely in complex algebra.

 

[thelema418]

If so the number, 1/i has the property that (1/i)^2 = -1 which is the definition of i.

This is where your problem lies. You have to be careful with square roots. x^2 = 25, but does that mean x is +5 or -5? I think you can figure it out from that.

 

[HallsofIvy]

Every number, real or complex, has two roots. -1 has the two roots i and -i. Your (1/i) is equal to -i.

And, while "i^2= -1" is often used as a defining property of "i", it cannot really be used as a "definition" because, as you say, -i has that same property.

If you want a rigorously correct definition of complex numbers, you have to do something like this:
"The complex numbers consist of pairs of real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication by (a, b)(c, d)= (ac- bc, ad+ bc). Notice that (a, 0)+ (c, 0)= (a+ c, 0) and (a, 0)(c, 0)= (ac, 0) so we can think of the real numbers as being those complex numbers of the form (a, 0). Further, (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0) so that the product of (0, 1) with itself is the complex number representing the real number -1. If we now define "i" to be (0, 1), we have "$i^2= -1$" We can also say that (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1). We are thinking of the complex number (1,0) as representing the real number 1 and since we have defined "i" to be (0, 1), we have (a, b) represented by a+ bi.

Now both (0, 1) and (0, -1) have the property that (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0) and (0, -1)(0,-1)= (0(0)- (-1)(-1), 0(-1)+ (-1)(0))= (-1, 0). That is "-1" has two complex roots but now we can distinguish between i= (0, 1) and -i= (0, -1).

 

[jbunniii]

Every number, real or complex, has two roots.

Exception: zero.