Question: If the square root of -1 is i, what is the square root of i?

How can you find the log of a negative number?

What is the log of -1?

Hi Kevin,

Square ^√2/₂ + i ^√2/₂ and see what you get.

Actually any complex number can be written in polar coordinates: z = r(cos(a) + i sin(a) ),
and it is then the square of √2 (cos( ^a/₂) + i sin( ^a/₂) ). You get two "square roots" of z that way,
one from selecting the angle a in [0,2) and the other by selecting a in [2,4).

Natural logarithms of negative numbers can be understood once imaginary exponents are explained:

The infinite series expansion of e^x is

e^x = 1 + x + ^x2/_2! + ^x3/_3! + ^x4/_4! + ...

and this can be applied to complex values:

e^ib = 1 + ib + ^(ib)2/_2! + ^(ib)3/_3! + ^(ib)4/_4! + ...

= (1 - ^b2/_2! + ^b4/_4! - ... ) + i(b - ^b3/_3! + ^b5/_5! - ...)

= cos(b) + i sin(b).

In particular eⁱ = cos() + i sin() = -1, hence ln(-1) = i .

Claude