help with imaginary numbers?

Question

i = sqrt (-1)
i ^ 2 = -1 [this I am certain of]

i ^ 2 = i * i
= sqrt(-1) * sqrt (-1)
= sqrt (-1 *-1 ) [line A]
= sqrt (1)
= + or - 1 [which can't be right]

I think there's a problem with [line A] but I can't work out why. Can anyone help?

Answer 1

sqrt(a) * sqrt(b) = sqrt(a*b) holds true if and only if at least one of a and b is non negative

Hence the following step in your method is wrong:
sqrt(-1) * sqrt (-1)
= sqrt (-1 *-1 ) [line A]

Answer 2

line A should be (sqrt(-1))^2
= -1

Answer 3

sqrt(-1) * sqrt (-1) =

[sqrt(-1)]^2 =

i^2 =

-1.

Answer 4

both +i = +sqrt(-1) and -i = -sqrt(-1) square to -1

without a sign, sqrt (-1) can mean +i or -i

write -1 = exp (i pi)
i = sqrt (-1) = (-1)^(1/2) = {e ^{ i pi} )^(1/2)
= e ^{ i pi/2}

i^2 = i* i = e ^{ i pi/2} e ^{ i pi/2} = e ^{i pi} = -1

But in your steps

i^2 = sqrt (1) = +1 or -1
the word 'or' here means ONLY one of the two answer is right, namely -1 and not +1

Answer 5

Hi,
Do it this way.
i² = (sqrt(-1))²
= -1
The square root symbol means the positive square root, and thus cannot be used the way you are using it with negative numbers.
sqrt (-1*-1) = 1

FE.

Answer 6

The square root of -1 is actually +i or -i. When you start taking square roots of things and not accounting for the two roots, you get into situations like the one you've encountered.