SOLUTION: prove that √ 2/2 + √ 2/2i is a square root of i

Question 1196798: prove that √ 2/2 + √ 2/2i is a square root of i
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

If x is a square root of y, then which rearranges into

Example: 7 is a square root of 49, so which turns into after squaring both sides.

The claim is is a square root of , where

A useful rule of complex numbers is this

I'll leave it to the student to prove this claim.

In this case, and

So,

This verifies that is indeed a square root of

In other words, one solution to is

The other solution is which I'll leave to the reader to prove.

Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.

The square of the modulus of this number, + , is = + = = + = 1. So, the modulus itself is r = = 1. The argument of this number, "a", satisfies tan(a) = = 1. So, the argument is a = = 45°. Hence, the square of this number has the modulus = 1 and the argument 2a = = 90°. It means that the square of the given number is the complex imaginary unit "i".

At this point, the proof is complete.

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