Evaluate:

sqrt(i)+sqrt(-i).

o There are at least 3 distinct methods

Using the model of the complex plane, both i and -i have magnitude 1, so the square root of each will also have magnitude 1.
The vector for i is pi/2 and for -i it is -pi/2.
The square root will have half the angle for the vector.  But taking a square root gives two solutions, a plus or minus value.  In the complex plane, the effect of negation is to reflect the point across the origin, or rotate the vector 180 degrees.

sqrt(i) = ±(√2/2 + √2/2 i)
sqrt(-i)= ±(√2/2 - √2/2 i)

sqrt(i)+sqrt(-i) =
√2/2 + √2/2 i + √2/2 - √2/2 i =   √2
-√2/2 - √2/2 i + √2/2 - √2/2 i = -√2 i
√2/2 + √2/2 i - √2/2 + √2/2 i =   √2 i
-√2/2 - √2/2 i - √2/2 + √2/2 i = -√2

So I'm getting 4 solutions:  ±√2 and ±√2 i
Are all 4 valid?

Posted by Larry on 2023-06-04 09:55:05