​ Consider the complex number z = 1. Write z in a + ib form. ​2. In the figure radius of the circle is 1.

Question

Consider the complex number z = \(\frac{1+i}{1-i}\)

1. Write z in a + ib form.

2. 

In the figure radius of the circle is 1. Write the polar form of the complex number represent by the points P and Q. 

3. Find the square root of i. 

Answer 1

1. \(\frac{1+i}{1-i}\) x \(\frac{1+i}{1-i}\) = \(\frac{1 + 2i - 1}{2}\) = i

2. Polar form of the point P is 1(cos\(\frac{\pi}{2}\)+isin\(\frac{\pi}{2}\))

Polar form of the point Q is 1(cos\(\frac{\pi}{2}\)+isin\(\frac{\pi}{2}\))

3. i = 0 + i ⇒ √i = x + iy ⇒ i = x² + y² + 2xyi x² + y² = 0; 2xy = 1

(x² + y²)² = (x² – y²)² + 4x²y²

(x² + y²)² = 0 + (1)² = 1

x² + y² = 1; x² + y² = 0

x = ± \(\frac{1}{\sqrt 2}\); y =  ± \(\frac{1}{\sqrt 2}\)

Rots are \(\frac{1}{\sqrt 2}\) + i\(\frac{1}{\sqrt 2}\); - \(\frac{1}{\sqrt 2}\) - i\(\frac{1}{\sqrt 2}\)