How do we find the polar representation of a)1-i ;and b) -1 + i?

Answer 1

In each case the modulus of the complex number is sqrt(1+1), i.e. sqrt(2). So a) is sqrt(2)[1/sqrt(2) -i/sqrt(2)]. We need the angle whose cosine is 1/sqrt(2) and whose sine is -1/sqrt(2). This angle is -pi/4, so the polar representation is
sqrt(2)[cos(-pi/4) + sin(-pi/4)]
i.e. (r, theta) is [sqrt(2), (-pi/4)]

Similarly, in case b) the polar representation is [sqrt(2), (3pi/4)]

Answer 2

Think of a square. Put in the axe the points (1,0) and (0,-1) for the first question. The module will be the diagonal of the square, the angle, -45

I think that you can do the second by yourself, use this time the points (-1,0) and (0,1)

Later

Ana

PS: Use the Pythagoras theorem to calculate the diagonal of the square

Answer 3

Cartesian form
z = x + i*y

Polar form
z = r*e^i*(theta)

r = sqrt(x^2 + y^2)
theta = tan^-1(y/x)

For (a) x=1 and y =-1, for (b) x=-1 and y =1

Let me know if you need further clarification.