Chapter 11: Complex Numbers

i^2 = -1, so i = sqrt(-1)

z = x + iy

Modulus = |z| = sqrt (x² + y²)

Modulus always gives the magnitude.

 

Argument is the angle theta between the positive real axis and the position vector. It gives the direction of the position vector.

 

4 Cases of calculating the argument.

Case 1: Quadrant 1 (x > 0, y > 0).
Theta is equal to tan^-1(y/x)

Case 2: Quadrant 2 (x < 0, y > 0)

Theta is equal to tan^-1(y/x)

Case 3: Quadrant 3 (x < 0, y < 0)

Theta is equal to  tan^-1(y/x)

Case 4: Quadrant 4 (x > 0, y < 0)

Theta is equal to tan^-1(y/x)

 

The principal argument is usually in the interval ( - ]. The meaning is that -  is not included in the set, while ‘]’  is included.

According to the complex conjugate root theorem, if one of the factors of a cubic equation is a complex number. Then its conjugate has to be definitely the second factor of the cubic equation.

z = x + iy --> Cartesian Form

a = angle in radians, r = |z|

z = r(cos a + sin a) where (r, a) are polar coordinates.

z = re^(ia)

For loci, if < or > (single inequality), then the line is broken/dotted. If <= or >= (double inequality), then the line is solid.

|z| = r

r^2 = x^2 + y^2 (Circumference of a circle)