Complex Numbers

If a and b are real numbers and i is the imaginary

unit, then a + bi is called a complex number.

▪ a is the real part

▪ bi is the imaginary part.

Definition of Complex Numbers

Definition: The number i, called the imaginary unit, is the number such that

i = ____√-1__ and i2 = __-1______

Powers of i

Let a + bi and c + di be complex numbers.1. Add/Subtract the Real parts.

2. Add/Subtract the Imaginary parts. (3 + 4i) + (2 - i) = (3 + 2) + (4i - i) = (5 + 3i)

(7 + i) - (3 - i) = (7 - 3) + i(1 - (-1)) = 4 + 2i

Let a + bi and c + di be complex numbers.1. Multiply the binomials.

2. Convert i2 to -1 and add the like terms.(3 + 2i)(4 + 5i) = (3 × 4) + (3 × (5i)) + ((2i) × 4) + ((2i) × (5i))                             = 12 + 15i + 8i + 10i²                             = 12 + 23i -10 (Remenber that 10i² = 10(-1) = -10)                             = 2 + 23i

 Therefore, (3 + 2i)(4 + 5i) = 2+23i

A complex number z is a number of the form z = x + yi. Its conjugate is a number of the form = x - yi. The complex number and its conjugate have the same real part. Re(z) = Re( ). The sign of the imaginary part of the conjugate complex number is reversed. Im(z) = - Im( ).

The conjugate numbers have the same modulus and opposite arguments.|z| = | |, arg(z) = - arg( ). Any complex number multiplied by its complex conjugate is a real number, equal to the square of the modulus of the complex numbers z. z = (x + yi)(x - yi) = x2+ y2 = |z|2

Division Of Complex Numbers

Let a + bi and c + di be complex numbers.

dic

bia

Multiply the numerator and denominator of the fraction by the Complex Conjugate of the Denominator.

Then to perform the operation

2+6i x 4-i = (2+6i) (4-i) = 14+22i = 14 + 22 i 4+i 4-i (4+i) (4-i) 17 17 17

RealAxis

Imaginary Axis

yixz

The magnitude or modulus of z denoted by z is the distance from the origin to the point (x, y).

yx

The angle formed from the real axis and a line from the origin to (x, y) is called the argument of z, with requirement that 0 < 2.

modified for quadrant and so that it is between 0 and 2

yixz Let a complex number be Z such that :

22 yxz

x

y1tan

z

Modulus and Argument of Complex Numbers

The Principal Argument is between - and

RealAxis

Imaginary Axis

y

x

z = r 13

3

1tan 1 but in Quad

II

6

5

6

5arg

6

5arg

principalz

The unique value of θ such that –π < θ <π is called principle value of the argument.

The magnitude or modulus of z is the same as r.

We can take complex numbers given asand convert them to polar form :

RealAxis

Imaginary Axis

y

x

z = r

irrz sincos

Plot the complex number: iz 3

Find the polar form of this number.

13

2413 22r

3

1tan 1 but in Quad

II 6

5

6

5sin

6

5cos2

iz

sincos ir factor r out

yixz

sinry cosrx