The imaginary unit, i, is defined by i = -1.

A complex number is a number that can be written in the form a + bi , where a and b are real numbers, and i is the imaginary unit. a is called the real part of a + bi and b is called the imaginary part.

Powers of i

i = i

i = -1

i = i· i = -1 · i = -i

i = (i)= (-1) = 1

i = i· i = 1 · i = i

i = i· i = 1 · -1 = -1

i = i· i = 1 · -i = -i

i = = (i) = 1 = 1

The powers if i continue to cycle through the numbers i, -1, -i, l. If k is a multiple of 4, i = 1.

Example. Evaluate: a) i b) i

Solution. a) i = i · i = 1 · -i = -i

b) i = i· i = 1 · i = i

Definition of Addition, Subtraction, and Multiplication

For complex numbers a + bi and c + di:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) – (c + di) = (a – c) + (b – d)i

(a + bi)(c + di) = (ac – bd) + (ad + bc)i

Example. a) (3 – 2i) + (-4 + 7i)

b) (7 + 3i) – (8 – 7i)

c) (1 + i)(-3 + 5i)

Solution. a) (3 – 2i) + (-4 + 7i) = (3 + (-4)) + (-2 + 7)i

= -1 + 5i

b) (7 + 3i) – (8 – 7i) = (7 – 8) + (3 – (-7))i

= -1 + 10i

c) Instead of using the definition, just use FOIL as you would in

multiplying two binomials

(1 + i)(-3 + 5i) = 1(-3) + 1(5i) + i(-3) + i(5i) Multiply using FOIL

= -3 + 5i – 3i + 5i

= -3 + 5i – 3i + 5(-1) Substitute –1 for i

= -3 + 5i – 3i – 5

= -8 + 2i Combine like terms

The conjugate of the complex number a + bi is a – bi.

For any two complex conjugates a + bi and a – bi: (a + bi)(a – bi) = a + b

To divide two complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

Example. Divide:

Solution: · = Multiply, using FOIL in the numerator;

use conjugate product rule in denominator

= Substitute –1 for i

=

=

=