Lesson 2.4

Read: Pages 139-143

Page 137: #1-73 (EOO)

Complex Numbers

Objectives

Students will use the imaginary unit i to write complex numbers, add subtract and multiply complex numbers, use complex conjugates to write the quotient of two complex numbers in standard form, and plot complex numbers in the complex plane.

abi

The Imaginary Unit

1i

𝑖2=−1

The set of real numbers is a subset of complex numbers.

RealNumbers

ImaginaryNumbers

ComplexNumbers

Addition and Subtraction of Complex Numbers

If are two complex numbers written in standard form, their sum and difference are defined as follows.

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

Difference: (a + bi) - (c + di) = (a - c) + (b - d)i

)32()3( ii

)5()32(3 ii

Perform the indicated operation.

444

Multiplying Complex Numbers

= (2i)(4i)

= 8

= 8

= -8

= 8 + 6i - 4i -3

= 8 + 6i - 4i -3

= 8 + 3 + 6i - 4i

= 11 + 2i

Complex Conjugates

(3 + 2i)(3 – 2i) = 9 - 6i + 6i -4

= 9 – 4(-1)

= 9 + 4

= 13

The product of two complex numbers can be a real number. This occurs with pairs of numbers of the forms a + bi and a – bi, called complex conjugates

Plotting Complex Numbers

The Complex Number Plane

Plot each number in the Complex Number Plane

a. -4 + 2i

b. 2 – 3i

c. 3 or 3 + 0i

d. 4i or 0 + 4i

e. 3 + 4i