Complex Numbers

Introduction

What is a complex number?

A complex number is made up of a real number and an imaginary number.

The standard form is a+bi where a is the real term and bi is the imaginary term.

Powers of i

i =

i2 =

i3 =

i4 =

√-1

i i = √-1 √-1 = -1

i i2 = i -1 = -i

i2 i2 = -1 -1 = 1

Practice

Simplify

i5 =

i6 =

i7 =

i8 =

Any power of i that is a multiple of 4 is also one. We can find any power of i by writing it as the product of the highest power that is a multiple of 4 and i, i2, and i3.

More Practice

Simplify

i25 =

i36 =

i11 =

i19 =

Solve quadratic equations

with complex roots Ex.1

Solve quadratic equations

with complex roots Ex.2

Solve quadratic equations

with complex roots Ex.3

Practice

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Operations with complex numbers

Ex. 4 Multiply or divide. Simplify.

a.) b.)

c.) d.)

Operations with complex numbers

Ex. 5 Write in standard form a+bi.

Operations with complex numbers

Addition & Subtraction

(a+bi) + (c+di)

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

(a+bi) – (c+di)

= (a+c) – (b+d)i

Operations with complex numbers

Ex. 6 Find each sum or difference.

a.) (4 – 5i) + (-5 + 8i) =

b.) (-6 + 3i) + (12 – 9i) =

c.) (-10 + 7i) – (5 – 3i) =

d.) (15 – 8i) – (-10 + 4i) + (-25 + 12i) =

Operations with complex numbers

Multiplication

(a+bi)(c+di)

= (ac – bd) + (ad + bc)i

FOI

L

Operations with complex numbers

Ex. 7 Find each product.

a.) (5 + 3i)(2 – 7i) b.) (4 – 5i)2

c.) (3 – i)(-3 + i) d.) (9 – 8i)(9 + 8i)

Operations with complex numbers

Ex. 8

Operations with complex numbers

Ex. 9

Practice

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# 95-106, 109, 110