Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex NumbersDefinitions, operations, and calculator shortcuts

Definition

The lowercase letter “i” is equal to the square root of negative one.

1i

Examples

9 64216x

Examples

9 64216x

i3

Examples

9 64216x

i3 i8

Examples

9 64216x

i3 i8 xi4

Powers of i

i2 = -1

i3 = -i

i4 = 1

http://supermanjaviolivares.iespana.es/poster1.jpg

Powers of i

i2 = -1

i3 = -i

i4 = 1

Higher powers of i?

http://supermanjaviolivares.iespana.es/poster1.jpg

Let’s exploit 74i

14 i

Let’s exploit 74i 2184 ii

14 i

Let’s exploit 74i 2184 ii

2181 i

14 i

Let’s exploit 74i 2184 ii

2181 i21 i

14 i

Let’s exploit 74i 2184 ii

2181 i21 i

2i

14 i

Let’s exploit 74i 2184 ii

2181 i21 i

2i 1

14 i

Complex NumbersWhat is a complex number?

A number with a real and an imaginary component

Standard form:a + bi

Adding and Subtractingii 23 ii 97

ii 3265 ii 2752

Adding and Subtractingii 23 ii 97

ii 3265 ii 2752

i5

Adding and Subtractingii 23 ii 97

ii 3265 ii 2752

i5 i2

Adding and Subtractingii 23 ii 97

ii 3265 ii 2752

i5 i2

i93

Adding and Subtractingii 23 ii 97

ii 3265 ii 2752

i5 i2

i93 i39

FOIL-ing Complex Numbers Remember: i2=-1 Example:

ii 3265

FOIL-ing Complex Numbers Remember: i2=-1 Example:

ii 3265 218121510 iii

FOIL-ing Complex Numbers Remember: i2=-1 Example:

ii 3265 218121510 iii

218310 ii

FOIL-ing Complex Numbers Remember: i2=-1 Example:

ii 3265 218121510 iii

218310 ii 18310 i

FOIL-ing Complex Numbers Remember: i2=-1 Example:

ii 3265 218121510 iii

218310 ii 18310 ii328

Conjugates of Complex Numbers

The conjugate of a+bi is a-bi.

The conjugate of a-bi is a+bi.

Dividing Complex NumbersWe cannot leave i in the

denominator of a fraction. (Why not?)

Multiply the numerator and denominator of the fraction by the conjugate of the denominator.

Sample Problem

i

i

32

65

Sample Problem i

i

32

65

ii

32

32

Sample Problem i

i

32

65

ii

32

32

2

2

9664

18121510

iii

iii

Sample Problem i

i

32

65

ii

32

32

2

2

9664

18121510

iii

iii

94

182710

i

Sample Problem i

i

32

65

ii

32

32

2

2

9664

18121510

iii

iii

94

182710

i

13

278 i

Sample Problem i

i

32

65

ii

32

32

2

2

9664

18121510

iii

iii

94

182710

i

13

278 i i

13

27

13

8

Graphing Calculator

Press MODE.Select a+bi mode.

The i key is the decimal key.

Sample Problem

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator.

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

Press ENTER.

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

Press ENTER. 1.7-1.1i

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

Press ENTER. 1.7-1.1i

Prefer fractions? Press MathFrac.

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

Press ENTER. 1.7-1.1i

Prefer fractions? Press MathFrac. 17/10-11/10i

i

i

3

54

Sample Problem

Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)

Press ENTER. 1.7-1.1i

Prefer fractions? Press MathFrac. 17/10-11/10i

i

i

3

54

IMPORTANT NOTE:

You must WRITE the answer as

i

10

11

10

17

Practice Questions1. (3 + 2i) + (4 – 5i)2. (5 – 6i) – (3 – 2i)3. (2 + 4i)(3 – 5i)4. (4 + i)2 5. 4 + 3i

2 – 3i