Section 2.4 Complex Numbers What you should learn How to use the imaginary unit i to write complex...

Section 2.4 Complex Numbers

What you should learn

• How to use the imaginary unit i to write complex numbers

• How to add, subtract, and multiply complex numbers

• How to use complex conjugates to write the quotient of two complex numbers in standard form

• How to find complex solutions to quadratic equations

Real Number System

Natural {1, 2, 3, 4,…}

How many

natural numbers

are there?

Real Number System

Whole

Natural {0, 1, 2, 3, 4,…}

How many whole

numbers are

there?

Real Number System

Integers

Whole

Natural {...-3, -2, -1, 0, 1, 2, 3, …}

How many integers numbers are there?

Real Number System

Rational

Integers

Whole

Natural Fractions

How many rational numbers are there?

0,, bIba

b

a

Real Number System

Rational

Integers

Whole

Natural

How many irrational numbers are there?

e,,2

Irrational

Real Number System

Rational

Integers

Whole

Natural Each set is a subset of the Real Number System.

The union of all these sets forms the real number system.

The number line is our model for the real number system.

Irrational

Real

Numbers

Definition of Square Root

If a2 = n then a is a square root of n.

42 = (4)(4) = 16

4 is a square root of 16

(-4)2 = (-4)(-4) = 16

-4 is a square root of 16

What square root of -16?

Whatever it is it is not on the real number line.

Definition of i

b bi1 i

The number i is such that 1i

221 i

21 i 16 16i 4i

Imaginary Unit

ImaginaryREAL

ComplexComplex

Complex Numbers

bia

i23 i5

2

8

3 i07

Definition of a Complex Number

• If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.

• If b = 0 then the number a + bi = a is a real number.

• If b ≠ 0, then the number a + bi is called an imaginary number.

• A number of the form bi, where b ≠ 0 is called a pure imaginary number.

Examples

16

81

7

i4

i9

7i

If you square a radical you get the radicand

1 i 25 5

12i

2 2

Whenever you have i2 the next turn you will have -1 and no

i.

Equality of Complex numbers

If a + bi = c + di, then a = c and b = d.

yiix 75

7x 5y

Is a negative times a negative always positive?

259

)5)(3( ii 215i 15

Trick question. This is not a negative times a negative.

Example

77 77 ii 27i

7

Example

105 525 ii

25 2i

25

Example

215 215 i

30i

Example

2

32

2

32

i

i

16

4

Cancel the i

factor

Add

)74()53( ii

7 i2

Collect like terms.

Subtract

9 i13

)204()75( ii

ii 20475

First distribute the negative sign.

Now collect like terms.

Multiplication

)23( i )54( i

F O I L

i1512 i8 210i10712 i

i722

Simplify each expression. Express your answer in form.

)73)(45( ii228123515 iii

Combine like terms.

282315 i i2343

Recall i2=-1

F-O-I-L

Combine like terms.

Write in the form

i23

26

.bia

i

i

23

23249

)23(26

i

i

13

)23(26 i )23(2 i

Multiply by the conjugate factor.

2i46

Powers of i0i11i2i3i

Anything other than 0 raised to the 0 is 1.

Anything raised to the 1 is itself.i12 i1

iii 23 i)1( ii

Simplify as much as possible.

4i 2 2i i ( 1)( 1) 1

30i 4 7 2( )i i (1)( 1) 1

Use the Quadratic Formula

03769 2 xx9a

6b37c

a

acbbx

2

42

)9(2

)37)(9(4)6()6( 2 x

18

1332366

18

12966

18

366 i

i18

36

18

6 i2

3

1

Homework Section 2.41-79, 83 odd