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College Algebra

Linear and Quadratic Functions(Chapter2)

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2

Section(2-4)

Complex Numbers

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ObjectivesAfter completing this Section, you should be able to:

1. Take the principle square root of a negative number.

2. Write a complex number in standard form.

3. Add and subtract complex numbers.

4. Multiply complex numbers.

5. Divide complex numbers.

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The equation x2 = - 1 has no real number solution.

To remedy this situation, we define a new number that solves this equation, called the imaginary unit, which is not a real number.

The solution to x2 = - 1 is the imaginary unit I where i2 = - 1, or

1i

Imaginary Unit

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1

12

i

i

The imaginary unit i is defined as

Example

ii 99

81181

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Standard Form of Complex Numbers

• A complex number has a real part & an imaginary part.

• Standard form is:

bia

Real part Imaginary part

Example: 5+4iExample: 5+4i

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• The set of all numbers in the form a+bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number a+bi.

Standard Form of Complex Numbers

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Complex Number System

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a + bi = c + di

if and only if a = c and b = d

In other words, complex numbers are equal if and only if there real and imaginary parts are equal.

Equality of Complex Numbers

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Addition with Complex Numbers

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

Example:

(2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i

= 1 + 10i

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Subtraction with Complex Numbers

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

Example

(3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i

= 2 + 3i

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a bi c di ac bd ad bc i Multiplying Complex Numbers

Step 1:  Multiply the complex numbers in the same manner as polynomials.

 

 

Step 2:  Simplify the expression.

Add real numbers together and imaginary numbers together. Whenever you have an    , use the definition and replace it with -1.

   

Step 3:  Write the final answer in standard form.

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Example1: Multiply using the distributive property

4 3 1 4 i i

Multiplying Complex Numbers

4 1 4 4 3 1 3 4( ) ( ) ( ) ( )i i i i

4 16 3 12 2i i i

4 19 12 1i ( )

4 19 12i

8 19i

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Example2: Multiply using the distributive property

Multiplying Complex Numbers

          

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Example 1

Perform the indicated operation, writing the result in standard form.

-)5 + 7i-) - (11 - 6i(

Combine the real and imaginary parts :

-) 5-)-11) + ((7-)-6((i = (-5+11) + (7+6)i

= 6+13i

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Example 2: Multiply

i

i

iii

ii

55

352

362

)31)(2(2

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Example 2 (continued)

i

i

iii

ii

55

352

362

)31)(2(2

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Example 2 (continued)

i

i

iii

ii

55

352

362

)31)(2(2

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i

i

iii

ii

55

352

362

)31)(2(2

Example 2 (continued)

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If z = a + bi is a complex number, then its conjugate, denoted by

, is defined asz

z a bi a bi

TheoremThe product of a complex number and its conjugate is a nonnegative real number. Thus, if z = a + bi, then

zz a b2 2

Conjugate

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Division with Complex Numbers

To divide by a complex number, multiply the dividend (numerator) and divisor (denominator) by the conjugate of the divisor.

Example1: 1 43

ii

1 43

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ii

ii

3 12 49 1

2i i i

7 1110

i

710

1110

i

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Example 2

Divide:

i1

2

23

ii

i

i

ii

i

ii

12

2211

22

1

221

1

1

2

1

2

2

Example 2 (continued)

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Example 2 (continued)

ii

i

i

ii

i

ii

12

2211

22

1

221

1

1

2

1

2

2

25

ii

i

i

ii

i

ii

12

2211

22

1

221

1

1

2

1

2

2

Example 2 (continued)

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Principal Square Root of a Negative Number

For any positive real number b, the principal square root of the negative number -b is defined by

(-b) = ib

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Example 1: Simplify

1212

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916

2

i

ii

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Example 1 (continued)

1212

34

916

2

i

ii

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1212

34

916

2

i

ii

Example 1 (continued)

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In the complex number system, the solutions of the quadratic equation where a, b, and c are real numbers and a 0, are given by the formula

02 cbxax

xb b ac

a 2 4

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Since we now have a way of evaluating the square root of a negative number, there are now no restrictions placed on the quadratic formula.

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0122 2 xxExample: Find all solutions to the equation real or complex.

a = 2, b = -2, c = 1

b ac2 24 2 4 2 1 4 8 4 ( ) ( )( )

x ( )( )

2 42 2

2 24

i 12

12

i

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Property of the square root of negative numbers

• If r is a positive real number, then

r ri

Examples:

3 3i 4 4i i2

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then,1- If i

12 i

ii 3

14 i

ii 5

16 i

ii 7

18 i

*For larger exponents, divide the exponent by 4, then use the

remainder as your exponent instead.

Example: ?23 i3 ofremainder a with 5

4

23

.etcii - which use So, 3

ii 23

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Examples2)3( 1. i

22 )3(i)3*3(1

)3(13

26103 Solve 2. 2 x

363 2 x122 x

122 x

12ix 32ix

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The Complex plane

Imaginary Axis

Real Axis

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Graphing in the complex plane

i34 .

i52 .i22 .

i34

.

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Adding and Subtracting(add or subtract the real parts, then add or

subtract the imaginary parts)

Ex: )33()21( ii )32()31( ii

i52

Ex: )73()32( ii )73()32( ii

i41

Ex: )32()3(2 iii )32()23( iii

i21

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MultiplyingTreat the i’s like variables, then change any i2 to -1

Ex :)3( ii 23 ii

)1(3 i

i31

Ex: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

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i

i

i

iEx

21

21*

21

113 :

)21)(21(

)21)(113(

ii

ii

2

2

4221

221163

iii

iii

)1(41

)1(2253

i

41

2253

i

5

525 i

5

5

5

25 i

i 5

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Absolute Value of a Complex Number

• The distance the complex number is from the origin on the complex plane.

• If you have a complex number

the absolute value can be found using:

) ( bia

22 ba

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Examples

1 .i52

22 )5()2(

254 29

2. i622 )6()0(

360

366

Which of these 2 complex numbers is closest to the origin? -2+5i

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