Complex Numbers

OBJECTIVES

•Use the imaginary unit i to write complex numbers

•Add, subtract, and multiply complex numbers

•Use quadratic formula to find complex solutions of quadratic equations

Consider the quadratic equation x2 + 1 = 0. What is the discriminant ?a = 1 , b = 0 , c = 1 therefore the discriminant is02 – 4 (1)(1) = – 4 If the discriminant is negative, then the quadratic

equation has no real solution. (p. 114) Solving for x , gives x2 = – 1

12 x

1xWe make the following definition:

1i

Note that squaring both sides yields

Real NumbersImaginary Numbers

Real numbers and imaginary numbers are subsets of the set of complex numbers.

Complex Numbers

12 i

Definition of a Complex Number

If a and b are real numbers, the number a + bi is a complex number written in standard form.

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

0b

If , the number a + bi is called an imaginary number. A number of the form bi, where , is called a pure imaginary number.

Write the complex number in standard form

81 81 i 241 i 221 i

Equality of Complex Numbers

Two complex numbers a + bi and c + di, are equal to each other if and only if a = c and b = d

Find real numbers a and b such that the equation ( a + 6 ) + 2bi = 6 –5i .a + 6 = 6 2b = – 5a = 0 b = –5/2

dicbia

Addition and Subtraction of Complex Numbers,

If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.

idbcadicbia )()()()(

idbcadicbia )()()()(

Sum:

Difference:

Perform the subtraction and write the answer in standard form.

Ex. 1 ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i

Ex. 2 234188 i

234298 ii

234238 ii

4

Properties for Complex Numbers

• Associative Properties of Addition and Multiplication• Commutative Properties of Addition and Multiplication• Distributive Property of MultiplicationMultiplying complex numbers is similar to multiplying

polynomials and combining like terms. #28 Perform the operation and write the result in standard

form. ( 6 – 2i )( 2 – 3i )F O I L12 – 18i – 4i + 6i2

12 – 22i + 6 ( -1 )6 – 22i

Consider ( 3 + 2i )( 3 – 2i )9 – 6i + 6i – 4i2

9 – 4( -1 )9 + 4 13

This is a real number. The product of two complex numbers can be a real number.

Complex Conjugates and Division

Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers.

( a + bi )( a – bi )a 2 – abi + abi – b 2 i 2

a 2 – b 2( -1 )a 2 + b 2

The product of a complex conjugate pair is a positive real number.

To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

dic

bia

dic

dic

dic

bia

22

2

dc

bdibciadiac

22 dc

iadbcbdac

Perform the operation and write the result in standard form. (Try p.131 #45-54)

i

i

21

76

i

i

i

i

21

21

21

76

22

2

21

147126

iii

41

5146

i

5

520 i

5

5

5

20 i i4

Principle Square Root of a Negative Number,

If a is a positive number, the principle square root of the negative number –a is defined as

.aiiaa

Use the Quadratic Formula to solve the quadratic equation.

9x2 – 6x + 37 = 0 a = 9 , b = - 6 , c = 37What is the discriminant?

( - 6 ) 2 – 4 ( 9 )( 37 )36 – 1332 -1296

Therefore, the equation has no real solution.

9x2 – 6x + 37 = 0 a = 9 , b = - 6 , c = 37

92

379466 2 x

18

1332366 x

18

12966 x i

18

1296

18

6

18

36

3

1 ix i2

3

1