Complex Numbers - MATH 160, Precalculust2010125/MATH30203/complex.pdf · Complex Numbers MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan

Complex NumbersMATH 160, Precalculus

J. Robert Buchanan

Department of Mathematics

Fall 2011

J. Robert Buchanan Complex Numbers

Objectives

In this lesson we will learn to:use the imaginary unit i to write complex numbers,add, subtract, and multiply complex numbers,use complex conjugates to write the quotient of twocomplex numbers in standard form,find complex solutions to quadratic equations.


Motivation

We would like to be able to describe all the solutions of allpolynomial equations, yet a very simple one has no realnumber solutions.

x2 + 1 = 0x2 = −1

Since x2 ≥ 0 for all real numbers x , there is no real solution tothis equation.

Thus we must expand our number system by using theimaginary unit,

i =√−1.

Thus i2 = −1 and the solutions to the equation above can bewritten as x = i and x = −i .


Motivation

We would like to be able to describe all the solutions of allpolynomial equations, yet a very simple one has no realnumber solutions.

x2 + 1 = 0x2 = −1

Since x2 ≥ 0 for all real numbers x , there is no real solution tothis equation.

Thus we must expand our number system by using theimaginary unit,

i =√−1.

Thus i2 = −1 and the solutions to the equation above can bewritten as x = i and x = −i .


Complex Numbers

DefinitionIf a and b are real numbers, the number a + bi is a complexnumber, and it is said to be written in standard form. If b = 0,the number a + bi = a is a real number. If b 6= 0, the numbera + bi is called an imaginary number. A number of the form biwith b 6= 0 is called a pure imaginary number.

The real numbers R are a subset of the complex numbers C.


Complex Numbers

DefinitionIf a and b are real numbers, the number a + bi is a complexnumber, and it is said to be written in standard form. If b = 0,the number a + bi = a is a real number. If b 6= 0, the numbera + bi is called an imaginary number. A number of the form biwith b 6= 0 is called a pure imaginary number.

The real numbers R are a subset of the complex numbers C.


Arithmetic of Complex Numbers (1 of 2)

EqualityTwo complex numbers a + bi and c + di , written in standardform, are equal to each other

a + bi = c + di

if and only if a = c and b = d .

Addition and SubtractionIf a + bi and c + di are two complex numbers, written instandard form, their sum and difference are defined as follows.

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



EqualityTwo complex numbers a + bi and c + di , written in standardform, are equal to each other

a + bi = c + di

if and only if a = c and b = d .

Addition and SubtractionIf a + bi and c + di are two complex numbers, written instandard form, their sum and difference are defined as follows.

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



Identities and InversesThe additive identity element in the complex number systemis 0 = 0 + 0i . The additive inverse of the complex numbera + bi is −a− bi .


Examples

Perform the addition or subtraction as appropriate and write theresult in standard form.

(13− 2i) + (−5 + 6i) =

8 + 4i

(3 + 2i)− (6 + 13i) =

−3− 11i

(8 +√−18)− (4 + 3

√2 i) =

4

25 + (−10 + 11i) + 15i =

15 + 26i


Examples


(13− 2i) + (−5 + 6i) = 8 + 4i(3 + 2i)− (6 + 13i) =

−3− 11i

(8 +√−18)− (4 + 3

√2 i) =

4

25 + (−10 + 11i) + 15i =

15 + 26i


Examples


(13− 2i) + (−5 + 6i) = 8 + 4i(3 + 2i)− (6 + 13i) = −3− 11i

(8 +√−18)− (4 + 3

√2 i) = 4

25 + (−10 + 11i) + 15i = 15 + 26i


Multiplication

Multiplication of complex numbers is carried out using the FOILMethod.

(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i2

= (ac) + (ad + bc)i − (bd)

= (ac − bd) + (ad + bc)i

The usual properties of arithmetic hold for complex numbers:associative propertycommutative propertydistributive property


Multiplication

Multiplication of complex numbers is carried out using the FOILMethod.

(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i2

= (ac) + (ad + bc)i − (bd)

= (ac − bd) + (ad + bc)i

The usual properties of arithmetic hold for complex numbers:associative propertycommutative propertydistributive property


Examples

Multiply the numbers below and write the result in standardform.

(13− 2i)(−5 + 6i) =

−53 + 88i

(3 + 2i)(6 + 13i) =

−8 + 51i

(8 + 3√

2i)(4 + 3√

2 i) =

14 + 36√

2i

(−10 + 11i)(15i) =

−165− 150i


Examples


(13− 2i)(−5 + 6i) = −53 + 88i(3 + 2i)(6 + 13i) =

−8 + 51i

(8 + 3√

2i)(4 + 3√

2 i) =

14 + 36√

2i

(−10 + 11i)(15i) =

−165− 150i


Examples


(13− 2i)(−5 + 6i) = −53 + 88i(3 + 2i)(6 + 13i) = −8 + 51i

(8 + 3√

2i)(4 + 3√

2 i) = 14 + 36√

2i(−10 + 11i)(15i) = −165− 150i


Complex Conjugates

DefinitionThe complex conjugate of the complex number a + bi is thecomplex number a− bi .

Note: (a + bi)(a− bi) = a2 + b2 a real number.


Complex Conjugates

DefinitionThe complex conjugate of the complex number a + bi is thecomplex number a− bi .

Note: (a + bi)(a− bi) = a2 + b2 a real number.


Quotients of Complex Numbers

The quotient of two complex numbers can be written instandard form by multiplying both numerator and denominatorby the complex conjugate of the denominator.

a + bic + di

=a + bic + di

c − dic − di

=(a + bi)(c − di)(c + di)(c − di)

=(ac + bd) + (bc − ad)i

c2 + d2


Example

Perform the indicated operation and write the result in standardform.

2 + i3− 2i

− 1 + i3 + 8i

=

(2 + i)(3 + 8i)(3− 2i)(3 + 8i)

− (1 + i)(3− 2i)(3− 2i)(3 + 8i)

=−2 + 19i25 + 18i

− 5 + i25 + 18i

=−2 + 19i − (5 + i)

25 + 18i

=−7 + 18i25 + 18i

=(−7 + 18i)(25− 18i)(25 + 18i)(25− 18i)

=149 + 576i625 + 324

=149949

+576949

i


Example

Perform the indicated operation and write the result in standardform.

2 + i3− 2i

− 1 + i3 + 8i

=(2 + i)(3 + 8i)(3− 2i)(3 + 8i)

− (1 + i)(3− 2i)(3− 2i)(3 + 8i)

=−2 + 19i25 + 18i

− 5 + i25 + 18i

=−2 + 19i − (5 + i)

25 + 18i

=−7 + 18i25 + 18i

=(−7 + 18i)(25− 18i)(25 + 18i)(25− 18i)

=149 + 576i625 + 324

=149949

+576949

i


Complex Solution to Quadratic Equations

When using the Quadratic Formula to solve a quadraticequation, we can use complex numbers and the imaginary rootto express the solutions.

0 = ax2 + bx + c ⇐⇒ x =−b ±

√b2 − 4ac

2a

ExampleUse the Quadratic Formula to solve the following equation.

0 = 2x2 − 5x + 7

x =−(−5)±

√(−5)2 − 4(2)(7)

2(2)

=5±√

25− 564

=5±√−31

4=

5±√

31 i4




0 = ax2 + bx + c ⇐⇒ x =−b ±

√b2 − 4ac

2a


0 = 2x2 − 5x + 7

x =−(−5)±

√(−5)2 − 4(2)(7)

2(2)

=5±√

25− 564

=5±√−31

4=

5±√

31 i4




0 = ax2 + bx + c ⇐⇒ x =−b ±

√b2 − 4ac

2a


0 = 2x2 − 5x + 7

x =−(−5)±

√(−5)2 − 4(2)(7)

2(2)

=5±√

25− 564

=5±√−31

4=

5±√

31 i4


Homework

Read Section 2.4.Exercises: 1, 5, 9, 13, . . . , 81, 85


Documents

Complex Numbers - MATH 160, Precalculust2010125/MATH30203/complex.pdf · Complex Numbers MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan