mth 306 spring term 2007 - college of science, oregon ...leejoh/306h_001_s2007/...... ..., 5, 4, 3,...

$: MTH 306 Spring Term 2007 - College of Science, Oregon ...leejoh/306H_001_S2007/...... ..., 5, 4, 3, 2, 1,0,1,2,3,4,5,... . rational numbers (fractions): numbers of the form p/q where$
MTH 306 Spring Term 2007Lesson 1

John Lee

Oregon State University

(Oregon State University) 1 / 26

Complex Numbers, Variables, and Functions

Goals:

Be able to add, subtract, multiply, and divide complex numbers

Represent complex numbers as points or vectors in the complex planeand illustrate addition and subtraction with the parallelogram (ortriangle) law

Learn the algebraic properties of absolute values and complexconjugates given in the lesson

Write equations or inequalities that de�ne any circle, disk, or theinterior or exterior of a disk in the complex plane

Learn the terms real part, imaginary part, and complex conjugate of acomplex number and be able to illustrate them geometrically

Express complex numbers in polar form and use polar forms tocalculate powers and roots of complex numbers


The Complex Number System

The number systems you use today evolved over several thousand years.You should be familiar will most of the kinds of numbers listed below (inthe order of their historical development).

natural numbers: 1, 2, 3, 4, 5, ... .

integers: ..., �5,�4,�3,�2,�1, 0, 1, 2, 3, 4, 5, ... .rational numbers (fractions): numbers of the form p/q where pand q are integers and q 6= 0.real numbers: numbers which can be represented in decimal formwith a possibly in�nite number of decimal digits following the decimalpoint.

complex numbers: numbers of the form a+ ib where a and b arereal numbers and i , called the imaginary unit, satis�es i2 = �1.




natural numbers: 1, 2, 3, 4, 5, ... .

integers: ..., �5,�4,�3,�2,�1, 0, 1, 2, 3, 4, 5, ... .

rational numbers (fractions): numbers of the form p/q where pand q are integers and q 6= 0.real numbers: numbers which can be represented in decimal formwith a possibly in�nite number of decimal digits following the decimalpoint.





natural numbers: 1, 2, 3, 4, 5, ... .

integers: ..., �5,�4,�3,�2,�1, 0, 1, 2, 3, 4, 5, ... .rational numbers (fractions): numbers of the form p/q where pand q are integers and q 6= 0.

real numbers: numbers which can be represented in decimal formwith a possibly in�nite number of decimal digits following the decimalpoint.





natural numbers: 1, 2, 3, 4, 5, ... .

integers: ..., �5,�4,�3,�2,�1, 0, 1, 2, 3, 4, 5, ... .rational numbers (fractions): numbers of the form p/q where pand q are integers and q 6= 0.real numbers: numbers which can be represented in decimal formwith a possibly in�nite number of decimal digits following the decimalpoint.



Why Were New Number Systems Developed?

Answer: To solve increasingly sophisticated problems that wereencountered as human societies became increasingly complex.

For example, almost no quadratic equations can be solved in the rationalnumber system.

What is the most famous example here?

Which quadratic equations can be solved in the rational numbersystem?

What if you want to solve all quadratic equations?Answer: You need the complex number system. Read on!







What if you want to solve all quadratic equations?Answer:

You need the complex number system. Read on!







What if you want to solve all quadratic equations?

Answer:








What if you want to solve all quadratic equations?Answer:








What if you want to solve all quadratic equations?Answer: You need the complex number system. Read on!


The Quadratic Formula

Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.

The QF yields real number solutions only if b2 � 4ac � 0. Why onlyif?

What can you do if b2 � 4ac < 0?Answer:Invent the complex number system.



Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.

The QF yields real number solutions only if b2 � 4ac � 0.

Why onlyif?




Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.





Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.


What can you do if b2 � 4ac < 0?

Answer:

Invent the complex number system.



Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.


What can you do if b2 � 4ac < 0?Answer:

Invent the complex number system.



Ifax2 + bx + c = 0, a 6= 0,

then

x =�b�

pb2 � 4ac2a

.




The Fundamental Theorem of Algebra

Once complex numbers are available, all quadratic equations can besolved.

The complex number system is even more powerful when it comes tosolving equations.

Every polynomial of degree � 1 with complex coe¢ cients can besolved in the complex number system. That is, all the roots of anysuch polynomial equation are complex numbers.

What does solve mean?


The Misnomer of Imaginary Numbers

Historical confusion preceded a clear understanding of the complexnumber system

The confusion led to unfortunate terminology that persists to thisday. For example:

The complex number �3+ 4i has real part the real number �3Imaginary part the real number 4And i is the imaginary unitThere is nothing �imaginary�about the imaginary part of a complexnumber

Nor is the imaginary unit somehow mathematically suspect





The complex number �3+ 4i has real part the real number �3

Imaginary part the real number 4And i is the imaginary unitThere is nothing �imaginary�about the imaginary part of a complexnumber






The complex number �3+ 4i has real part the real number �3Imaginary part the real number 4

And i is the imaginary unitThere is nothing �imaginary�about the imaginary part of a complexnumber






The complex number �3+ 4i has real part the real number �3Imaginary part the real number 4And i is the imaginary unit

There is nothing �imaginary�about the imaginary part of a complexnumber






The complex number �3+ 4i has real part the real number �3Imaginary part the real number 4And i is the imaginary unitThere is nothing �imaginary�about the imaginary part of a complexnumber



The Field of Complex Numbers

The complex numbers behave in many respects just like the more familiarrational or real number systems:

All three systems are �elds

Each has a number zero such that 0+ a = a for all aEach has a number one such that 1 � a = a for all aEach number a has an additive inverse, �a, such that a+ (�a) = 0Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic




All three systems are �eldsEach has a number zero such that 0+ a = a for all a

Each has a number one such that 1 � a = a for all aEach number a has an additive inverse, �a, such that a+ (�a) = 0Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic




All three systems are �eldsEach has a number zero such that 0+ a = a for all aEach has a number one such that 1 � a = a for all a

Each number a has an additive inverse, �a, such that a+ (�a) = 0Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic




All three systems are �eldsEach has a number zero such that 0+ a = a for all aEach has a number one such that 1 � a = a for all aEach number a has an additive inverse, �a, such that a+ (�a) = 0

Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic




All three systems are �eldsEach has a number zero such that 0+ a = a for all aEach has a number one such that 1 � a = a for all aEach number a has an additive inverse, �a, such that a+ (�a) = 0Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1

The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic




All three systems are �eldsEach has a number zero such that 0+ a = a for all aEach has a number one such that 1 � a = a for all aEach number a has an additive inverse, �a, such that a+ (�a) = 0Each number a 6= 0 has a multiplicative inverse, a�1, such thata � a�1 = 1The familiar rules of arithmetic hold �numbers in any �eld can beadded, subtracted, multiplied, and divided according to the usual rulesof arithmetic


i has period 4

i = i i2 = �1 i3 = �i i4 = 1i5 = i i6 = �1 i7 = �i i8 = 1

and so on where the powers of i repeat in a cycle of length 4.

Consequently after any of the usual arithmetic operations are performed,the resulting expressions can be simpli�ed and put in the standard form

(real) + i(real)


ExampleExpress in standard form:

(a) 2+ 3i + (�6+ 7i)(b) 2+ 3i � (�6+ 7i)(c) (2+ 3i) (�6+ 7i)

(d)�6+ 7i2+ 3i


Rationalization into Standard Form

The process used to carry out division in (d) is called rationalization.

Given a complex number c = a+ ib with a and b realthe complex number c̄ = a� ib is called the complex conjugate of c .

To puta0 + ib0

a+ ib

in standard form multiply numerator and denominator by c̄ = a� ib, thecomplex conjugate of the denominator.


Rationalization into Standard Form

The process used to carry out division in (d) is called rationalization.

Given a complex number c = a+ ib with a and b realthe complex number c̄ = a� ib is called the complex conjugate of c .To put

a0 + ib0

a+ ib

in standard form multiply numerator and denominator by c̄ = a� ib, thecomplex conjugate of the denominator.


Vocabulary

If c = a+ ib is a complex number written in standard form so that a andb are real, then

a is called the real part of c

b is called the imaginary part of c .Brie�y

a = Re (c) and b = Im (c) .

The absolute value (modulus or length) of a complex numberc = a+ ib, denoted jc j , is the real number

jc j =pa2 + b2.

Two complex numbers are equal (by de�nition) when they have thesame real and imaginary parts.


Vocabulary


a is called the real part of cb is called the imaginary part of c .

Brie�ya = Re (c) and b = Im (c) .


jc j =pa2 + b2.



Vocabulary


a is called the real part of cb is called the imaginary part of c .Brie�y

a = Re (c) and b = Im (c) .


jc j =pa2 + b2.



What About Inequalities among Complex Numbers?

There is no way to de�ne a notion of inequalityamong complex numbers that preservesall the familiar rules of inequalities.

In general, it makes no sense to say one complex number is greater than orless than some other complex number.


Handy Algebraic Properties of Absolute Values andConjugates

For complex numbers z and w

jz j � 0

jzw j = jz j jw j�� zw

�� = jz jjw j w 6= 0

jz j2 = zz̄z � w = z̄ � w̄zw = z̄ w̄� zw

�= z̄

w̄ w 6= 0z = z̄ if and only if z is real




jz j � 0jzw j = jz j jw j

�� zw

�� = jz jjw j w 6= 0


�= z̄





jz j � 0jzw j = jz j jw j�� zw

�� = jz jjw j w 6= 0


�= z̄






�� = jz jjw j w 6= 0

jz j2 = zz̄

z � w = z̄ � w̄zw = z̄ w̄� zw

�= z̄






�� = jz jjw j w 6= 0

jz j2 = zz̄z � w = z̄ � w̄

zw = z̄ w̄� zw

�= z̄






�� = jz jjw j w 6= 0

jz j2 = zz̄z � w = z̄ � w̄zw = z̄ w̄

� zw

�= z̄






�� = jz jjw j w 6= 0


�= z̄

w̄ w 6= 0

z = z̄ if and only if z is real





�� = jz jjw j w 6= 0


�= z̄



The Complex Plane

Real numbers are represented geometrically as points on a numberline.

Complex numbers are represented as points (or vectors) in a numberplane, called the complex plane:

c = a + ib

imaginary axis

real axisa

ib


Complex Numbers Represented as Vectors

Let z = x + iy be a complex number with real part x = Re (z) andimaginary part y = Im (z) .

z = x + iy

x

iy

The Complex Plane


Why Represent Complex Numbers as Vectors?

The vector representation �ts together with the way addition andsubtraction of complex numbers are de�ned and gives us a geometricvisualization of these two algebraic operations.

Ifz = x + iy and z 0 = x 0 + iy 0

are in standard form, then

z + z 0 =�x + x 0

�+ i

�y + y 0

�,

z � z 0 =�x � x 0

�+ i

�y � y 0

�.

These rules for addition and subtraction look just like the correspondingrules for addition and subtraction of two-dimensional vectors.

What is the geometric expression of these results?


Absolute Value �the Geometric View

The vector point of view also helps you to visualize the absolute value of acomplex number. If z = x + iy is a complex number with x and y real,then jz j =

px2 + y2 is simply the length of the vector that represents the

complex number z .

z

| z |

The Complex Plane


Distance in the Complex Plane

The distance between z = x + iy and w = r + is in the complex plane isthe length of the vector z � w = (x � r) + i (y � s) :

jz � w j =q(x � r)2 + (y � s)2

w

z | z w |

jz � w j the distance between z and w(Oregon State University) 19 / 26

Polar Form of a Complex Number

Use of polar coordinates r and θ in the complex plane leads to the polarform for complex numbers.

Which is?

Who cares?

Multiplication and division of complex numbers have revealing geometricinterpretations when the computations are expressed in polar form. Use ofpolar forms make it easy to calculate powers and roots of complexnumbers. The key isDeMovire�s Theorem: If z = r (cos θ + i sin θ) , then

zn = rn (cos nθ + i sin nθ)

for n any rational number.


DEF. If c 2 C, then z 2 C is an n th root of c if zn = c .

ExampleFind the cube roots of i . Express the roots both in polar and standardform. Plot them.


Circles in the Complex Plane

Let z be a complex variable; that is, a complex number that can vary.

Since jz j = jz � 0j , the graph of jz j = 1 consists of all complex numbersz whose distance from the origin is 1 :

2 2

2

2

Unit Circle jz j = 1


Disks in the Complex Plane

Evidently the graph of jz j < 1 consists of all complex numbers inside theunit circle. The graph of jz j < 1 is called the open unit disk. (Openrefers to the fact that the circular boundary of the disk is not part of thedisk.)

2 2

2

2

Open Unit Disk jz j < 1(Oregon State University) 23 / 26

Let c be a �xed complex number and r > 0 a �xed real number. Then theequation jz � c j = r describes the points z in the complex plane whosedistance from c is r . Thus, the graph of jz � c j = r is the circle withcenter c and radius r in the complex plane. What is

the graph ofjz � c j � r?

ExampleDescribe the graph of (a) the equation jz � 4j = 3 and (b) the inequalityj2z � 8+ 6i j = 4 in the complex plane.


Let c be a �xed complex number and r > 0 a �xed real number. Then theequation jz � c j = r describes the points z in the complex plane whosedistance from c is r . Thus, the graph of jz � c j = r is the circle withcenter c and radius r in the complex plane. What is the graph ofjz � c j � r?

ExampleDescribe the graph of (a) the equation jz � 4j = 3 and (b) the inequalityj2z � 8+ 6i j = 4 in the complex plane.


Complex Functions

Complex variables are often denoted by z and w and, unless we sayotherwise, x and y will always stand for real variables.

Functions involving complex numbers are de�ned in the same fashion asfor the more familiar functions of algebra and calculus that involve onlyreal numbers and variables. Thus,

f (z) = z2,

with the understanding that z is a complex variable, de�nes acomplex-valued function of a complex variable. Which means?The rule

g (x) = 2x � ix2,with x understood to be real, de�nes a complex-valued function of areal variable. Which means?The rule

h (z) = Re (z)� Im (z)

de�nes a real-valued function of a complex variable. Which means?


Complex Functions



f (z) = z2,

with the understanding that z is a complex variable, de�nes acomplex-valued function of a complex variable. Which means?

The rule


h (z) = Re (z)� Im (z)



Complex Functions



f (z) = z2,


g (x) = 2x � ix2,with x understood to be real, de�nes a complex-valued function of areal variable. Which means?

The ruleh (z) = Re (z)� Im (z)



Complex Functions



f (z) = z2,



h (z) = Re (z)� Im (z)

de�nes a real-valued function of a complex variable. Which means?(Oregon State University) 25 / 26

All the common ways of expressing functions used earlier in calculus areused in the context of complex functions. For example,

the functionf (z) = z2 is also expressed by

w = z2.

Perhaps you are wondering what f (z) = sin z means when z is a complexvariable. Your calculator knows. So should you. Stay tuned.


All the common ways of expressing functions used earlier in calculus areused in the context of complex functions. For example, the functionf (z) = z2 is also expressed by

w = z2.

Perhaps you are wondering what f (z) = sin z means when z is a complexvariable.

Your calculator knows. So should you. Stay tuned.


All the common ways of expressing functions used earlier in calculus areused in the context of complex functions. For example, the functionf (z) = z2 is also expressed by

w = z2.

Perhaps you are wondering what f (z) = sin z means when z is a complexvariable. Your calculator knows. So should you. Stay tuned.


mth 306 spring term 2007 - college of science, oregon ...leejoh/306h_001_s2007/...... ..., 5, 4, 3,...

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