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We will be defining complex numbers. an introduction to iota and its use in complex numbers.

1.
The Complex Number System

1. Let a and b be real numbers with a 0.

There is a real number r that satisfies the

equation b

ax + b = 0; r .

a

The equation ax + b = 0 is a linear equation

in one variable.

1. Let a and b be real numbers with a 0.

There is a real number r that satisfies the

equation b

ax + b = 0; r .

a

The equation ax + b = 0 is a linear equation

in one variable.

2.
2. Let a, b, and c be real numbers with

a 0. Does there exist a real

number r which satisfies the

equation 2

ax bx c 0 ?

Answer: Not necessarily; sometimes

“yes”, sometimes “no”.

2

ax bx c 0

The equation

is a quadratic equation in one variable.

a 0. Does there exist a real

number r which satisfies the

equation 2

ax bx c 0 ?

Answer: Not necessarily; sometimes

“yes”, sometimes “no”.

2

ax bx c 0

The equation

is a quadratic equation in one variable.

3.
2

1. x 5 x 6 0; roots : r1 2, r2 3.

2

2. x 2 x 5 0; no real roots!

3. Simple case:

2

x 1 0; no real roots

1. x 5 x 6 0; roots : r1 2, r2 3.

2

2. x 2 x 5 0; no real roots!

3. Simple case:

2

x 1 0; no real roots

4.
The imaginary number i

DEFINITION: The imaginary number i

is a root of the equation

2

x 1 0.

(– i is also a root of this equation.)

ALTERNATE DEFINITION: i2 = 1 or

i 1.

DEFINITION: The imaginary number i

is a root of the equation

2

x 1 0.

(– i is also a root of this equation.)

ALTERNATE DEFINITION: i2 = 1 or

i 1.

5.
The Complex Number System

DEFINITION: The set C of complex

numbers is given by

C = {a + bi| a, b R}.

NOTE: The set of real numbers is a subset

of the set of complex numbers; R C,

a = a + 0i for every a R.

DEFINITION: The set C of complex

numbers is given by

C = {a + bi| a, b R}.

NOTE: The set of real numbers is a subset

of the set of complex numbers; R C,

a = a + 0i for every a R.

6.
Some terminology

Given the complex number z = a + bi.

•The real number a is called the real

part of z.

•The real number b is called the

imaginary part of z.

•The complex number z a bi

is called the conjugate of z.

Given the complex number z = a + bi.

•The real number a is called the real

part of z.

•The real number b is called the

imaginary part of z.

•The complex number z a bi

is called the conjugate of z.

7.
Arithmetic of Complex Numbers

Let a, b, c, and d be real

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

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

(a bi)(c di ) (ac bd ) (ad bc)i

Let a, b, c, and d be real

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

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

(a bi)(c di ) (ac bd ) (ad bc)i

8.
a bi a bi c di

c di c di c di

(ac bd ) (bc ad )i

2 2

c d

ac bd bc ad

2 2 2 2i

c d c d

2 2

provided c d 0

c di c di c di

(ac bd ) (bc ad )i

2 2

c d

ac bd bc ad

2 2 2 2i

c d c d

2 2

provided c d 0

9.
Field Axioms

The set of complex numbers C satisfies

the field axioms:

•Addition is commutative and associative,

0 = 0 + 0i is the additive identity, a bi is

the additive inverse of a + bi.

•Multiplication is commutative and

associative, 1 = 1 + 0i is the multiplicative

identity, a b

i is the

a2 b2 a2 b2

multiplicative inverse of a + bi.

The set of complex numbers C satisfies

the field axioms:

•Addition is commutative and associative,

0 = 0 + 0i is the additive identity, a bi is

the additive inverse of a + bi.

•Multiplication is commutative and

associative, 1 = 1 + 0i is the multiplicative

identity, a b

i is the

a2 b2 a2 b2

multiplicative inverse of a + bi.

10.
• the Distributive Law holds. That is,

if , , and are complex numbers, then

( + ) = +

if , , and are complex numbers, then

( + ) = +

11.
“Geometry” of the Complex Number

A complex number is a number of the

form a + bi, where a and b are real

numbers.

If we “identify” a + bi with the ordered

pair of real numbers (a,b) we get a point

in a coordinate plane – which we call the

complex plane.

A complex number is a number of the

form a + bi, where a and b are real

numbers.

If we “identify” a + bi with the ordered

pair of real numbers (a,b) we get a point

in a coordinate plane – which we call the

complex plane.

12.
The Complex Plane

13.
Absolute Value of a Complex Number

Recall that the absolute value of a real

number a is the distance from the point a

(on the real line) to the origin 0.

The same definition is used for complex

Recall that the absolute value of a real

number a is the distance from the point a

(on the real line) to the origin 0.

The same definition is used for complex

14.
2 2

| a bi | a b

| a bi | a b

15.
Fundamental Theorem of Algebra

A polynomial of degree n 1

n n 1 2

an x an 1 x a2 x a1 x a0

has exactly n (complex) roots.

A polynomial of degree n 1

n n 1 2

an x an 1 x a2 x a1 x a0

has exactly n (complex) roots.