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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.
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.
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
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.
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.
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.
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
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
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.
10.
• the Distributive Law holds. That is,
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.
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
14.
2 2
| 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.
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