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Transcript
```Math 102
0.8 "Complex Numbers"
Objectives:
*
Express the square root of a negative number in terms of i:
*
Add, subtract, multiply and divide complex numbers.
We need to study the complex numbers because there are some very simple equations that do not have real solutions.
For example, x2 + 1 = 0 has no real solutions. To solve such equations, we need to extend the real number system. In this
section we will introduce the de…nition of complex numbers.
De…nition:
"Complex Numbers "
A complex number is any number that can be expressed in the form
p
where a and b are real numbers, and i is the imaginary unit i =
1
;
:
Example:
The form a + bi is called the standard form of a complex number. The real number a is called the real part of the
complex number, and b is called the imaginary part. (b is a real number even though it is called the imaginary part).
(a + bi) + (c + di) =
(a + bi)
(c + di) =
Note:
The set of complex numbers is closed with respect to addition; that is, the sum of two complex numbers is
a complex number. Moreover, the commutative and associative properties of addition hold for all complex numbers. The
additive identity element is 0 + 0i, or simply the real number 0.
Example 1: (Adding and subtracting complex numbers)
a) ( 9 + 3i) + (4 + 5i)
c)
5
8
+ 12 i
7
8
+ 51 i
b) (5 + 3i) + (7
2i) + ( 8
i)
d) (5
2i)
2i)
Page: 1
7i)
(6
( 1
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
0.8
Multiplying and Dividing Complex Numbers
Recall that i =
roots (i and
p
1 =)
:
Therefore, in the set of complex numbers,
i) : This is expressed symbolically as
:
1 has two square
Let’s extend the de…nition so
that in the set of complex numbers, every negative real number has two square roots. For any positive real number b;
p
Therefore, the principal square root of b is denote by
b and de…ne it to be
:
p
2
b =
For this reason we obtain that
:
:
Example 2: (Using the de…nition of i)
Write each expression in terms of i and simplify.
p
p
a)
49
b)
8
We must be careful with the use of the symbol
c)
p
q
4
9
p
d) 6
8
b; where b > 0. Some properties that are true in the set of real
numbers involving the square root symbol do not hold if the square root symbol does not represent a real number. For
example,
does not hold if a and b are both negative numbers.
Correct
Incorrect
Example 3: (Using the de…nition of i)
Write each in terms of i, perform the indicated operations, and simplify.
p
p p
p
25
9
b)
7
9
a)
p
64
c) p
16
Note:
p
54
d) p
9
The two complex numbers
and
are called conjugates of
each other. The product of a complex number and its conjugate is a real number.
Page: 2
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
0.8
Example 4: (Multiplying complex numbers)
Find each product and express the answer in standard form.
2
a) (4 + 5i) (2
9i)
b) (4
2i)
c) (5 + 3i) (5
3i)
d) (7 + 3i) (8 + 4i)
Example 5: (Dividing complex numbers)
Find each quotient and express the answer in standard form.
3i
3 5i
a)
b)
6 + 2i
4i
c)
1 i
2 3i
d)
Page: 3
3 + 9i
4 i
Notes by Bibiana Lopez
```