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Math 35
7.7 "Complex Numbers"
Objectives:
*
Express square roots of negative numbers in terms of i:
*
Write complex numbers in the form a + bi:
*
Add, subtract, multiply and divide complex numbers.
*
Perform operations involving powers of i:
Express Square Roots of Negative Numbers in Terms of i
Recall that the square root of a negative number is not a real number. However, an expanded number system, called
p
9: To de…ne complex numbers,
the complex number system, gives meaning to square roots of negative numbers, such as
we use a number that is denoted by the letter i.
De…nition:
" The Number i "
The imaginary number i is de…ned as i =
p
Square Root of a Negative Number:
p
p
For any positive real number b;
b=i b
1
:
From the de…nition, it follows that i2 =
1
:
:
Example 1: (Writing expressions in terms of i)
Write each expression in terms of i.
p
a)
9
b)
p
45
c)
r
50
81
Write Complex Numbers in the Form a+bi
De…nition:
"Complex Numbers"
A complex number is any number that can be written in the form a + bi ; where a and b are real numbers
p
and i =
1 : Complex numbers of the form a + bi; where b 6= 0, are also called imaginary numbers.
Example 2: (Writing complex numbers)
Write each number in the form a + bi.
a)
5
b)
p
36
c) 1 +
p
24
Add and Subtract Complex Numbers
Addition and Subtraction of Complex Numbers:
1: To add complex numbers, add their real parts and add their imaginary parts.
2: To subtract complex numbers, add the opposite of the complex number being subtracted.
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.7
Example 3: (Adding and subtracting complex numbers)
Perform each operation. Write the answer in the form a + bi.
a) ( 6 + 4i)
(3 + 2i)
b)
7
p
16 + 9 +
p
4
Multiply Complex Numbers
Example 4: (Multiplying complex numbers)
Perform each operation. Write the answer in the form a + bi.
p p
2
20
b)
a)
c) (2 + 3i) (3
2i)
5i (4
8i)
d) ( 4 + 2i) (2 + i)
Divide Complex Numbers
Complex Conjugates:
The complex numbers a + bi
and a
bi are called complex conjugates.
Division of Complex Numbers:
kTo divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.k
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.7
Example 5: (Dividing complex numbers)
Perform each operation. Write the answer in the form a + bi.
3
1 + 2i
a)
b)
6+i
3 4i
p
4+
16
p
c)
2+
4
d)
7
2i
Perform Operations Involving Powers of i
Powers of i
If n is a natural number that has a remainder of r when divided by 4, then in = ir
:
Example 6: (Performing operations involving powers of i)
Simplify.
a) i55
b) i62
Page: 3
Notes by Bibiana Lopez