1. Equations and Inequalities
Chapter 1
Equations and Inequalities
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Complex Numbers
SECTION 1.3
Define complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers. 1 2 3 4
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Definition of i
The square root of −1 is called i.
The number i is called the imaginary unit.
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Complex Numbers
A complex number is a number of the form
where a and b are real numbers and i2 = –1.
The number a is called the real part of z, and we write Re(z) = a.
The number b is called the imaginary part of z and we write Im(z) = b.
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Definitions
A complex number z written in the form
a + bi is said to be in standard form.
A complex number with a = 0 and b ≠ 0, written as bi, is called a pure imaginary number.
If b = 0, then the complex number a + bi is a real number. Real numbers form a subset of complex numbers (with imaginary part 0).
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6. Square Root of a Negative Number
For any positive number, b
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Identifying the Real and the Imaginary Parts of a Complex Number
EXAMPLE 1
Identify the real and the imaginary parts of each complex number.
Solution
real part 2; imaginary part 5
real part 7; imaginary part
real part 0; imaginary part 3
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Identifying the Real and the Imaginary Parts of a Complex Number
EXAMPLE 1
Solution continued
real part –9; imaginary part 0
real part 0; imaginary part 0
real part 3; imaginary part 5
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9. Equality of Complex Numbers
Two complex numbers z = a + bi and w = c + di are equal if and only if
a = c and b = d
That is, z = w if and only if Re(z) = Re(w) and Im(z) = Im(w).
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EXAMPLE 2
Equality of Complex Numbers
Find x and y assuming that
(3x – 1) + 5i = 8 + (3 – 2y)i.
Solution
Let z = (3x – 1) + 5i and w = 8 + (3 – 2y)i.
Then Re(z) = Re(w) and Im(z) = Im(w).
So, x – 1 = 8 and 5 = 3 – 2y.
3x = – 3 = –2y
x = –1 = y
So, x = 3 and y = –1.
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11. ADDITION AND SUBTRACTION OF COMPLEX NUMBERS
For all real numbers a, b, c, and d, let
z = a + bi and w = c + di.
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EXAMPLE 3
Adding and Subtracting Complex Numbers
Write the sum or difference of two complex numbers in standard form.
Solution
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EXAMPLE 3
Adding and Subtracting Complex Numbers
Solution continued
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14. MULTIPLYING COMPLEX NUMBERS
For all real numbers a, b, c, and d,
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EXAMPLE 4
Multiplying Complex Numbers
Write the following products in standard form.
Solution F O I L
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WARNING
Recall that if a and b are positive real numbers,
However, this property is not true for nonreal numbers. For example,
but
Thus
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Multiplication Involving Roots of Negative Numbers
EXAMPLE 5
Perform the indicated operation and write the result in standard form.
Solution
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Multiplication Involving Roots of Negative Numbers
EXAMPLE 5
Solution continued b. c.
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Multiplication Involving Roots of Negative Numbers
EXAMPLE 5
Solution continued d.
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20. CONJUGATE OF A COMPLEX NUMBER
If z = a + bi, then the conjugate (or complex conjugate) of z is denoted by and defined by
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Multiplying a Complex Number by Its Conjugate
EXAMPLE 6
Find the product for each complex number .
Solution
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22. COMPLEX CONJUGATE PRODUCT THEOREM
If z = a + bi, then
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23. DIVIDING COMPLEX NUMBERS
To write the quotient of two complex numbers w and z (z ≠ 0), and write
and then write the right side in standard form.
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EXAMPLE 7
Dividing Complex Numbers
Write the following quotients in standard form.
Solution
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EXAMPLE 7
Dividing Complex Numbers
Solution continued
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