Download Complex Numbers - MATH 160, Precalculus

Complex Numbers
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Complex Numbers
Objectives
In this lesson we will learn to:
use the imaginary unit i to write complex numbers,
add, subtract, and multiply complex numbers,
use complex conjugates to write the quotient of two
complex numbers in standard form,
find complex solutions to quadratic equations.
J. Robert Buchanan
Complex Numbers
Motivation
We would like to be able to describe all the solutions of all
polynomial equations, yet a very simple one has no real
number solutions.
x2 + 1 = 0
x 2 = −1
Since x 2 ≥ 0 for all real numbers x, there is no real solution to
this equation.
J. Robert Buchanan
Complex Numbers
Motivation
We would like to be able to describe all the solutions of all
polynomial equations, yet a very simple one has no real
number solutions.
x2 + 1 = 0
x 2 = −1
Since x 2 ≥ 0 for all real numbers x, there is no real solution to
this equation.
Thus we must expand our number system by using the
imaginary unit,
√
i = −1.
Thus i 2 = −1 and the solutions to the equation above can be
written as x = i and x = −i.
J. Robert Buchanan
Complex Numbers
Complex Numbers
Definition
If a and b are real numbers, the number a + bi is a complex
number, and it is said to be written in standard form. If b = 0,
the number a + bi = a is a real number. If b 6= 0, the number
a + bi is called an imaginary number. A number of the form bi
with b 6= 0 is called a pure imaginary number.
J. Robert Buchanan
Complex Numbers
Complex Numbers
Definition
If a and b are real numbers, the number a + bi is a complex
number, and it is said to be written in standard form. If b = 0,
the number a + bi = a is a real number. If b 6= 0, the number
a + bi is called an imaginary number. A number of the form bi
with b 6= 0 is called a pure imaginary number.
The real numbers R are a subset of the complex numbers C.
J. Robert Buchanan
Complex Numbers
Arithmetic of Complex Numbers (1 of 2)
Equality
Two complex numbers a + bi and c + di, written in standard
form, are equal to each other
a + bi = c + di
if and only if a = c and b = d.
J. Robert Buchanan
Complex Numbers
Arithmetic of Complex Numbers (1 of 2)
Equality
Two complex numbers a + bi and c + di, written in standard
form, are equal to each other
a + bi = c + di
if and only if a = c and b = d.
Addition and Subtraction
If a + bi and c + di are two complex numbers, written in
standard form, their sum and difference are defined as follows.
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i
J. Robert Buchanan
Complex Numbers
Arithmetic of Complex Numbers (2 of 2)
Identities and Inverses
The additive identity element in the complex number system
is 0 = 0 + 0i. The additive inverse of the complex number
a + bi is −a − bi.
J. Robert Buchanan
Complex Numbers
Examples
Perform the addition or subtraction as appropriate and write the
result in standard form.
(13 − 2i) + (−5 + 6i) =
(3 + 2i) − (6 + 13i) =
√
√
(8 + −18) − (4 + 3 2 i) =
25 + (−10 + 11i) + 15i =
J. Robert Buchanan
Complex Numbers
Examples
Perform the addition or subtraction as appropriate and write the
result in standard form.
(13 − 2i) + (−5 + 6i) = 8 + 4i
(3 + 2i) − (6 + 13i) =
√
√
(8 + −18) − (4 + 3 2 i) =
25 + (−10 + 11i) + 15i =
J. Robert Buchanan
Complex Numbers
Examples
Perform the addition or subtraction as appropriate and write the
result in standard form.
(13 − 2i) + (−5 + 6i) = 8 + 4i
(3 + 2i) − (6 + 13i) = −3 − 11i
√
√
(8 + −18) − (4 + 3 2 i) = 4
25 + (−10 + 11i) + 15i = 15 + 26i
J. Robert Buchanan
Complex Numbers
Multiplication
Multiplication of complex numbers is carried out using the FOIL
Method.
(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i 2
= (ac) + (ad + bc)i − (bd)
= (ac − bd) + (ad + bc)i
J. Robert Buchanan
Complex Numbers
Multiplication
Multiplication of complex numbers is carried out using the FOIL
Method.
(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i 2
= (ac) + (ad + bc)i − (bd)
= (ac − bd) + (ad + bc)i
The usual properties of arithmetic hold for complex numbers:
associative property
commutative property
distributive property
J. Robert Buchanan
Complex Numbers
Examples
Multiply the numbers below and write the result in standard
form.
(13 − 2i)(−5 + 6i) =
(3 + 2i)(6 + 13i) =
√
√
(8 + 3 2i)(4 + 3 2 i) =
(−10 + 11i)(15i) =
J. Robert Buchanan
Complex Numbers
Examples
Multiply the numbers below and write the result in standard
form.
(13 − 2i)(−5 + 6i) = −53 + 88i
(3 + 2i)(6 + 13i) =
√
√
(8 + 3 2i)(4 + 3 2 i) =
(−10 + 11i)(15i) =
J. Robert Buchanan
Complex Numbers
Examples
Multiply the numbers below and write the result in standard
form.
(13 − 2i)(−5 + 6i) = −53 + 88i
(3 + 2i)(6 + 13i) = −8 + 51i
√
√
√
(8 + 3 2i)(4 + 3 2 i) = 14 + 36 2i
(−10 + 11i)(15i) = −165 − 150i
J. Robert Buchanan
Complex Numbers
Complex Conjugates
Definition
The complex conjugate of the complex number a + bi is the
complex number a − bi.
J. Robert Buchanan
Complex Numbers
Complex Conjugates
Definition
The complex conjugate of the complex number a + bi is the
complex number a − bi.
Note: (a + bi)(a − bi) = a2 + b2 a real number.
J. Robert Buchanan
Complex Numbers
Quotients of Complex Numbers
The quotient of two complex numbers can be written in
standard form by multiplying both numerator and denominator
by the complex conjugate of the denominator.
a + bi
c + di
=
=
=
a + bi c − di
c + di c − di
(a + bi)(c − di)
(c + di)(c − di)
(ac + bd) + (bc − ad)i
c2 + d 2
J. Robert Buchanan
Complex Numbers
Example
Perform the indicated operation and write the result in standard
form.
2+i
1+i
−
=
3 − 2i
3 + 8i
J. Robert Buchanan
Complex Numbers
Example
Perform the indicated operation and write the result in standard
form.
2+i
1+i
(2 + i)(3 + 8i)
(1 + i)(3 − 2i)
−
=
−
3 − 2i
3 + 8i
(3 − 2i)(3 + 8i) (3 − 2i)(3 + 8i)
−2 + 19i
5+i
=
−
25 + 18i
25 + 18i
−2 + 19i − (5 + i)
=
25 + 18i
−7 + 18i
=
25 + 18i
(−7 + 18i)(25 − 18i)
=
(25 + 18i)(25 − 18i)
149 + 576i
=
625 + 324
149 576
=
+
i
949 949
J. Robert Buchanan
Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadratic
equation, we can use complex numbers and the imaginary root
to express the solutions.
√
−b ± b2 − 4ac
2
0 = ax + bx + c ⇐⇒ x =
2a
J. Robert Buchanan
Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadratic
equation, we can use complex numbers and the imaginary root
to express the solutions.
√
−b ± b2 − 4ac
2
0 = ax + bx + c ⇐⇒ x =
2a
Example
Use the Quadratic Formula to solve the following equation.
0 = 2x 2 − 5x + 7
J. Robert Buchanan
Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadratic
equation, we can use complex numbers and the imaginary root
to express the solutions.
√
−b ± b2 − 4ac
2
0 = ax + bx + c ⇐⇒ x =
2a
Example
Use the Quadratic Formula to solve the following equation.
0 = 2x 2 − 5x + 7
p
−(−5) ± (−5)2 − 4(2)(7)
x =
2(2)
√
5 ± 25 − 56
=
√
√4
5 ± −31
5 ± 31 i
=
=
4
4
J. Robert Buchanan
Complex Numbers
Homework
Read Section 2.4.
Exercises: 1, 5, 9, 13, . . . , 81, 85
J. Robert Buchanan
Complex Numbers