complex numbers - math 160, precalculust2010125/math30203/complex.pdf · complex numbers math 160,...
TRANSCRIPT
Complex NumbersMATH 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 twocomplex 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 allpolynomial equations, yet a very simple one has no realnumber solutions.
x2 + 1 = 0x2 = −1
Since x2 ≥ 0 for all real numbers x , there is no real solution tothis equation.
Thus we must expand our number system by using theimaginary unit,
i =√−1.
Thus i2 = −1 and the solutions to the equation above can bewritten as x = i and x = −i .
J. Robert Buchanan Complex Numbers
Motivation
We would like to be able to describe all the solutions of allpolynomial equations, yet a very simple one has no realnumber solutions.
x2 + 1 = 0x2 = −1
Since x2 ≥ 0 for all real numbers x , there is no real solution tothis equation.
Thus we must expand our number system by using theimaginary unit,
i =√−1.
Thus i2 = −1 and the solutions to the equation above can bewritten as x = i and x = −i .
J. Robert Buchanan Complex Numbers
Complex Numbers
DefinitionIf a and b are real numbers, the number a + bi is a complexnumber, 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 numbera + bi is called an imaginary number. A number of the form biwith 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
Complex Numbers
DefinitionIf a and b are real numbers, the number a + bi is a complexnumber, 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 numbera + bi is called an imaginary number. A number of the form biwith 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)
EqualityTwo complex numbers a + bi and c + di , written in standardform, are equal to each other
a + bi = c + di
if and only if a = c and b = d .
Addition and SubtractionIf a + bi and c + di are two complex numbers, written instandard 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 (1 of 2)
EqualityTwo complex numbers a + bi and c + di , written in standardform, are equal to each other
a + bi = c + di
if and only if a = c and b = d .
Addition and SubtractionIf a + bi and c + di are two complex numbers, written instandard 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 InversesThe additive identity element in the complex number systemis 0 = 0 + 0i . The additive inverse of the complex numbera + bi is −a− bi .
J. Robert Buchanan Complex Numbers
Examples
Perform the addition or subtraction as appropriate and write theresult 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
Examples
Perform the addition or subtraction as appropriate and write theresult 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
Examples
Perform the addition or subtraction as appropriate and write theresult 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 FOILMethod.
(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i2
= (ac) + (ad + bc)i − (bd)
= (ac − bd) + (ad + bc)i
The usual properties of arithmetic hold for complex numbers:associative propertycommutative propertydistributive property
J. Robert Buchanan Complex Numbers
Multiplication
Multiplication of complex numbers is carried out using the FOILMethod.
(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i2
= (ac) + (ad + bc)i − (bd)
= (ac − bd) + (ad + bc)i
The usual properties of arithmetic hold for complex numbers:associative propertycommutative propertydistributive property
J. Robert Buchanan Complex Numbers
Examples
Multiply the numbers below and write the result in standardform.
(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
Examples
Multiply the numbers below and write the result in standardform.
(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
Examples
Multiply the numbers below and write the result in standardform.
(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
DefinitionThe complex conjugate of the complex number a + bi is thecomplex number a− bi .
Note: (a + bi)(a− bi) = a2 + b2 a real number.
J. Robert Buchanan Complex Numbers
Complex Conjugates
DefinitionThe complex conjugate of the complex number a + bi is thecomplex 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 instandard form by multiplying both numerator and denominatorby the complex conjugate of the denominator.
a + bic + di
=a + bic + di
c − dic − di
=(a + bi)(c − di)(c + di)(c − di)
=(ac + bd) + (bc − ad)i
c2 + d2
J. Robert Buchanan Complex Numbers
Example
Perform the indicated operation and write the result in standardform.
2 + i3− 2i
− 1 + i3 + 8i
=
(2 + i)(3 + 8i)(3− 2i)(3 + 8i)
− (1 + i)(3− 2i)(3− 2i)(3 + 8i)
=−2 + 19i25 + 18i
− 5 + i25 + 18i
=−2 + 19i − (5 + i)
25 + 18i
=−7 + 18i25 + 18i
=(−7 + 18i)(25− 18i)(25 + 18i)(25− 18i)
=149 + 576i625 + 324
=149949
+576949
i
J. Robert Buchanan Complex Numbers
Example
Perform the indicated operation and write the result in standardform.
2 + i3− 2i
− 1 + i3 + 8i
=(2 + i)(3 + 8i)(3− 2i)(3 + 8i)
− (1 + i)(3− 2i)(3− 2i)(3 + 8i)
=−2 + 19i25 + 18i
− 5 + i25 + 18i
=−2 + 19i − (5 + i)
25 + 18i
=−7 + 18i25 + 18i
=(−7 + 18i)(25− 18i)(25 + 18i)(25− 18i)
=149 + 576i625 + 324
=149949
+576949
i
J. Robert Buchanan Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadraticequation, we can use complex numbers and the imaginary rootto express the solutions.
0 = ax2 + bx + c ⇐⇒ x =−b ±
√b2 − 4ac
2a
ExampleUse the Quadratic Formula to solve the following equation.
0 = 2x2 − 5x + 7
x =−(−5)±
√(−5)2 − 4(2)(7)
2(2)
=5±√
25− 564
=5±√−31
4=
5±√
31 i4
J. Robert Buchanan Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadraticequation, we can use complex numbers and the imaginary rootto express the solutions.
0 = ax2 + bx + c ⇐⇒ x =−b ±
√b2 − 4ac
2a
ExampleUse the Quadratic Formula to solve the following equation.
0 = 2x2 − 5x + 7
x =−(−5)±
√(−5)2 − 4(2)(7)
2(2)
=5±√
25− 564
=5±√−31
4=
5±√
31 i4
J. Robert Buchanan Complex Numbers
Complex Solution to Quadratic Equations
When using the Quadratic Formula to solve a quadraticequation, we can use complex numbers and the imaginary rootto express the solutions.
0 = ax2 + bx + c ⇐⇒ x =−b ±
√b2 − 4ac
2a
ExampleUse the Quadratic Formula to solve the following equation.
0 = 2x2 − 5x + 7
x =−(−5)±
√(−5)2 − 4(2)(7)
2(2)
=5±√
25− 564
=5±√−31
4=
5±√
31 i4
J. Robert Buchanan Complex Numbers
Homework
Read Section 2.4.Exercises: 1, 5, 9, 13, . . . , 81, 85
J. Robert Buchanan Complex Numbers