Download 171S3.1q The Complex Numbers - Cape Fear Community College

171S3.1q The Complex Numbers
February 20, 2013
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value
This program calculates the sum, difference, product, and quotient of two complex numbers.
http://cfcc.edu/mathlab/geogebra/complexnumbers.html
Complex Numbers & TI calculator
i 4 = 1
(5+3i)(­6­i) = ­27 ­ 23i
This 4­minute Youtube video gives an introduction to the Use of TI­83 Calculator to perform complex number arithmetic.
This program will be used in Section 3.2 when graphing quadratic functions. http://cfcc.edu/mathlab/geogebra/quadratic2ss.html
Sep 27­3:10 PM
Sep 28­7:02 AM
2.2 The Complex Numbers
• Perform computations involving complex numbers.
We can define a non­ real number that is a solution of the equation x2 + 1 = 0.
The complex numbers are formed by adding real numbers and multiples of i.
The Complex­
Number System
Some functions have zeros that are not real numbers.
The complex­number system is used to find zeros of functions that are not real numbers.
When looking at a graph of a function, if the graph does not cross the x­axis, then it has no x­intercepts, and thus it has no real­number zeros.
Sep 27­3:10 PM
Feb 19­8:16 AM
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171S3.1q The Complex Numbers
February 20, 2013
Complex Numbers
Example
Express each number in terms of i.
A complex number is a number of the form a + bi, where a and b are real numbers. The number a is said to be the real part of a + bi and the number bi is said to be the imaginary part of a + bi.
Imaginary Number a + bi, a ≠ 0, b ≠ 0
Solution
Pure Imaginary Number a + bi, a = 0, b ≠ 0
Addition and Subtraction
Complex numbers obey the commutative, associative, and distributive laws.
We add or subtract them as we do binomials.
We collect the real parts and the imaginary parts of complex numbers just as we collect like terms in binomials.
Example
Add or subtract and simplify each of the following.
a. (8 + 6i) + (3 + 2i) b. (4 + 5i) – (6 – 3i)
Solution
a. (8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i)
= 11 + (6 + 2)i = 11 + 8i
b. (4 + 5i) – (6 – 3i) = (4 – 6) + [5i − (−3i)]
= − 2 + 8i
Sep 27­3:10 PM
Sep 27­3:10 PM
Multiplication
When and are real numbers, This is not true when and are not real numbers.
Note: Remember i2 = –1
Example
Simplifying Powers of i
Recall that −1 raised to an even power is 1, and −1 raised to an odd power is −1.
Simplifying powers of i can then be done by using the fact that i2 = –1 and expressing the given power of i in terms of i2.
Multiply and simplify each of the following.
Note that powers of i cycle through i, –1, –i, and 1.
Solution
Conjugates
The conjugate of a complex number a + bi is a − bi. The numbers a + bi and a − bi are complex conjugates.
Examples:
−3 + 7i and −3 − 7i
14 − 5i and 14 + 5i
8i and −8i
The product of a complex number and its conjugate is a real number.
Sep 27­3:10 PM
Sep 27­3:10 PM
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171S3.1q The Complex Numbers
Multiplying Conjugates ­ Example
Multiply each of the following.
a. (5 + 7i)(5 – 7i) b. (8i)(–8i)
February 20, 2013
Express the number in terms of i : (By hand and by TI)
241/4. 241/6. Solution
a. (5 + 7i)(5 ­ 7i) = 52 ­ (7i)2
= 25 ­ 49i2
= 25 ­ 49(­1)
= 25 + 49
= 74
b. (8i)(–8i) = ­64i2
= ­64(­1)
= 64
Dividing Using Conjugates ­ Example
Divide 2 − 5i by 1 − 6i.
Solution: Write fraction notation. Multiply by 1, using the conjugate of the denominator to form the symbol for 1.
Express the number in terms of i : (By hand and by TI)
241/9. 241/10. This program calculates the sum, difference, product, and quotient of two complex numbers.
http://cfcc.edu/mathlab/geogebra/complexnumbers.html
Sep 27­3:10 PM
Sep 27­3:12 PM
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
241/14. Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
241/22. (­3 ­ 4i) ­ (8 ­ i)
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/30. (10 ­ 4i) ­ (8 + 2i)
241/19. Sep 27­3:12 PM
Sep 27­3:12 PM
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171S3.1q The Complex Numbers
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/36. 3i(6 + 4i)
242/42. (3 ­ 5i)(8 ­ 2i)
February 20, 2013
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/56. (5 ­ 4i)2 242/58. (­3 + 2i)2
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/47. 242/50. (5 + 9i)(5 ­ 9i)
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/60. (2 ­ 4i)2 Sep 27­3:12 PM
242/64. (6 + 5i)2 Sep 27­3:12 PM
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/70.
242/72.
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/82. i24 242/86. (­i)6
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/74.
242/76.
Sep 27­3:12 PM
Simplify. Write answers in the form a + bi, where a and b are real numbers: (By hand and by TI)
242/87. (5i)4
242/88. (2i)5
Sep 27­8:34 PM
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