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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Exponents and Radicals7
Exponents and Radicals
7.1 Radical Expressions and Functions
7.2 Rational Numbers as Exponents
7.3 Multiplying Radical Expressions
7.4 Dividing Radical Expressions
7.5 Expressions Containing Several Radical Terms
7.6 Solving Radical Equations
7.7 Geometric Applications
The Complex Numbers
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3. 7.8 The Complex Numbers Imaginary and Complex Numbers
Addition and Subtraction
Multiplication
Conjugates and Division
Powers of i
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4. Imaginary and Complex Numbers
Negative numbers do not have square roots in the real-number system. A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. The complex-number system makes use of i, a number that is, by definition, a square root of –1.
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5. The Number i i is the unique number for which and i 2 = –1.
We can now define the square root of a negative number as follows:
for any positive number p.
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6. Example Express in terms of i: Solution
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7. Imaginary Numbers
An imaginary number is a number that can be written in the form a + bi, where a and b are real numbers and
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8. The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers.
Complex Numbers
A complex number is any number that can be written in the form a + bi, where a and b are real numbers. (Note that a and b both can be 0.)
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9. The following are examples of imaginary numbers:
Here a = 7, b =2.
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10. Copyright © 2006 Pearson Education, Inc
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11. Addition and Subtraction
The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials.
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12. Example Add or subtract and simplify. Solution
Combining the real and the imaginary parts
Solution
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13. Multiplication
To multiply square roots of negative real numbers, we first express them in terms of i. For example,
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14. Caution
With complex numbers, simply multiplying radicands is incorrect when both radicands are negative:
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15. Multiply and simplify. When possible, write answers in the form a + bi.
Example
Solution
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16. Solution continued
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17. Conjugate of a Complex Number
The conjugate of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi.
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18. Example Find the conjugate of each number. Solution
The conjugate is 4 – 3i.
The conjugate is –6 + 9i.
The conjugate is –i.
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19. Conjugates and Division
Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators.
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20. Example Divide and simplify to the form a + bi. Solution
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21. Solution continued
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22. Powers of i
Simplifying powers of i can be done by using the fact that i 2 = –1 and expressing the given power of i in terms of i 2. Consider the following:
i 23 = (i 2)11i1
= (–1)11i = –i
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23. Example Simplify: Solution
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