1. 8.7 Complex Numbers Simplify numbers of the form where b > 0.
Recognize subsets of the complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Find powers of i.

2. Simplify numbers of the form .
Objective 1
Simplify numbers of the form
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3. Simplify numbers of the form where b > 0. Imaginary Unit i
The imaginary unit i is defined as
That is, i is the principal square root of –1.
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4. Simplify numbers of the form where b > 0.
For any positive real number b,
It is easy to mistake for with the i under the radical. For this reason, we usually write as as in the definition of
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5. Write each number as a product of a real number and i.
CLASSROOM EXAMPLE 1
Simplifying Square Roots of Negative Numbers
Write each number as a product of a real number and i.
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6. Multiply. Multiplying Square Roots of Negative Numbers
CLASSROOM EXAMPLE 2
Multiplying Square Roots of Negative Numbers
Multiply.
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7. Divide. Dividing Square Roots of Negative Numbers CLASSROOM EXAMPLE 3
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8. Recognize subsets of the complex numbers.
Objective 2
Recognize subsets of the complex numbers.
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9. Complex Number Recognize subsets of the complex numbers.
If a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part.
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10. Recognize subsets of the complex numbers.
For a complex number a + bi, if b = 0, then a + bi = a, which is a real number.
Thus, the set of real numbers is a subset of the set of complex numbers.
If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number.
For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number.
A complex number written in the form a + bi is in standard form.
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11. Recognize subsets of the complex numbers.
The relationships among the various sets of numbers.
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12. Add and subtract complex numbers.
Objective 3
Add and subtract complex numbers.
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13. CLASSROOM EXAMPLE 4
Adding Complex Numbers
Add.
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14. Subtract. Subtracting Complex Numbers CLASSROOM EXAMPLE 5
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15. Multiply complex numbers.
Objective 4
Multiply complex numbers.
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16. Multiply. Multiplying Complex Numbers CLASSROOM EXAMPLE 6
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17. Multiply. Multiplying Complex Numbers (cont’d) CLASSROOM EXAMPLE 6
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18. Multiply. Multiplying Complex Numbers (cont’d) CLASSROOM EXAMPLE 6
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19. Multiply complex numbers.
The product of a complex number and its conjugate is always a real number.
(a + bi)(a – bi) = a2 – b2( –1)
= a2 + b2
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20. Divide complex numbers.
Objective 5
Divide complex numbers.
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21. Find the quotient. Dividing Complex Numbers CLASSROOM EXAMPLE 7
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22. Find the quotient. Dividing Complex Numbers (cont’d)
CLASSROOM EXAMPLE 7
Dividing Complex Numbers (cont’d)
Find the quotient.
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23. Objective 6
Find powers of i.
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24. i3 = i6 = i4 = i7 = i5 = i8 = Find powers of i.
Because i2 = –1, we can find greater powers of i, as shown below.
i3 = i6 =
i4 = i7 =
i5 = i8 =
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25. Find each power of i. Simplifying Powers of i CLASSROOM EXAMPLE 8
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