1. 10.7 Complex Numbers

2. Simplify numbers of the form where b > 0.
Objective 1
Simplify numbers of the form where b > 0.
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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.
Solution:
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6. Multiply. CLASSROOM EXAMPLE 2
Multiplying Square Roots of Negative Numbers
Multiply.
Solution:
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7. Divide. CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers
Solution:
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8. Recognize subsets of the complex numbers.
Objective 2
Recognize subsets of the complex numbers.
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9. Recognize subsets of the complex numbers. Complex Number
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. Add. CLASSROOM EXAMPLE 4 Adding Complex Numbers Solution:
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14. Subtract. CLASSROOM EXAMPLE 5 Subtracting Complex Numbers Solution:
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15. Multiply complex numbers.
Objective 4
Multiply complex numbers.
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16. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers Solution:
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17. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)
Solution:
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18. Multiply. CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)
Solution:
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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. CLASSROOM EXAMPLE 7 Dividing Complex Numbers
Solution:
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22. Find the quotient. CLASSROOM EXAMPLE 7
Dividing Complex Numbers (cont’d)
Find the quotient.
Solution:
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23. Objective 6
Find powers of i.
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24. Find powers of i.
Because i2 = –1, we can find greater powers of i, as shown below. i3 = i · i2 = i · ( –1) = –i i4 = i2 · i2 = ( –1) · ( –1) = 1 i5 = i · i4 = i · 1 = i i6 = i2 · i4 = ( –1) · (1) = –1 i7 = i3 · i4 = (i) · (1) = –I i8 = i4 · i4 = 1 · 1 = 1
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25. Find each power of i. CLASSROOM EXAMPLE 8 Simplifying Powers of i
Solution:
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