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Complex Numbers Objectives Students will learn:
Basic Concepts of Complex Numbers Operations on Complex Numbers
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Basic Concepts of Complex Numbers
There are no real numbers for the solution of the equation To extend the real number system to include such numbers as, the number i is defined to have the following property;
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Basic Concepts of Complex Numbers
The number i is called the imaginary unit. Numbers of the form a + bi, where a and b are real numbers are called complex numbers. In this complex number, a is the real part and b is the imaginary part.
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THE EXPRESSION
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Write as the product of a real number and i, using the definition of
Example 1 WRITING AS Write as the product of a real number and i, using the definition of a. Solution:
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Write as the product of a real number and i, using the definition of
Example 1 WRITING AS Write as the product of a real number and i, using the definition of c. Solution: Product rule for radicals
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Operations on Complex Numbers
Caution When working with negative radicands, use the definition… before using any of the other rules for radicands.
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Operations on Complex Numbers
Caution In particular, the rule is valid only when c and d are not both negative. while so
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First write all square roots in terms of i.
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Example 2 Multiply or divide, as indicated. Simplify each answer. a. Solution: First write all square roots in terms of i. i 2 = −1
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Multiply or divide, as indicated. Simplify each answer.
FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Example 2 Multiply or divide, as indicated. Simplify each answer. c. Solution: Quotient rule for radicals
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Write in standard form a + bi. Solution:
SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Example 3 Write in standard form a + bi. Solution:
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Be sure to factor before simplifying
SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Example 3 Write in standard form a + bi. Solution: Be sure to factor before simplifying Factor. Lowest terms
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Find each sum or difference.
ADDING AND SUBTRACTING COMPLEX NUMBERS Example 4 Find each sum or difference. a. Add imaginary parts. Solution: Add real parts. Commutative, associative, distributive properties
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Find each sum or difference.
ADDING AND SUBTRACTING COMPLEX NUMBERS Example 4 Find each sum or difference. b. Solution:
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Find each product. a. Solution: MULTIPLYING COMPLEX NUMBERS Example 5
FOIL i2 = −1
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Remember to add twice the product of the two terms.
MULTIPLYING COMPLEX NUMBERS Example 5 Find each product. b. Solution: Square of a binomial Remember to add twice the product of the two terms. i 2 = −1
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Simplifying Powers of i
Powers of i can be simplified using the facts
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Powers of i and so on.
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Ex 5c. showed that… The numbers differ only in the sign of their imaginary parts and are called complex conjugates. The product of a complex number and its conjugate is always a real number. This product is the sum of squares of real and imaginary parts.
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Property of Complex Conjugates
For real numbers a and b,
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Write each quotient in standard form a + bi.
DIVIDING COMPLEX NUMBERS Example 7 Write each quotient in standard form a + bi. a. Solution: Multiply by the complex conjugate of the denominator in both the numerator and the denominator. Multiply.
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Write each quotient in standard form a + bi.
DIVIDING COMPLEX NUMBERS Example 7 Write each quotient in standard form a + bi. a. Solution: Multiply. i 2 = −1
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Write each quotient in standard form a + bi.
DIVIDING COMPLEX NUMBERS Example 7 Write each quotient in standard form a + bi. a. Solution: i 2 = −1
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