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Objectives Students will learn: Complex Numbers 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 So… 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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Complex numbers a + bi, a and b real Nonreal complex numbers a + bi, b 0 Real numbers a + bi, b = 0 Irrational numbers Rational numbers Integers Non- integers

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Basic Concepts of Complex Numbers Two complex numbers are equal provided that their real parts are equal and their imaginary parts are equal; if and only ifand

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Basic Concepts of Complex Numbers For complex number a + bi, if b = 0, then a + bi = a So, the set of real numbers is a subset of complex numbers.

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Basic Concepts of Complex Numbers If a = 0 and b 0, the complex number is pure imaginary. A pure imaginary number or a number, like 7 + 2i with a 0 and b 0, is a nonreal complex number. The form a + bi (or a + ib) is called standard form.

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THE EXPRESSION

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Example 1 WRITING AS Write as the product of a real number and i, using the definition of a. Solution:

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Example 1 Write as the product of a real number and i, using the definition of b. Solution: WRITING AS

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Example 1 Write as the product of a real number and i, using the definition of c. Solution: Product rule for radicals WRITING AS

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Operations on Complex Numbers Products or quotients with negative radicands are simplified by first rewriting for a positive number. Then the properties of real numbers are applied, together with the fact that

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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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Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS 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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Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Multiply or divide, as indicated. Simplify each answer. b. Solution:

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Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Multiply or divide, as indicated. Simplify each answer. c. Solution: Quotient rule for radicals

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Example 3 SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Writein standard form a + bi. Solution:

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Example 3 SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Writein standard form a + bi. Solution: Factor. Be sure to factor before simplifying Lowest terms

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Addition and Subtraction of Complex Numbers For complex numbers a + bi and c + di, and

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Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. a. Solution: Add real parts. Add imaginary parts. Commutative, associative, distributive properties

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Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. b. Solution:

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Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. c. Solution:

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Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. d. Solution:

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Multiplication of Complex Numbers The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i 2 = – 1, as follows. FOIL Distributive property; i 2 = – 1

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Multiplication of Complex Numbers For complex numbers a + bi and c + di,

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Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. a. Solution: FOIL i 2 = 1

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Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. b. Solution: Square of a binomial i 2 = 1 Remember to add twice the product of the two terms.

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Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. c. Solution: Product of the sum and difference of two terms i 2 = 1 Standard form

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Simplifying Powers of i Powers of i can be simplified using the facts

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Example 6 SIMPLIFYING POWERS OF i Simplify each power of i. a. Solution: Since i 2 = – 1 and i 4 = 1, write the given power as a product involving i 2 or i 4. For example, Alternatively, using i 4 and i 3 to rewrite i 15 gives

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Example 6 SIMPLIFYING POWERS OF i Simplify each power of i. b. Solution:

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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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Example 7 DIVIDING COMPLEX NUMBERS 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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Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: Multiply. i 2 = 1

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Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: i 2 = 1

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Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: Lowest terms; standard form

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Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. b. Solution: – i is the conjugate of i.

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Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. b. Solution: i 2 = 1(1) = 1 Standard form

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