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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 1.6: The Complex Number System HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Objectives o The imaginary unit i and its properties. o The algebra of complex numbers. o Roots and complex numbers. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Imaginary Unit i and its Properties Consider the following question: For a given non-zero real number a, how many solutions does the equation x 2 a have? As we have noted before, this equation has two solutions: x a and x a if a is positive. However, if a is negative, it has no (real) solutions . The following definitions of complex numbers explain this situation in more detail. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists The Imaginary Unit i and its Properties The Imaginary Unit i The imaginary unit i is defined as i 1. In other words, i has the property that its square is 1: i 2 1 . Square Roots of Negative Numbers If a is a positive real number, a i a . Note that by this definition, and by a logical extension of 2 2 2 exponentiation, i a i a a. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1:The Imaginary Unit i and its Properties Write the following radicals in simplified form in terms of i. a. 9 i 9 3i We write a constant, such as 3, before letters in algebraic expressions, even if the letter is not a 2 variable. Remember: i is NOT a variable:i 1 . b. 18 i 18 3i 2 We write the radical factor last. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 2: The Imaginary Unit i and its Properties Write the following expressions in simplified form. c. i 5 i 2 i 2 i 1 1 i i 6 2 2 2 i i i i d. 1 1 1 1 4 2 2 i 1 i i i i 1 e. 4 4 4 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Imaginary Unit i and its Properties Complex Numbers For any two real numbers a and b, the sum a bi is a complex number. The collection a bi | a and b are both real is called the set of complex numbers. Ex: The number 2 3i is a complex number, where a = 2 and b = –3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Imaginary Unit i and its Properties Complex Numbers The number a is called the real part of a bi , and the number b is called the imaginary part. If the imaginary part of a given complex number is 0, the number is simply a real number. Ex. 4 0 i 4 (4 is a real number) If the real part of a given complex number is 0, the number is a pure imaginary number. Ex. 0 7 i 7i (7i is a pure imaginary number) HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists The Algebra of Complex Numbers Some notes about Complex Numbers: Every real number is a complex number with 0 as the imaginary part. Furthermore, the set of real numbers is a subset of the complex numbers. Real Numbers ( ) Complex Numbers ( ) Rational Numbers ( ) Integers ( ) Whole Numbers Natural Numbers ( ) Irrational Numbers The field properties (closure, commutate, associative, distributive and the existence of an identity and inverse), discussed in Section 1.2, still apply to complex numbers. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Algebra of Complex Numbers When faced with a complex number, the goal is to write it in the form a bi . Simplifying Complex Expressions Step 1: Add, subtract, or multiply the complex numbers, as required, by treating every complex number a bi as a polynomial expression. Remember, though, that i is not actually a variable. Treating a bi as a binomial in i is just a handy device. Step 2: Complete the simplification by using the fact that i 2 1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: The Algebra of Complex Numbers Simplify the following complex number expression. a. 6 2i 3 5i 6 3 5i 2i 3 7i b. 1 4i 2 4i 1 2 4i 4i 1 0i 1 Treating the two complex numbers as polynomials, we combine the real and then the imaginary parts and simplify. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: The Algebra of Complex Numbers Simplify the following complex number expressions. 2 i 1 2i 2 i 4i 2i 2 2 1 4 i 2 1 2 3i 2 4 3i The product of two complex numbers leads to four products via the distributive property. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: The Algebra of Complex Numbers Simplify the following complex number expression. 3 2i 2 3 2i 3 2i 9 6i 6i 4i 2 9 6 6 i 4 1 5 12 i 5 12i Squaring a complex number leads to four products via the distributive property. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. The Algebra of Complex Numbers Division of Complex Numbers: In order to rewrite a quotient in the standard form a bi, we make use of the following observation: 2 2 2 2 2 a bi a bi a abi abi b i a b Given any complex number a bi , the complex number a bi is called its complex conjugate. We simplify the quotient of two complex numbers by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: The Algebra of Complex Numbers Simplify the quotient. Find the complex conjugate of the denominator and 1 2i 2 3i a. multiply that by both the numerator and the 2 3i 2 3i denominator. 2 2 2 a bi a bi a b Remember that , which 2 3i 4i 6i makes solving the product in the denominator 49 much easier than using the distributive property. 2 7i 6 49 This would be an acceptable answer however, we 4 7i have stated that all simplified complex numbers 13 should be in the form a bi . 4 7 i 13 13 HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 5: The Algebra of Complex Numbers Simplify the quotients. a. 2 i 1 1 2i 2i 2i 2i 4 1 2i 5 2 1 i 5 5 Find the complex conjugate of the denominator. 1 i b. i i i 2 i i 1 i HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Roots and Complex Numbers We earlier said that if a and b are real numbers, then a a a b ab and . If a and b are not real b b numbers, then these properties do not necessarily hold. For instance: 9 4 9 4 3i 2i 36 6 6i 2 6 In order to apply either of these two properties, first simplify any square roots of negative numbers by rewriting them as pure imaginary numbers. HAWKES LEARNING SYSTEMS Copyright © 2010 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 6: Roots and Complex Numbers Simplify the following expressions. 2 a. 3 2 b. 9 9 3 2 3 2 3 3i 9 6 2 2 2 1 2 9 6i 2 i 2 i 9 6i 2 2 7 6i 2 i