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OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 1 Complex Numbers Define complex numbers. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. SECTION

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The square root of 1 is called i. Definition of i The number i is called the imaginary unit. 2 © 2010 Pearson Education, Inc. All rights reserved

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A complex number is a number of the form Complex Numbers where a and b are real numbers and i 2 = –1. The number a is called the real part of z, and we write Re(z) = a. The number b is called the imaginary part of z and we write Im(z) = b. 3 © 2010 Pearson Education, Inc. All rights reserved

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Definitions A complex number z written in the form a + bi is said to be in standard form. A complex number with a = 0 and b 0, written as bi, is called a pure imaginary number. If b = 0, then the complex number a + bi is a real number. Real numbers form a subset of complex numbers (with imaginary part 0). 4 © 2010 Pearson Education, Inc. All rights reserved

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For any positive number, b Square Root of a Negative Number 5 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 1 Identifying the Real and the Imaginary Parts of a Complex Number Identify the real and the imaginary parts of each complex number. 6 © 2010 Pearson Education, Inc. All rights reserved To be discussed in class

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Equality of Complex Numbers Two complex numbers z = a + bi and w = c + di are equal if and only if a = c and b = d That is, z = w if and only if Re(z) = Re(w) and Im(z) = Im(w). 7 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 2 Equality of Complex Numbers Find a and b assuming that (1 – 2a) + 3i = 5 – (2b – 5)i. Solution 8 © 2010 Pearson Education, Inc. All rights reserved To be solved in class

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ADDITION AND SUBTRACTION OF COMPLEX NUMBERS For all real numbers a, b, c, and d, let z = a + bi and w = c + di. 9 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 3 Adding and Subtracting Complex Numbers Textbook Exercises Page © 2010 Pearson Education, Inc. All rights reserved To be discussed in class

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MULTIPLYING COMPLEX NUMBERS For all real numbers a, b, c, and d, 11 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Multiplying Complex Numbers 12 © 2010 Pearson Education, Inc. All rights reserved Textbook Exercises Page 115 To be discussed in class

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CONJUGATE OF A COMPLEX NUMBER If z = a + bi, then the conjugate (or complex conjugate) of z is denoted by and defined by 13 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 5 Multiplying a Complex Number by Its Conjugate Find the product for each complex number. a. z = 4 + 5ib. z = 1 – 2i 14 © 2010 Pearson Education, Inc. All rights reserved To be discussed in class

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COMPLEX CONJUGATE PRODUCT THEOREM If z = a + bi, then 15 © 2010 Pearson Education, Inc. All rights reserved

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DIVIDING COMPLEX NUMBERS To write the quotient of two complex numbers w and z (z 0), and write and then write the right side in standard form. 16 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 6 Dividing Complex Numbers Write the following quotients in standard form. 17 © 2010 Pearson Education, Inc. All rights reserved To be discussed in class

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