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**© 2010 Pearson Education, Inc. All rights reserved**

Complex Numbers SECTION 1.3 Define complex numbers. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. 1 2 3 4 © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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**© 2010 Pearson Education, Inc. All rights reserved**

Complex Numbers A complex number is a number of the form where a and b are real numbers and i2 = –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. © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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). © 2010 Pearson Education, Inc. All rights reserved

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**Square Root of a Negative Number**

For any positive number, b © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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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). © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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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. © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

EXAMPLE 3 Adding and Subtracting Complex Numbers Textbook Exercises Page 115 To be discussed in class © 2010 Pearson Education, Inc. All rights reserved

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**MULTIPLYING COMPLEX NUMBERS**

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

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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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 © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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**COMPLEX CONJUGATE PRODUCT THEOREM**

If z = a + bi, then © 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. © 2010 Pearson Education, Inc. All rights reserved

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**© 2010 Pearson Education, Inc. All rights reserved**

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

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