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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals B.3 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Complex Numbers System KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Algebra of complex numbers Definitions Properties

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Introduction KFUPM - Prep Year Math Program (c) 2009 All Right Reserved We introduce a number called the imaginary unit, denoted by i, whose square is (by definition) equals. The number system that results intruding the imaginary unit i is called the complex number system,.

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved A complex number, in standard form, is written as Where and are real numbers, that is

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Definition

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Algebra of complex numbers Algebra of complex numbers KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Equality, addition, subtraction, and multiplication of complex numbers are defined in such ways that the familiar algebra rules for real numbers are preserved. The property where a non-zero complex number has a multiplicative inverse (or reciprocal), requires the definition of the conjugate.

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Theorem:

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals The quotient of two complex numbers KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Note that KFUPM - Prep Year Math Program (c) 2009 All Right Reserved For an integer n > 0, we have where r is the remainder of dividing n by 4.

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Note that KFUPM - Prep Year Math Program (c) 2009 All Right Reserved But rather: In general, for any real number a > 0, we have

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Note that: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved 1. For any and ifand thus 2. It is required to write the result of performing the algebraic operations on complex numbers in standard form, that is:

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Example KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Perform the indicated operations and write the answers in the standard form.

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Example KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Properties of complex numbers KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Example KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Evaluate each of the following and put your answer in the standard form.

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Example KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Find two complex numbers Solution we want to find x and y so that which is equivalent to the equations

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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals A1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals.

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals A1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals.

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