Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.2

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.2

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8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS In Section 8.1, it was mentioned that the complex zeros of a polynomial with real coeffi- cients occur in conjugate pairs. For instance, you saw that the zeros of are and In this section, you will examine some additional properties of complex conjugates. You will begin with the definition of the conjugate of a complex number. From this definition, you can see that the conjugate of a complex number is found by changing the sign of the imaginary part of the number, as demonstrated in the following example. EXAMPLE 1 Finding the Conjugate of a Complex Number Complex Number Conjugate (a) (b) (c) (d) REMARK : In part (d) of Example 1, note that 5 is its own complex conjugate. In gen- eral, it can be shown that a number is its own complex conjugate if and only if the number is real. (See Exercise 29.) Geometrically, two points in the complex plane are conjugates if and only if they are reflections about the real (horizontal) axis, as shown in Figure 8.5. z 5 5 z 5 5 z 5 2 i z 52 2 i z 5 4 1 5 i z 5 4 2 5 i z 2 2 3 i z 2 1 3 i 3 2 2 i . 3 1 2 i p s x d 5 x 2 2 6 x 1 13 SECTION 8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS 477 Definition of the Conjugate of a Complex Number The conjugate of the complex number is denoted by and is given by z 5 a 2 bi . z z 5 a 1 bi , Figure 8.5 Imaginary axis Real axis z = 2 + 3 i z = 2 3 i 2 2 3 3 4 11 2 3 Imaginary axis Real axis z = 4 5 i z = 4 + 5 i 2 2 2 3 4 5 33 5 7 6 1 2 3 4 5 Conjugate of a Complex Number
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Complex conjugates have many useful properties. Some of these are given in Theorem 8.1. Proof To prove the first property, let Then and The second and third properties follow directly from the first. Finally, the fourth property follows the definition of complex conjugate. That is, EXAMPLE 2 Finding the Product of Complex Conjugates Find the product of and its complex conjugate. Solution Because you have The Modulus of a Complex Number Because a complex number can be represented by a vector in the complex plane, it makes sense to talk about the length of a complex number. This length is called the modulus of the complex number.
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.2

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