Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.3

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.3

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SECTION 8.3 POLAR FORM AND DEMOIVRE’S THEOREM 483 8.3 POLAR FORM AND DEMOIVRE’S THEOREM At this point you can add, subtract, multiply, and divide complex numbers. However, there is still one basic procedure that is missing from the algebra of complex numbers. To see this, consider the problem of finding the square root of a complex number such as i . When you use the four basic operations (addition, subtraction, multiplication, and division), there seems to be no reason to guess that That is, To work effectively with powers and roots of complex numbers, it is helpful to use a polar representation for complex numbers, as shown in Figure 8.6. Specifically, if is a nonzero complex number, then let u be the angle from the positive x -axis to the radial line passing through the point ( a, b ) and let r be the modulus of So, and and you have from which the following polar form of a complex number is obtained. a 1 bi 5 s r cos d 1 s r sin d i r 5 ! a 2 1 b 2 b 5 r sin , a 5 r cos , a 1 bi . a 1 bi 1 1 1 i ! 2 2 2 5 i . ! i 5 1 1 i ! 2 . Figure 8.6 Imaginary axis Real axis Complex Number: a + bi Rectangular Form: ( a , b ) Polar Form: ( r , ) b ( a , b ) r a 0 θ Definition of Polar Form of a Complex Number The polar form of the nonzero complex number is given by where and The number r is the modulus of z and is the argument of z . tan 5 b y a . a 5 r cos , b 5 r sin , r 5 ! a 2 1 b 2 , z 5 r s cos 1 i sin d z 5 a 1 bi
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REMARK :The polar form of is given by where u is any angle. Because there are infinitely many choices for the argument, the polar form of a complex number is not unique. Normally, the values of that lie between and are used, though on occasion it is convenient to use other values. The value of that satisfies the inequality Principal argument is called the principal argument and is denoted by Arg( z ). Two nonzero complex numbers in polar form are equal if and only if they have the same modulus and the same principal argument. EXAMPLE 1 Finding the Polar Form of a Complex Number Find the polar form of each of the complex numbers. (Use the principal argument.) (a) (b) (c) i Solution (a) Because and which implies that From and and So, and (b) Because and then which implies that So, and and it follows that So, the polar form is (c) Because and it follows that and so The polar forms derived in parts (a), (b), and (c) are depicted graphically in Figure 8.7. z 5 1 1 cos p 2 1 i sin 2 2 . 5 y 2, r 5 1 b 5 1, a 5 0 < ! 13 f cos s 0.98 d 1 i sin s 0.98 dg . z 5 ! 13 ± cos 1 arctan 3 2 2 1 i sin 1 arctan 3 2 2 4 5 arctan s 3 y 2 d . sin 5 b r 5 3 ! 13 cos 5 a r 5 2 ! 13 r 5 ! 13 . r 2 5 2 2 1 3 2 5 13, b 5 3, a 5 2 z 5 ! 2 ± cos 1 2 4 2 1 i sin 1 2 4 2 4 . 52 y 4 sin 5 b r 1 ! 2 ! 2 2 . cos 5 a r 5 1 ! 2 5 ! 2 2 b 5 r sin , a 5 r cos r 5 ! 2 . b 1, then r 2 5 1 2 1 s 2 1 d 2 5 2, a 5 1 2 1 3 i 1 2 i 2 < 2 z 5 0 s cos 1 i sin d z 5 0 484 CHAPTER 8 COMPLEX VECTOR SPACES
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EXAMPLE 2 Converting from Polar to Standard Form Express the complex number in standard form.
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.3

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