UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

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1 Prerequste Sklls Ths lesson requres the use of the followng sklls: understandng that multplyng the numerator and denomnator of a fracton by the same quantty produces an equvalent fracton multplyng complex numbers smplfyng fractons by reducng the numerator and denomnator by a common factor rewrtng a quotent as a fracton fndng complex conjugates Introducton Recall that the magnary unt s equal to 1. A fracton wth n the denomnator does not have a ratonal denomnator, snce 1 s not a ratonal number. Smlar to ratonalzng a fracton wth an rratonal square root n the denomnator, fractons wth n the denomnator can also have the denomnator ratonalzed. Key Concepts Any powers of should be smplfed before dvdng complex numbers. After smplfyng any powers of, rewrte the dvson of two complex numbers n the form a + b as a fracton. To dvde two complex numbers of the form a + b and c + d, where a, b, c and d are real numbers, rewrte the quotent as a fracton. c d ( c+ d) ( a+ b) + a+ b Ratonalze the denomnator of a complex fracton by usng multplcaton to remove the magnary unt from the denomnator. The product of a complex number and ts conjugate s a real number, whch does not contan. Multply both the numerator and denomnator of the fracton by the complex number n the denomnator. Smplfy the ratonalzed fracton to fnd the result of the dvson. U4-106

2 In the followng equaton, let a, b, c, and d be real numbers. ( ) ( ) + + a + b c+ d c d a b a+ b + a+ b a b ac bd ad bc Common Errors/Msconceptons multplyng only the denomnator by the complex conjugate ncorrectly determnng the complex conjugate of the denomnator U4-107

3 Guded Practce Example 1 Fnd the result of (3 + ). 1. Rewrte the expresson as a fracton. 3 ( 3+ ) +. Ratonalze the fracton. To ratonalze the denomnator, multply both the numerator and denomnator by. 3+ ( 3+ ) U4-108

4 Example Fnd the result of (10 + ) ( ). 1. Rewrte the expresson as a fracton. 10+ ( 10+ ) ( ). Fnd the complex conjugate of the denomnator. The complex conjugate of a b s a + b, so the complex conjugate of s Ratonalze the fracton by multplyng both the numerator and denomnator by the complex conjugate of the denomnator. ( + ) ( + ) If possble, smplfy the fracton. The answer can be left as a fracton, or smplfed by dvdng both terms n the numerator by the quantty n the denomnator U4-109

5 Example 3 Fnd the result of (4 4 ) (3 4 3 ). 1. Smplfy any powers of. 3. Smplfy any expressons contanng a power of ( ) Rewrte the expresson as a fracton, usng the smplfed expresson. Both numbers should be n the form a + b. 4 4 ( 4 4 ) ( 3+ 4 ) Fnd the complex conjugate of the denomnator. The complex conjugate of a + b s a b, so the complex conjugate of s Ratonalze the fracton by multplyng both the numerator and denomnator by the complex conjugate of the denomnator ( 4 4) ( 3 4) ( 3+ 4) ( 3 4) U4-110

6 6. If possble, smplfy the fracton. The answer can be left as a fracton, or smplfed by dvdng both terms n the numerator by the quantty n the denomnator Example 4 The mpedance of an element can be represented usng the complex number V + I, where V s the element s voltage and I s the element s current n mllamperes. If two elements are n a crcut n parallel, the total mpedance s the sum of the recprocals of each mpedance. If the mpedance of element 1 s Z 1, and the mpedance of element s Z, the total mpedance of the two elements n 1 1 Z1+ Z parallel s +. The followng dagram of a crcut contans two elements, 1 and, Z1 Z Z1 Z n parallel. 1 Element 1 has a voltage of 0 volts and a current of 4 mllamperes. Element has a voltage of 35 volts and a current of 5 mllamperes. What s the total mpedance of the crcut? Use a calculator to estmate the real and magnary parts of the total mpedance. Round your answer to the nearest ten thousandth. 1. Wrte each mpedance as a complex number. Impedance: Z V + I Element 1: Z Element : Z U4-111

7 . Fnd the total mpedance by replacng Z 1 and Z wth the determned values. Z1+ Z ( 0+ 4)+ ( 35+ 5) Z Z ( + ) If the two terms n the denomnator have a common factor, factor the denomnator Fnd the complex conjugate of the complex factor n the denomnator. The complex conjugate of a + b s a b, so the complex conjugate of 17 + s 17. U4-11

8 5. Ratonalze the fracton by multplyng both the numerator and denomnator by the complex conjugate of the complex factor n the denomnator. ( ) ( ) 40( 17+ ) ( 17 ) ( 54) , Fnd decmal approxmatons for the real part and the magnary multple ,000 13,000 13, U4-113