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READING Note that while a component’s resistance or reactance is a real number, its impedance is a complex number. AVOID ERRORS When simplifying an expression that involves complex numbers, be sure to simplify i 2 to 21. E XAMPLE 3 Use addition of complex numbers in real life ELECTRICITY Circuit components such as resistors, inductors, and capacitors all oppose the flow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. A circuit’s total opposition to current flow is impedance. All of these quantities are measured in ohms (Ω). Component and symbol Resistor Inductor Capacitor Resistance or reactance R L C Impedance R Li 2Ci The table shows the relationship between a component’s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit shown above. Solution The resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor has a reactance of 3 ohms, so its impedance is 3i ohms. The capacitor has a reactance of 4 ohms, so its impedance is 24i ohms. Impedance of circuit 5 5 1 3i 1 (24i) Add the individual impedances. 5 5 2 i Simplify. c The impedance of the circuit is 5 2 i ohms. MULTIPLYING COMPLEX NUMBERS To multiply two complex numbers, use the distributive property or the FOIL method just as you do when multiplying real numbers or algebraic expressions. E XAMPLE 4 Multiply complex numbers Write the expression as a complex number in standard form. a. 4i(26 1 i) b. (9 2 2i)(24 1 7i) Solution a. 4i(26 1 i) 5224i 1 4i 2 Distributive property 5224i 1 4(21) Use i 2 521. 5224i 2 4 Simplify. 524 2 24i Write in standard form. b. (9 2 2i)(24 1 7i) 5236 1 63i 1 8i 2 14i 2 Multiply using FOIL. 5236 1 71i 2 14(21) Simplify and use i 2 521. 5236 1 71i 1 14 Simplify. 3 Ω 5 Ω Alternating current source 5222 1 71i Write in standard form. 4 Ω 4.6 Perform Operations with Complex Numbers 277
REWRITE QUOTIENTS When a quotient has an imaginary number in the denominator, rewrite the denominator as a real number so you can express the quotient in standard form. COMPLEX CONJUGATES Two complex numbers of the form a 1 bi and a 2 bi are called complex conjugates. The product of complex conjugates is always a real number. For example, (2 1 4i)(2 2 4i) 5 4 2 8i 1 8i 1 16 5 20. You can use this fact to write the quotient of two complex numbers in standard form. E XAMPLE 5 Divide complex numbers Write the quotient 7 1 5i } 5 7 1 5i } p 1 1 4i } 1 2 4i 1 2 4i 1 1 4i 278 Chapter 4 Quadratic Functions and Factoring 5 5 5 7 1 5i } 1 2 4i in standard form. 7 1 28i 1 5i 1 20i 2 } 1 1 4i 2 4i 2 16i 2 7 1 33i 1 20(21) } 1 2 16(21) 213 1 33i } 17 Multiply numerator and denominator by 1 1 4i, the complex conjugate of 1 2 4i. Multiply using FOIL. Simplify and use i 2 5 1. Simplify. 52 13 } 17 1 33 } 17 i Write in standard form. ✓ GUIDED PRACTICE for Examples 3, 4, and 5 10. WHAT IF? In Example 3, what is the impedance of the circuit if the given capacitor is replaced with one having a reactance of 7 ohms? Write the expression as a complex number in standard form. 11. i(9 2 i) 12. (3 1 i)(5 2 i) 13. 5 } 1 1 i 14. 5 1 2i } 3 2 2i COMPLEX PLANE Just as every real number corresponds to a point on the real number line, every complex number corresponds to a point in the complex plane. As shown in the next example, the complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis. E XAMPLE 6 Plot complex numbers Plot the complex numbers in the same complex plane. a. 3 2 2i b. 22 1 4i c. 3i d. 24 2 3i Solution a. To plot 3 2 2i, start at the origin, move 3 units to the right, and then move 2 units down. b. To plot 22 1 4i, start at the origin, move 2 units to the left, and then move 4 units up. c. To plot 3i, start at the origin and move 3 units up. d. To plot 24 2 3i, start at the origin, move 4 units to the left, and then move 3 units down. 22 1 4i 24 2 3i i imaginary 3i 1 real 3 2 2i
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