EXAMPLE 3 Use addition of complex numbers in

EXAMPLE 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 ( ).

EXAMPLE 3 Use addition of complex numbers in real life 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.

EXAMPLE 3 Use addition of complex numbers in real life 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 3 i ohms. The capacitor has a reactance of 4 ohms, so its impedance is – 4 i ohms. Impedance of circuit = 5 + 3 i + (– 4 i) =5–i Add the individual impedances. Simplify.

EXAMPLE 3 Use addition of complex numbers in real life ANSWER The impedance of the circuit is = 5 – i ohms.

EXAMPLE 4 Multiply complex numbers Write the expression as a complex number in standard form. a. 4 i(– 6 + i) b. (9 – 2 i)(– 4 + 7 i) SOLUTION a. 4 i(– 6 + i) = – 24 i + 4 i 2 Distributive property = – 24 i + 4(– 1) Use i 2 = – 1. = – 24 i – 4 Simplify. = – 4 – 24 i Write in standard form.

EXAMPLE 4 Multiply complex numbers b. (9 – 2 i)(– 4 + 7 i) = – 36 + 63 i + 8 i – 14 i 2 = – 36 + 71 i – 14(– 1) = – 36 + 71 i + 14 = – 22 + 71 i Multiply using FOIL. Simplify and use i 2 = – 1. Simplify. Write in standard form.

EXAMPLE 5 Divide complex numbers Write the quotient 7 + 5 i 1 4 i 7 + 5 i = 7 + 5 i 1 – 4 i 1 + 4 i 2 7 + 28 i + 5 i + 20 i = 1 + 4 i – 16 i 2 in standard form. Multiply numerator and denominator by 1 + 4 i, the complex conjugate of 1 – 4 i. Multiply using FOIL. 7 + 33 i + 20(– 1) = 1 – 16(– 1) Simplify and use i 2 = 1. = – 13 + 33 i 17 Simplify.

EXAMPLE 5 Divide complex numbers 13 + 33 i = – 17 17 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? 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 3 i ohms. The capacitor has a reactance of 7 ohms, so its impedance is – 7 i ohms. Impedance of circuit = 5 + 3 i + (– 7 i) Add the individual impedances. =5– 4 i Simplify. ANSWER The impedance of the circuit is = 5 – 4 i ohms.

GUIDED PRACTICE for Examples 3, 4 and 5 11. i(9 – i) SOLUTION i(9 – i) = 9 i – i 2 = 9 i + (– 1)2 = 9 i + 1 = 1 + 9 i Distributive property Use i 2 = – 1. Simplify. Write in standard form.

GUIDED PRACTICE for Examples 3, 4 and 5 12. (3 + i) (5 – i) = 15 – 3 i + 5 i – i 2 = 15 – 3 i + 5 i– (1)2 = 15 – 3 i + 5 i + 1 = 16 + 2 i Multiply using FOIL. Simplify and use i 2 = – 1. Simplify. Write in standard form.

GUIDED PRACTICE 13. for Examples 3, 4 and 5 5 1+i 5 = 5 1+i = = 1–i 5 – 5 i 1 –i + i –i 2 5 – 5 i Multiply numerator and denominator by 1 – i, the complex conjugate of 1 + i. Multiply using FOIL. Simplify and use i 2 = 1. 1 + 1 = 5 – 5 i 2 Simplify.

GUIDED PRACTICE = – 25 i for Examples 3, 4 and 5 Write in standard form.

GUIDED PRACTICE for Examples 3, 4 and 5 14. 5 + 2 i 3 – 2 i 5 + 2 i = 5 + 2 i 3 – 2 i 3 + 2 i 2 15 + 10 i + 6 i + 4 i = 9 + 6 i – 4 i 2 Multiply numerator and denominator 3 + 2 i, the complex conjugate of 3 – 2 i. Multiply using FOIL. 15+ 16 i + 4(– 1) = 9 – 4(– 1)2 Simplify and use i 2 = 11 + 16 i 13 Simplify.

GUIDED PRACTICE 11 + 16 i = – 13 13 for Examples 3, 4 and 5 Write in standard form.