College algebra . i sin ^i), c + di = r2(cos 62 + i sin 6^. Abts. 82-83] CONJUGATE COMPLEX NUMBERS 115 By actual multiplication, (a + &i)(c + di) = rir2[cos $1 cos 62 -f i(sin 61 cos d^ + cos ^1 sin 62)— sin di sin 62]= (ei + 62) + t sin(«i + ^2)]. Hence, the modulus of the productof two complex numbers is the productof their moduli and the argument isthe sum of their arguments. The point P, which represents(a + bi){c + di) may then be con-structed by drawing through 0 a linemaking an angle 6 = 6^ + 0^ with theline OX, and constructing on thislength a segment OP, whose lengthis riTi-
College algebra . i sin ^i), c + di = r2(cos 62 + i sin 6^. Abts. 82-83] CONJUGATE COMPLEX NUMBERS 115 By actual multiplication, (a + &i)(c + di) = rir2[cos $1 cos 62 -f i(sin 61 cos d^ + cos ^1 sin 62)— sin di sin 62]= (ei + 62) + t sin(«i + ^2)]. Hence, the modulus of the productof two complex numbers is the productof their moduli and the argument isthe sum of their arguments. The point P, which represents(a + bi){c + di) may then be con-structed by drawing through 0 a linemaking an angle 6 = 6^ + 0^ with theline OX, and constructing on thislength a segment OP, whose lengthis riTi- 83. Conjugate complex numbers. Numbers which differ only in thesign of the imaginary parts are calledconjugate numbers. Thus, 3-l-2i and 3—2 i are conjugate. -Since {a + bi) + {a — M)=2a,(a + bi){a — bi)= a^ + ¥,and (a + bi)—{a — bi)=2bi, we see that the sum and the product of two conjugate complexnumbers are real numbers, but the difference of two conjugatecomplex numbers is an imaginary Fig. 21. EXERCISES Multiply both analytically and graphically, finding the arguments andmoduli of the products. 1. (S + V3i)(2 + 2i). Solution. (3 + VSi)(2 + 2i) = 6+ 6i + 2y/3i + 2\/31^ = 6 - 2^3 + i(6 + ZVS). Putting the numbers in the polar form, we have, 3 + V3 i = 2-n/3(oos30 + isin30°),2 + 2 i = 2 V2 (cos 45° + isin45°). 116 COMPLEX NUMBERS [Chap. XIL r Hence, oP n = 2v3, ra = 2V2, Si = 30°, ffj = 45°.The modulus of the product is, then, ri»-2 = 4v6,and the argument is 75°. Let Pi and P2 in Fig. 22 represent the twogiven numbers. Through 0 draw a line mak-ing an angle of 75° with the line OX. On thishue measure off the distance /y?^ OP = 4 VS. The point P then represents the product of the l^^vfX V two numbers. 2. (3 + v3i)(2 + fV3i). 0 A 3. (l+V3i)(4 + |V3i). Fig. 22. 4. (V2 + V-2)2. 5. (1 + i)^. 6. (1 + Q2(2- 2V3i). 7. (-2 + 2i)(2 + 2J). 8. (-2-2i)(2 + 2i). 9. (l + i)(l + 2i)(l+3i). 1( 3. (0 + 3i)(2 + i). ai. (0 + 2i)(0-2i). * 84. De Moivres theorem. If t
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