Higher algebra for the undergraduate by Marie Weiss

By Marie Weiss

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Find the three cube roots of 8i. Write 8i = 8(cos 1r/ 2 + i sin r / 2). Let p(cos 8 + i sin 8) be a cube root. T hen p3 (cos 38 + i sin 38) = 8(cos 1r/ 2 + i sin' ,r/ 2), and p3 = 8 and 30 = r / 2 + 2k11'. Hence p = 2, and 8 = 1r/6 + 2k1r/3. The three cube roots of 8i are: 2 ( cos; 2 ( COS = v'3 + i, 511' + t. Sill . 511') = - V_~3 + t,. 6 6 311' 2 ( cos - -2 Example 2. + i sin ~) + . sin. -3r) 2 i = - 2i.. Find the n nth roots of 1. These roots are often called then nth roots of unity. Using the I notation introduced above, ,ve have r = 1, and "' = 0, and hence p = 1, and O = 2k11'/ n.

8 • De Moivre's theorem The product and quotient of t wo complex numbers when written in polar form give us some interesting results. Let z1 = P1 (cos 81 + i sin 81) and z2 = P2(cos 82 + i sin 82). Then z1z2= P1P2[cos 81 cos 82-sin 81 sEn t72+i(sin 81 cos t72+cos 81 sin 82)) = P1P2[cos (/11 + 82) + i sin (/11 + 112)). THE n nth ROOTS O F A COM PLEX NUMBER Similarly 35 z1 p1(cos 81 + i sin 81 ) (cos 82 - i sin 62 ) - = • z2 P2(cos 82 + i sin 82) (cos 82 - i sin 82) = Pi [cos P2 (81 - 82) + i sin (81 - 82)].

Primitive nth roots of unity. An nth root z of 1 is a primitive nth root of 1 if zn = I and if zm ;re 1, when O < m < n. + Theorem 3. Let R = cos 21r/ n i sin 21r/ n. then Rk is a primitive n / dth root of unity. If (k, n) = d, Let k = k 1d and n = n1d so that (ki, n 1) = I. Then Rk = cos 2k 1d,r/ n 1d + i sin 2k1d1r/ n1d = cos 2k11r/ n 1 + i sin 2k11r/ n 1. Now Rk is an n 1 = n / dth root of unity, for (Rk)"' -:- cos 2k 11r + i sin 2krir. Moreover, Rk is a primitive n/dth root of unity, for if (Rk)m = I = cos 2k 1m ,r/ n1 + i sin 2k1m1r/ n 1, k 1m / n 1is an integer.

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