Chap 1 The Number System

Then z c2 x iy a ib2 x a iy b2

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Unformatted text preview: as well as all the roots of (17) To study (16), note that 1 + roots of (16) are given by z= in other words, z1 = √ 3 2 cos π π + i sin , 9 9 √ z2 = √ 3 2 cos 7π 7π + i sin 9 9 , z3 = √ 3 2 cos 13π 13π + i sin 9 9 . √ 3 √ z3 = 1 − √ 3i. z3 = 1 + √ 3i, 3i = 2(cos(π/3) + i sin(π/3)). It follows from Proposition 1F that the 2 cos π 2kπ + 9 3 + i sin π 2kπ + 9 3 , where k = 0, 1, 2; To study (17), note that 1 − roots of (17) are given by z= in other words, z4 = √ 3 2 cos 5π 5π + i sin 9 9 √ 3 3i = 2(cos(5π/3) + i sin(5π/3)). It follows from Proposition 1F that the 2 cos 5π 2kπ + 9 3 + i sin 5π 2kπ + 9 3 , where k = 0, 1, 2; , z5 = √ 3 2 cos 11π 11π + i sin 9 9 , z6 = √ 3 2 cos 17π 17π + i sin 9 9 . 1.8. Analytic Geometry In classical analytic geometry, we express the equation of a locus as a relation between x and y . If we write z = x + iy , then such an equation can be equally well described as a relation between z and z . However, it is important to bear in mind that a complex equation is usually equivalent to two rea...
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