20 1. The Complex Numbers r θ r e Figure 1.4.1. The Polar Form r eiθ of a Complex Number. Products, Powers, and Roots. Polar form is particularly useful in dealing with products of complex numbers, since the product has such a simple expression if the numbers are given in polar form. If z1 = r1 eiθ1 and z2 = r2 eiθ2, then the law of exponents implies that (1.4.1) z1z2 = r1r2 ei(θ1+θ2) . Thus, z1z2 is the complex number whose norm (distance from the origin) |z1z2| is the product of |z1| and |z2| and whose argument is the sum of the arguments of z1 and z2. Similarly, the quotient of z1 and z2 is given by (1.4.2) z1/z2 = (r1/r2) ei(θ1−θ2) . Example 1.4.3. Find z1z2 and z1/z2 if z1 = 2 eπi/3 and z2 = 3 e2πi/3. Solution: By (1.4.1) and (1.4.2), we have z1z2 = 6 eπi = −6 and z1/z2 = (2/3) e−πi/3 = 1/3 i 3/3. It is evident from (1.4.1) that the nth power of a complex number z = r eiθ is (1.4.3) zn = rn einθ . From this we conclude that if z = r eiθ, and we choose w = r1/n eiθ/n, then wn = z. Thus, w is an nth root of z. It is not the only one, however. Since z can also be written as z = ei(θ+2πk) for any integer k, each of the numbers wk = r1/n ei(θ/n+2πk/n), where k is an integer, is also an nth root of z. Of course, these numbers are not all different. Those whose arguments differ by an integral multiple of are the same. The numbers w0,w1,w2, · · · , wn−1 are all distinct, but every other wk is equal to one of these. This proves the following theorem. Theorem 1.4.4. If z = r eiθ is a non-zero complex number, then z has exactly n nth roots. They are the numbers r1/n ei(θ/n+2πk/n) for k = 0, 1, 2, · · · , n 1.
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