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**Unformatted text preview: **sentially, Theorem 11.17 says that to ﬁnd the nth roots of a complex
number, we ﬁrst take the nth root of the modulus and divide the argument by n. This gives the
π
ﬁrst root w0 . Each succeessive root is found by adding 2n to the argument, which amounts to
π
rotating w0 by 2n radians. This results in n roots, spaced equally around the complex plane. As
an example of this, we plot our answers to number 2 in Example 11.7.4 below.
Imaginary Axis
2i
w1 w0
i −2 0 −1 1 Real Axis 2 −i
w2 w3
−2i The four fourth roots of z = −16 equally spaced 2π
4 = π
2 around the plane. We have only glimpsed at the beauty of the complex numbers in this section. The complex plane
is without a doubt one of the most important mathematical constructs ever devised. Coupled with
Calculus, it is the venue for incredibly important Science and Engineering applications.17 For now,
the following Exercises will have to suﬃce. 17 For more on this, see the beautifully written epilogue to Section 3.4 found on page 226. 11.7 Polar Form of Complex Numbers 11.7.1 855 Exercises 1. Find a polar representation for each complex number z given below and then identify Re(z ),
Im(z ), |z |, arg(z ) and Arg(z ).
√ (a) z = 9 + 9i (b) z = −5i (c) z = − 3
2 − 1i
2 (d) z = −7 + 24i 2. Find the rectangular form of each complex number given below. Use whatever identities are
necessary to ﬁnd the exact values.
(a) z = 12cis − π
3 (c) z = 2cis 7π
8 π
(b) z = 7cis − 34 (d) z = 5cis arctan...

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