Solutions for assignment

# Solutions for assignment - Solutions for assignment 1 1 of...

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Section 1.1 Question 15: The simplest way to do this is to use polar co-ordinates, but I shall use the exponent laws. (The laws of exponents are part of the laws of algebra, and as such are just as valid for the complex numbers as they are for the reals or rational numbers - as long as all t he exponents are integers. If the exponents are fractional, irrational, or complex it gets a little trickier, as we shall see later on in the course). Since i^4 = 1, we see that i^{4k} = (i^4)^k = 1^k = 1, i^{4k+1} = i^{4k} i = 1 i = i, etc. Question 20: All the normal rules of algebra apply: (a) 2iz = 4, so z = 4/(2i) = -2i (b) z = (1-5i)(1-z), so (2-5i)z = (1-5i), so z = (1-5i)/(2-5i) = (-23 - 15i)/29 = -23/29 - 15i/29. (c) z(8z + (2-i)) = 0, so either z = 0 or z = -(2-i)/8 = -1/4 + i/8. (d) z^2 = -16, so z = +4i or -4i. For the purposes of this course, "solving for z" means expressing z in one of the two standard forms (Cartesian or polar). Unless the problem asks you to simplify, however, then you may get away with an unsimplified form such as (1-5i)/(2-5i). Just don't count on it.

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