Exercises
Compute the following complex numbers in the form \(z = a+ib\):
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\((1-i)(2+i)\)
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\((1+i)^3 (:=(1+i)(1+i)(1+i))\)
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\(i^{89}\)
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\(e^{-i\pi}\)
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\(3 (\cos (\pi / 2) + i \sin (\pi / 2))\)
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\(\frac{\sqrt 3 + \sqrt 2 i}{\sqrt 2 - \sqrt 3 i}\)
Depict these complex numbers on the complex plane.
Prove that a complex number \(z\) is a real number exactly if \(z = \overline z\).
Compute the trigonometric form of the following complex numbers:
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\(\sqrt 3-i\).
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\((1-i)^5\)
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\(\frac{1+i}{(1-i)(\sqrt 3+i)}\)
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\(i^n\), where \(n\) is a natural number.
Exercise 1.4
(See Solution 1.1.) Compute the algebraic and the trigonometric form of \(z = \left ( \frac{i-1}{i+1}\right )^3\).
Exercise 1.5
(See Solution 1.2.) Find all complex numbers \(z\) satisfying the equation
Exercise 1.6
(See Solution 1.3.) Compute the solutions of the equation \((\overline z)^3 = 8i\) in algebraic and in trigonmetric form. Draw a picture of these solutions.