Algebra Exercise Sheet 2 PDF
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Uploaded by KindlyFrancium3194
City St George's, University of London
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This document is an algebra exercise sheet covering complex numbers. It includes problems involving sketching complex numbers, converting between polar and Cartesian forms, finding roots of complex numbers, and applying De Moivre's theorem. The exercises are ideal for undergraduate level students.
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ALGEBRA EXERCISE SHEET 2 1. Sketch √ the√points representing √ the following complex numbers in the complex plane. a) 2, 2i, 2 + 2i, 2 − 2i, −2 + 3i, −2 − 3i. b) z, zi, zi2 , zi3 , zi4 where z = 1 + i. What does multiplication by i mean geometrica...
ALGEBRA EXERCISE SHEET 2 1. Sketch √ the√points representing √ the following complex numbers in the complex plane. a) 2, 2i, 2 + 2i, 2 − 2i, −2 + 3i, −2 − 3i. b) z, zi, zi2 , zi3 , zi4 where z = 1 + i. What does multiplication by i mean geometrically? 2. Find the set of points of the plane described by the following equations: a) (1 − i)z + (1 + i)z̄ = 2, b) (1 − αi)z = (1 + αi)z̄, for a real number α, c) |z|2 = 1, d) |z − 1|2 = 2. √ √ 6−i 2 3. a) Write the following complex numbers in polar form: 5, −1, −1 + i, , √ 2 2 + i 3, 2 − 7i. b) Write 1+i 1−i in polar form. 4. a) Write the following numbers in Cartesian form: 5eiπ/4 , e−iπ /7. b) Write (8eiπ/3 )/(2eiπ/2 ) in Cartesian form. c) Write 2eiπ/4 + 3eiπ/3 in Cartesian form. 5. a) Express the following in the form cos β + i sin β. i) (cos θ + i sin θ)7 ii) (cos(π/4) + i sin(π/4))−2 b) Use De Moivre’s theorem to find expressions for cos(3θ) and sin(3θ) in terms of powers of cos θ and/or sin θ. Then rewrite the answer so that cos(3θ) is written only in terms of cos(θ) and sin(3θ) only in terms of sin(θ). √ √ 6. (i) Write 1 + 3i in polar form and find the 3rd roots of 1 + 3i. √ (ii) Find all solutions of the equation (z̄ − i)3 = 1 + 3i. Express the solutions in Cartesian form. Your answer may involve trignometric expressions. 7. Find the nth roots of the following complex numbers and plot them on the complex plane a) z = −1 and n = 3; , b) z = 1 and n = 4; , c) z = i and n = 5.
