Chap 1 The Number System

Solve the equation z 2 z 1 0 if the solutions are z1

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Unformatted text preview: ent in part (b)? f) Let a, b ∈ R be fixed positive real numbers. Repeat steps (a)–(e) with a + bi in place of 3 + 4i. g) What can we say about the effect of multiplying a complex number by i? h) What can we say about the effect of multiplying a complex number by 2i? 9. Let z1 = 2 + 4i and z2 = 1 (1 − 5i). Find each of the following numbers: 2 a) z1 z2 b) (z1 + 2z2 )2 c) iz1 + 2z2 √ 10. Let z1 = 5, z2 = 3 + 4i and z3 = 1 − 3i. Evaluate each of the following numbers: a) z1 b) z2 c) z1 − z2 d) z2 z3 e) Re(z1 /z2 ) f) arg z3 11. Suppose that z = x + iy , where x, y ∈ R and i2 = −1. Write down each of the following numbers in terms of x, y and i: zz a) Re(z ) b) Im(z ) c) z d) |z |2 e) Im(z −1 ) f) |z |2 12. Express each of the following numbers in the form x + y i, where x, y ∈ R: a) (1 + 3i)3 b) (3 − 2i)2 − (3 + 2i)2 3 + 4i c) (1 + i + i2 + i3 )100 d) 5 + 6i 13. Find the real and imaginary parts of each of the following numbers: 1+i 1 + 2i x + yi a) , b) c) . 1−i 3 + 4i i 14. Suppose that a + bi = c + di,...
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