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Please submit written (or typeset) solutions to the textbook problems

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Please submit written (or typeset) solutions to the textbook problems. Justify your answers. 
Practice with complex numbers (show work. check answers in back) 1.1(e), 1.2(b), 1.3(b), 1.4(c), 1.5(a), 1.9 
1.11(d) [hint: use the result of 1.19] 1.19 1.25 [hint: use the ordinary triangle inequality in a clever way] 1.26 1.27 1.28 

Exercises 
1.1. Let z = 1 + 2i and w = 2 — i. Compute the following: 

(a) z + 3w (b) w — z 
(c) z3 (d) Re(w2 + w) 1.2. Find the real and imaginary parts of each of the following. (a) zz:--7.4 for any a E JR (c) -1+2'13 (b) 
1.3. Find the absolute value and conjugate of each of the following: 
(a) —2 + i (b) + + 3i) 
1.4. Write in polar form: (a) 2i (b) 1 + i (c) —3 + f i rocfnmnifar form: 

(e) z2+Z-+i (d) i" for any n E Z (d) (1+ 06 (g) f — 
(h) (1.71y 

 Write in rectangular form: (a) (b) 34eif 

(C) -e29). (d) 2 e4-i 1.6. Write in both polar and rectangular form: (a) et (5)i (b)T etfut 
1.7. Show that the quadratic formula works. That is, for a, b, c E R with a A 0, prove that the roots of the equation az2 + bz + c = 0 are 
—b 4ac 2a • 
Here we define Vb2 — 4ac —b2 4ac if the discriminant b2 — 4ac is negative. 1.8. Use the quadratic formula to solve the following equations. (a) z2 + 25 — 0 (c) 5z2 + 4z + 1 = 0 (e) zz = 2z (b) 2z2+2z+5-0 (d) — z 1 1.9. Find all solutions of the equation z2 + 2z + (1 — i) — 0. 1.10. Fix a E C and b e IR. Show that the equation1z21 + Re(az) + b 0 has a solution if and only if la21 > 4b. When solutions exist, show the solution set is a circle. 

(a) 26 = 1 (C) — v (d) (13) = —16 z6 — 0 — 2 = t) 1.12. Show that 121 = 1 if and only if = 2. 1.13

1.13. Show that (a) z is a real number if and only if z = 2; (b) z is either real or purely imaginary if and only if (2)2 = 1.14. Prove Proposition 1.1. 1.15. Show that if z, z2 = 0 then z, = 0 or z2 = 0. 1.16. Prove Proposition 1.3. 

1.17. Fix a positive integer s. Prove that the solutions to the equation z" — 1 are precisely z = where m E Z. (Hint: To show that every solution of z" =1 is of this form, first prove that it must be of the form z = ebri: for some a E R, then write a = m -1- b for some integer in and some real number 0 < b < 1, and then argue that b has to be zero.) 
L18. Show that 
z5 — 1 = (z — 1) (z2 +2zcos +1) — 2zcos aff +1) and deduce from this closed formulas for cos and cos 1.19. Fix a positive integer n and a complex number w. Find all solutions to zn = w. (Hist: Write w in terms of polar coordinates.) 1.20. Use Proposition 1.3 to derive the triple angle formulas: n