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Lecture 9

MATH135 Lecture 9: Exercises - Complex numbers


Department
Mathematics
Course Code
MATH135
Professor
Roxane Itier Itier
Lecture
9

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MATH 135, Winter 2015
Exercises: Complex Numbers
Part 1
1. Express the given numbers in standard form:
(a) p32iqp32iq(b) p32iqp32iq(c) p32iqp32iq
(d) 32i
32i(e) p43iq1(f) 1
p12iqp2iq
(g) 23i
4i(h) p13iq2(i) p25iq3
(j) 2
1i1
i(k) 1
i47 (l) p?2iq2
p?2iqp1?2iq
2. Let z, w PCwith w0. Prove that
z
w
z
w,
z
w
|z|
|w|.
(Hint: wpz{wq  z.) Keep these formulas in your toolbox!
3. Find the modulus of each of the following numbers.
(a) 3ip2iqp32iqp1iq, (b) p43iqp2iq
p1iqp13iq, (c) p34iq5.
4. Let zxyi, where x, y PR. Find the real and imaginary parts of the following numbers.
(a) z2(b) z3(c) z1
2z5(z5
2)
5. Let z, w PC. Are the following true or false? Justify.
(a) Repzwq  RepzqRepwq(b) RepizqImpzq(c) ImpzqImpzq
(d) 1
zw 1
z1
w(e) 1
zw1
z1
w(f)
zw
zw
1
(In (d), (e) and (f), assume that z0, w0 and z w.)
6. Prove that if two integers mand nare sums of two squares, i.e., ma2b2and nc2d2
for some integers a, b, c, d, then so is mn.
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7. Express the following numbers in polar form rcis θrpcos θisin θq, with r¡0 and
πθ¤π:
z12i, z2?6?2i, z3 55i.
Plot them on the complex plane.
8. Convert the complex numbers
w1?33i, w2?33i, w3 ?33i, w4 ?33i
into polar form rpcos θisin θq, where r¡0 and 0 ¤θ2π. Plot these numbers on the
complex plane.
9. Let
w1
2
cos π
6
isin π
6
1
2eiπ
6.
Convert w,w2and w3into standard form. Sketch these points on the complex plane.
10. Sketch the following regions of the complex plane.
(a) A zPC
|zi| ¥ 2(
(b) B zPC
1Repzq  2,Impzq ¡ 0(
(c) C zPC
Rep1iqz¤0(
(d) D zPC
0¤Repz2q ¤ 1(
(e) E zPC
0¤Impz2q ¤ 1(
(f) F zPC
Repz3q  0(
11. Find all complex numbers zxyi (where x, y PR) which satisfy the equation
|z2|  |z4i|.
Sketch these points together with 2 and 4ion the complex plane.
12. Describe the set of all zPCsuch that
z1
z1
2.
13. Solve the system of equations
|zi|  ?2,
|z1|  ?2
for zPC. Provide a geometric interpretation.
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14. Given z1, z2PC, show that
|z1z2|2 |z1|2|z2|22 Repz1z2q.
Hence, prove the parallelogram law
|z1z2|2|z1z2|22|z1|22|z2|2.
Make a sketch the points 0, z1,z2and z1z2to justify the name.
15. Prove the polarization identity: for any z, w PC,
4zw  |zw|2|zw|2i|ziw|2i|ziw|2.
16. Let z1and z2be non-zero complex numbers with respective arguments θ1and θ2.
(a) Prove that
Repz1z2q|z1||z2|cospθ1θ2q,Impz1z2q|z1||z2|sinpθ1θ2q.
(b) Using (a), deduce that z1z2PRif and only if the three points z1, z2and 0 are collinear,
i.e., they lie on a straight line. (Hint: When is sin x0?)
17. The unit circle
T tzPC| |z|  1ute|θPRu
has special significance in mathematics and physics. (The T stands for ‘torus’.)
(a) Given any zPCwith z0, prove that zPTif and only if z1z.
(b) Given any z, w PT, prove that zw PTand z1PT.
(c) Show that for every positive integer n,Tcontains all n-th roots of unity.
18. Assume that uand vare complex numbers such that |u|  |v|and that uv ris a positive
real number. Show that vu.
19. Find all square roots of 2i. Express your answers in standard form.
20. Find all sixth roots of 8. Express your answers in standard form.
21. Find all possible values of 4
?i(i.e., find all the fourth roots of i). Express your answers
in polar form rercis θ, where πθ¤π.
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