WTW 124 Assignment
1. Let z ∈ C. If |z| = Re(z), then z is a non-negative real number. Prove or disprove.
2. Describe the following sets in a complex (or Argand) plane.
(i) A = {z ∈ C : |z| = Re(z) + 2},
(ii) B = {z ∈ C : |z − i| < 2},
(iii) C = {z ∈ C : |z − 1| + |z + 1| = 7}.
3. Let |z| = 1, where z ∈ C, with z ̸= 1. Find Re
1
1−z
.
4. Find the following,
(i)
i(2+i)
(1−i)2
(ii) |(1 + i)(2 − 3i)(4i − 3)|
5. Let z1 , z2 ∈ C. Prove that;
(i) |z1 − z2 | ≤ |z1 | + |z2 |,
(ii) ||z1 | − |z2 || ≤ |z1 − z2 |.
6. Show geometrically (sketch) that the non-zero complex numbers z1 and z2 satisfies
|z1 + z2 | = |z1 | + |z2 | if and only if z1 and z2 have the same argument.
7. Let z1 = 2 − i and z2 = 1 + i. Use parallelogram law to construct the following
(i) 2z1 − 3z2 ,
(ii) z1 − z2 ,
(iii) z1 + z2 .
8. Prove that the vector z1 = a + ib is parallel to the vector z2 = c + id if and only if
Im(z1 z2 ) = 0.
9. Is the function cos(arccos(x)) a polynomial on R?
10 (a) Find all the complex number(s) that satisfies the equations below:
(i) (2 + 3i)z + 3i = 2.
1
(ii) z 2 − 2 = 3iz.
(b) Use induction to prove the following statement:
The complex number (1 + 2i)n has an odd real part for any integer n ⩾ 2.
(c) Let α be the last digit of your student number. Sketch the following sets in the Argand
plane (aka complex plane):
(i) A = {z ∈ C : |z| ⩽ α + 2, Im(z) ≥ α + 1}.
(ii) B = {z ∈ C : |z + 2i| = |z − 2|}.
√
11. Let z = 21 + 23 i and w =
√
√
2
2
+
2
2 i.
(a) Find zw in polar form, with principal argument.
7π
7π
(b) Use (a) to find the exact values cos( 7π
12 ), sin( 12 ) and tan( 12 ).
π
π
π
), sin( 12
) and tan( 12
).
(c) Find the exact values of cos( 12
12. Let x = 1 + 3i and y = 2 + i.
(a) Find
x
in standard form.
y
(b) Use (a) to show that
π
= arctan
4
3
1
− arctan
.
1
2
13. (a) Compute
(− √22 − √22 i)19 .
Write your answer in polar form with principal argument.
(b) Determine all roots of the polynomial 2z 3 − 15z 2 + 44z − 39 given that 3 − 2i is a root.
14. Let f (z) = z 4 − iz 3 + wz − 8i for some unknown complex number w.
(a) Find w, given that i is a root of f(z).
(b) Find all roots of f . Give all answers in polar form with principal argument.
2