Problem Sheet 2. Complex Numbers
Please attempt these questions before your second tutorial on Friday February 13th.
The first Course Test will take place in the tutorial on February 20th. The questions will
be quite similar to the questions on the first two example sheets. So make sure that you
can do them.
Questions marked with a dagger † can be skipped if you run out of time.
0. Write 8i in polar form and use this to find
√
8i. Compare with Question 6 of Ex Sheet 1.
1. Expand (1 + i)4 and (1 − i)3 , writing the answer in the form a + bi.
2. [From the 2009 Final Exam.] Let z = 1 − i and w = e3πi/7 . Find zw,
z
w
and z + w, writing your
answers in both Cartesian and polar form.
3. Let z = 3 cos
π
3
+ i sin
π
3
.
(a) What are z 2 , z 3 , and z 100 in polar form? (You do not need to simplify the last two).
(b) What are the possible values of
√
1
z ? What about z 4 ?
4. [From the 2009 Final Exam.] Find all solutions z of the equation z 5 = −i and plot your answers on
an Argand diagram.
5†. Let z = 3 + i. Write z in polar form and find the modulus and argument for the six points z, z̄, z 2 ,
z̄ 2 , 1/z̄ and z/z̄. (If you think about it, you really only need to find one argument.) Plot the points on
an Argand diagram.
6. (a) If z = 3e1.1i , write down z 2 in polar form.
(b†) If z = 3e1.1i , write down z 3 , z −1 and z̄ in polar form.
7†. (a) Let z = a + bi. Calculate z 2 and z̄ 2 , and show that (z̄)2 = (z 2 ).
(b) If z, w are complex numbers, can you guess a formula for zw ?
8.† Show that the real and imaginary parts of ex+yi are ex cos(y) and ex sin(y), respectively. Use this to
find the modulus of e1+2i .
1