Data Structures and Algorithms - COMS21103
Introductory Complex Numbers Problem Sheet
Disclaimer This sheet is intended to be completed with reference to the introductory material on
complex numbers from Brigham Young University, linked from the COMS21103 course website (http:
//morse.cs.byu.edu/450/lectures/lect13/complex.slides.printing.pdf).
A: Familiarity with Complex Numbers
1. Express the following in the form a+bi:
(a) (2 + 3i)+(−4+5i)−(9 −3i)/3.
(b) (−5+3i)(−4+8i)
(c) (−1−2i)/(−4+3i)
(d) (−5−i)(−7+8i)/(2 −4i)
2. Consider the complex number z=−1−√3i:
(a) Find z∗(the complex conjugate of z).
(b) Find |z|(the magnitude of z).
(c) Find φ(z) (the phase of z).
(d) Express zin polar form.
(e) Express z∗in polar form.
(f) Express z7in polar form.
(g) Express z7in cartesian form (a+bi).
B: Roots of Unity The N-th roots of unity are the complex numbers satisfying the equation zN= 1.
They have values given by:
ωj
N=e2πij
Nfor j= 0,1. . . , N −1 (where i=√−1 as above).
1. Express the 4-th roots of unity in polar form (simplifying your answers).
2. Let z=ω3
4. Verify that z4= 1 as required.
3. Express the 8-th roots of unity in polar form (simplifying your answers).
4. Draw the 4-th and 8-th roots of unity on the complex plane (an Argand diagram).
You should observe from your answers and diagram that all the even numbered 8-th roots of unity
align with a 4-th root of unity and further that
ω2j
8=ωj
4for j= 0,1. . . , 3.
5. Show that ωdj
dN =ωj
Nfor j= 0,1. . . , N −1 (refered to in the notes as the cancellation lemma).
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