m1001 - tute 2
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7/25/2019 M1001 - Tute 2
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The University of Sydney
MATH1001 Differential Calculus
(http://www.maths.usyd.edu.au/u/UG/JM/MATH1001/)
Semester1, 2012 Lecturer: H. Dullin, R. Marangell, D. Warren, Z. Zhang
Week 3 Tutorial
Assumed knowledge
Basic arithmetic involving complex numbers in Cartesian form (and you can check your fluency
by completing the first preparatory question below).
Objectives
By the end of Week 3 you should
(a) know and be able to use the definitions of modulus, argument and principal argument of a
complex number;
(b) be familiar with the polar form z = r (cosθ + i sinθ ) (which can be abbreviated to z = r cis θ )of a complex number z, be able to pass between polar and Cartesian forms, and be able to
multiply and divide complex numbers in polar form;
(c) understand the formula known as de Moivre’s theorem, which says that
r (cosθ + i sinθ )
n= r n(cos nθ + i sin nθ )
for all n ∈ Z, and be able to use it to write down nth powers of complex numbers;
(d) be able to find all nth roots of a complex number α which is given in polar form, and
understand that these nth roots are the solutions of the polynomial equation zn = α ;
(e) be able to solve simple polynomial equations involving complex numbers, and know thatnon-real solutions of real polynomial equations come in complex conjugate pairs.
Preparatory questions to do before the tutorial
1. Express the following complex numbers in Cartesian form:
(i) (2− i) + (3 + 5i) (ii) (2 − i) − (3 + 5i)
(iii) (2− i)(3 + 5i) (iv) 2 − i
3 + 5i
2. In each case find the modulus and principal argument of the given complex number:
(i) i (ii) 1 + i (iii) 1 − i (iv) −6i (v) 1 +√
3i (vi) −√ 3 − i.
3. Find the solutions to the following equations in Cartesian form:
(i) z2 + 4 = 0 (ii) z2 + z + 1 = 0 (iii) z2 + 2 z + 4 = 0.
Questions to do in the tutorial class
4. Write down polar forms for the following complex numbers (using your answers to Question 2
or otherwise):
(i) i (ii) 1 + i (iii) 1
−i.
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5. Express the following complex numbers in Cartesian form a + bi (where a and b are real):
(i) 2
cos(π /4) + i sin(π /4)
(ii) −4
cos(π /3) + i sin(π /3)
(iii)
cos(π /2) + i sin(π /2)
cos(π /3) + i sin(π /3)
cos(π /6) + i sin(π /6)
.
6. Find polar forms for zw, z/w and 1/ z where
(i) z = √ 3 + i and w = 1 + √ 3i,
(ii) z = 4√
3 − 4i and w = 8i.
7. Find the indicated power, expressing your final answer in Cartesian form:
(i) (1− i)24 (ii) (1 +√
3 i)7 (iii) (3√
3 + 3i)3.
8. Solve the following equations (giving your answers in polar form) and sketch the solutions in
the complex plane. (Note that each equation is of the form zn = α , and so the solutions are
the nth roots of α .)
(i)
z5
= 1
(ii
) z6
= −1
(iii) z3 = 4 − 4√
3i (iv) z4 = 9i
Questions for further practice
9. Write down polar forms for the following complex numbers (using your answers to Problem 2
or otherwise):
(i) −6i (ii) 1 +√
3i (iii) −√
3 − i.
10. Solve the following equations (giving your answers in polar form) and sketch the solutions in
the complex plane:
(i) z3 + i = 0 (ii) z4 = 8√ 2 + 8√ 2i
(iii) z5 + z3 − z2 − 1 = 0, given that z = i is a root.
11. Observe that z3 − 1 = ( z− 1)( z2 + z + 1) and z3 − 8 = ( z− 2)( z2 + 2 z + 4). Now (using your
answers to Problem 3) write down all complex cube roots of 1 and 8. If a is any real number,
can you easily write down all complex cube roots of a3?