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.

7/25/2019 M1001 - Tute 2

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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?