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Programme 2: Complex numbers 2
PROGRAMME 2
COMPLEX
NUMBERS 2
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Notation
Positive angles
Negative angles
Multiplication
Division
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Notation
The polar form of a complex number is
readily obtained from the Argand diagram
of the number in Cartesian form.
Given:
z  a  jb
then:
r 2  a2  b2 so r  a 2 b 2
and
tan  b so   tan 1 b
a
a
The length r is called the modulus of the complex number and the angle q is
called the argument of the complex number
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Positive angles
The shorthand notation for a positive angle (anti-clockwise rotation) is
given as, for example:
z r 
With the modulus outside the bracket and the angle inside the bracket.
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Negative angles
The shorthand notation for a negative angle (clockwise rotation) is given as,
for example:
z r 
With the modulus outside the bracket and the angle inside the bracket.
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Multiplication
When two complex numbers, written in polar form, are multiplied the
product is given as a complex number whose modulus is the product of the
two moduli and whose argument is the sum of the two arguments.
If z1  r1  cos1  j sin 1  and z2  r2  cos 2  j sin  2 

then z1z2  r1r2 cos1  2  j sin1  2 
STROUD

Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Division
When two complex numbers, written in polar form, are divided the quotient
is given as a complex number whose modulus is the quotient of the two
moduli and whose argument is the difference of the two arguments.
If z1  r1  cos1  j sin 1  and z2  r2  cos 2  j sin  2 
then
STROUD

z1 r1
 cos1  2  j sin1  2 
z2 r2

Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Roots of a complex number
De Moivre’s theorem
nth roots
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Roots of a complex number
De Moivre’s theorem
If a complex number is raised to the power n the result is a complex number
whose modulus is the original modulus raised to the power n and whose
argument is the original argument multiplied by n.
If z  r  cos  j sin  
then z n   r  cos  j sin    = r n  cos n  j sin n 


n
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Roots of a complex number
nth roots
There are n distinct values of the nth roots of a complex number z. Each root
has the same modulus and is separated from its neighbouring root by
2
radians
n
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Expansions
Trigonometric expansions
Since:
 cos  j sin  
n
 cos n  j sin n
then by expanding the left-hand side by the binomial theorem we can find
expressions for:
cos n and sin n in terms of powers of cos and sin
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Expansions
Trigonometric expansions
Let:
z  cos  j sin then
1
 cos  j sin
z
so that:
1
z   2cos
z
zn 
1
 2cos n
n
z
1
z   j 2sin 
z
zn 
1
 j 2sin n
n
z
from which we can expand cosn  and sinn  in terms of powers of cos and sin
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Polar-form calculations
Roots of a complex number
Expansions
Loci problems
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Loci problems
The locus of a point in the Argand diagram is the curve that a complex
number is constrained to lie on by virtue of some imposed condition.
That condition will be imposed on either the modulus of the complex
number or its argument.
For example, the locus of z
constrained by the condition
that
z 5
is a circle
STROUD
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Loci problems
The locus of z constrained by the condition that
arg z 
is a straight line
STROUD

4
Worked examples and exercises are in the text
Programme 2: Complex numbers 2
Learning outcomes
Use the shorthand form for a complex number in polar form
Write complex numbers in polar form using negative angles
Multiply and divide complex numbers in polar form
Use de Moivre’s theorem
Find the roots of a complex number
Demonstrate trigonometric identities of multiple angles using complex numbers
Solve loci problems using complex numbers
STROUD
Worked examples and exercises are in the text