Chapter 2: Polar form and roots of complex numbers
27
Similarly (r(cos +i sin )4 = r4 (cos 4+i sin 4). It is easy to see that for n cfw_1, 2, 3, . . .
(r(cos + i sin )n = rn (cos n + i sin n).
It is a useful exercise to prove this by induction. In fact,
Chapter 2: Polar form and roots of complex numbers
25
Subtraction is done in a similar way. Generally, there is little point in changing a complex
number from Cartesian form to polar form to perform addition or subtraction. Using polar form for addition a
Chapter 2: Polar form and roots of complex numbers
21
y
1
x
r
3i
arg(1 + 3i) lies in the third quadrant. Since tan = 3 then = 4/3. (We
could also write = 2/3 equally correctly.) Therefore 1 3i = 2(cos 4/3 +
i sin 4/3) in polar form.
iii) Find the modulus
Chapter 1: Numbers and sets
11
Example 1.5b The solutions to x2 +6x+25 = 0 must be complex since b2 4ac = 64 < 0.
Using the quadratic formula, the solutions are found to be 3 + 4i and 3 4i. These
solutions are complex conjugates of each other. This will b
Chapter 2: Polar form and roots of complex numbers
31
2i
11
12
2
3
4
2
3
2
2
3
2
5
12
2i
How many complex roots does a real number have? Let us look at the fourth roots of 16. You
already know that 24 = (2)4 = 16. Hence 2 and 2 are fourth roots of 16.
Chapter 1: Numbers and sets
9
iii) V = (3, 1) [2, 5] = cfw_x R | 3 < x < 1 or 2 x 5
-3
-1 0
5
2
(3, 1) [2, 5]
iv) T = (, 0) (0, ) = cfw_x R |x = 0 = R \cfw_0. As you can see there may be a
number of ways of writing down a set.
R \ cfw_0
0
The modulus or a
CHAPTER 2
Polar form and roots of complex numbers
2.1
Polar and Cartesian forms of complex numbers
In the last chapter we introduced the set of complex numbers and showed how such numbers
can be graphed on the complex plane. To graph any complex number z
Chapter 1: Numbers and sets
5
The set of real numbers, R, contains all the other number sets. In fact we can summarise the
numbers sets diagrammatically
Real Numbers, R
eg. , e, 5, 0.1010010001 . . .
Rational Numbers, Q
4
eg.
3 , 0.1, 7.2, 0.3
1
,
2
Inte
Chapter 1: Numbers and sets
7
Examples 1.2a
i)
A = cfw_x Q |x > 0
In words, this says A is the set of all rational numbers x such that x is positive.
Alternatively A is the set of all positive rational numbers. The vertical slash should
be read as such th
Chapter 2: Polar form and roots of complex numbers
23
Sometimes the polar form of a complex number, r(cos + i sin ) is abbreviated to r cis
where
cis = cos + i sin.
So, for example, 8 cis
6
= 8 cos
6
+ i sin
6
.
For a complex number z we can choose how t