Revision: Complex Numbers
Exercise
1.
For any complex number z = a + b i, where a, b
i) ii)
are real numbers and and i = -1, then a = Real part and b = imaginary part the conjugate is z* = a – bi – bi
iii)
the modulus is
2
z
a
2
If z1 = 2 + i and z 2 = 3 – 3 – 2i 2i , find a) 2 z1 + i z2
the argument is arg z = tan
v)
the Argand diagram is
c)
-1 b
tan = r
Real axis
x
3. 4.
If a + ib is is a complex root of a polynomial, then its conjugate a – a – ib ib is also a root of the polynomial.
5.
z2
r 1 r 2
[ cos(1 2 ) i sin(1 2 )]
4.
zn
6
]
6
The complex numbers z 1 and z 2 satisfy the 2
equation z = 2 - 2
2k 2k r n cos i sin n n 1
b) c)
5.
6
3 i.
Express z1 and z2 in the form of a + bi, where a and b are real numbers. Represent z1 and z2 in an Argand diagram. For each z1 and z 2, find the modulus, and the argument in radians.
[Z1 = 3 +i ; =-
By using De Moivre’s Theorem a) The power of a complex number z = r [ cosθ + n n i sinθ] is z = r [ cos nθ + I sin nθ ]. th b) The n roots of a complex number z = r(cos r(cos θ + i sin θ), with n as a positive integer, is 1
i +
[-1 2i]
z1 z2 + z1 + z2 = 3.
a)
If z1 = r 1( cos θ1 + i sinθ1) and z2 = r 2(cosθ (cosθ2 + i sin θ2) Then i) z1 z2 = r 1 r 2 [ cos( θ1 + θ2) + i sin (θ1 + θ2) ] z1
1+ i z 2
5
- 5 = 0 are z 1 and z2. Find z1 and z2 in the form a + ib, and show that
The polar form: z = r ( cos θ + i sinθ ) Where r = z and θ = Argument
ii)
[
3 2 The two non-real roots of the equation z + z + 3z
3.
vi) The angle is positive, if it is measure in the anticlockwise direction and is negative if it is measure in clockwise direction. 2.
z1 + 2 i
The complex number z is such that z - 2z* = 3 3i, where z*.denotes the conjugate of z. (a) Express z in the form a + bi, where a and b are real numbers. (b) Find the modulus and argument of z. (c) Convert z in polar form (d) Represent z and its conjugate conjugate in an Argand diagram.
2.
a
[ 65 ]
a
Z(a, b)
O
[6+5i]
z1 z2
b)
b2
iv)
s i x a y r a n i g a m I
1.
- i ; Z1 Z 2 =2; Arg(Z 1) =
3
5 6
; Arg(Z2)
]
The complex number z is given
by z
=
1 3i
(a) Find z and arg z. (b)
5
Using de Moivre's theorem, show that z = 16 3 i
– 16 – 16
4
(c)
where k = 0, 1, 2, …, n – 1 – 1
z
Express
in the form x + yi, where z* is
*
z
the conjugate of z and x, y
∈ R
Note 1
6.
th
i)
the modulus of the n root is r n
ii)
the argument of the first n root is
th
Express in the form a + bi where a, b
n
and the
a)
3 1 i 2 2
b)
sin 3 i cos 3
th
subsequent each n root exceed the argument of the previous root by iii)
2 n
.
[
1 2
3
2
i ]
6
1
th
all the n roots of z lie equally spaced on the 1
R
10
c)
circle with radius r n .
4 4i
8 cos
d)
2 cos
5
[
3
i sin
5
9
9
2
2
9
i sin
[-1]
9
1 1 256
[2 + i 2
i ]
3
]
7.
Find the smallest positive integer values of p and q for which p
cos i sin 8 8 q cos i sin 12 12 8.
5
Write down the five roots of the equation z = 1 , giving your answers in the polar form with - < θ < .
sin 0; [Ans: 1 or cos 0 i si
cos(
2 5
9.
[p=2; q=3]
) i sin(
cos
2
i sin
5 2
2 5
) ; cos(
5
4
; cos
4
4
5
) i sin(
5
i sin
5 4 5
)]
4
a) Find the roots of w = - 16i , and sketch the roots on the an Argand diagram. b) Find the fifth root of
[ (b)
z
k
3
i
2k 2k 51 2 cos 6 i sin 6 , 5 5
where k = 0, 1 , 2, 3, 4 ]
