complex number - Runnymede Mathematics Department
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Transcript complex number - Runnymede Mathematics Department
Complex numbers
What is
There is no number which squares to
make -1, so there is no ‘real’ answer!
1 ?
Mathematicians have realised that by defining the imaginary number i
many previously unsolvable problems could be understood and explored.
If
1,
i 1 , what is:
i 3 i
i 2 1
i4 1
100 10i
4 2i
3 3i
A number with both a real part and an imaginary one is called a complex number
Eg
Complex numbers are often
referred to as z, whereas real
numbers are often referred to as x
z 2 3i
The imaginary part
of z, called Im z is 3
The real part of z,
called Re z is 2
A complex number in the
form z x iy is said
to be in Cartesian form
Manipulation with complex numbers
Techniques used with real numbers can still be applied with complex numbers:
z = 5 – 3i,
WB1
w = 2 + 2i
Express in the form a + bi, where a and b are real constants,
(a) z2
(b)
z
w
a) z 5 3i 25 30i 9 16 30i
2
2
Expand & simplify as usual,
remembering that i2 = 1
z 5 3i 2 2i 10 10i 6i 6 4 16i 1 2i
b)
2
8
4
4
w 2 2i 2 2i
An equivalent complex number with a real
denominator can be found by multiplying by
the complex conjugate of the denominator
If z x iy then
its complex conjugate
is z* x iy
Modulus and argument
The complex number z x iy
can be represented on an Argand
diagram by the coordinates x, y
Eg z1 1 3i Eg
x2 y2
2
Eg
z1
Eg
z2
22 12
Eg
z3
22 22 2 2
3
5
4
4
Remember the definition of arg z
arg z 3
1
2
z2
2
3
2
3 tan 1 arg z 3
2
z1 1, 3
z1
3
z
12
z3
Eg z 3 2 2i
z2 2 i
The modulus of z,
z 3 2,2
Im
Re
z2 2,1
The principal argument arg z
is the angle from the positive real axis
to z x, y in the range
1 tan
1
3 3
arg z1
3
1 1
tan
2
0.463...
2
arg z2 0.46 (2dp)
WB2 The complex numbers z1 and z2 are given by
z1 2 8i
z2 1 i
z1 2 8i 1 i
z2
1 i
1 i
2 2i 8i 8
1 1
6 10i
3 5i
2
Find, showing your working,
(a)
z1
in the form a + bi, where a and b are real,
z2
z1
z2
(b) the value of
The modulus of
z x iy is z
(c) the value of arg
3,5
3 5i
32 52
x2 y2
z1
, giving your answer in radians to 2 decimal places.
z2
Im
The principal argument arg z
is the angle from the positive real axis
to z x, y in the range
34
Re
5
tan 1 1.03 arg z 2.11
3
WB3
z = 2 – 3i
(a) Show that z2 = −5 −12i.
z 2 2 3i 4 6i 6i 9 5 12i
Find, showing your working,
(b) the value of z2,
z 2 5 12i
2
52 122 13
(c) the value of arg (z2), giving your answer in radians to 2 decimal places.
Im
12
tan 1 1.176...
5
5,12
Re
arg z 1.97
(d) Show z and z2 on a single Argand diagram.
Im
Re
z
z2
Complex roots
In C1, you saw quadratic equations that had no roots.
Eg x 4 x 13 0
2
x
4
4 2 4 1 13
2 1
Quadratic formula
b b 2 4ac
If ax bx c 0 then x
2a
2
We can obtain
complex roots
though
4 6i
4 36
2 3i
2
2
We get no real answers because
the discriminant is less than zero
We could also obtain these roots
by completing the square:
This tells us the curve
x 2 4 x 13 0
y x 2 4 x 13
x 2 4 13 0
will have no
intersections with
the x-axis
x 2 9
2
2
x 2 3i
x 2 3i
WB4
z1 = − 2 + i
(a) Find the modulus of z1
2i
22 12
5
(b) Find, in radians, the argument of z1 , giving your answer to 2 decimal places.
Im
2,1
1
tan 0.463...
2
1
arg z 2.68
Re
The solutions to the quadratic equation z2 − 10z + 28 = 0 are z2 and z3
(c) Find z2 and z3 , giving your answers in the form p iq, where p and q are integers.
z 2 10 z 28 0
z 5 25 28 0
2
z 5 3
(d) Show, on an Argand diagram, the points
representing your complex numbers
Im
2
z 5 i 3
z 5i 3
Re
WB6
f x x 3 x 2 44 x 150
Given that f x x 3 x ax b , where a and b are real constants,
2
(a) find the value of a and the value of b.
x 3x2 ax b x 3 ax 2 bx 3 x 2 3ax 3b
x3 a 3x 2 b 3a x 3b
Comparing coefficients of x2 a 2
Comparing coefficients of x0 b 50
(b) Find the three roots of f(x) = 0.
f x x 3 x 2 2 x 50 0
either x 3 0 x 3
or x 2 x 50 0
2
(c) Find the sum of the
three roots of f (x) = 0.
1 7i 1 7i 3
So sum of the three roots is -1
x 1 1 50 0
2
x 1 49
x 1 7i
2
x 1 7i
Problem solving with roots
In C2 you met the Factor Theorem:
If a is a root of f(x)
then ( x a ) is a factor
Eg Given that x = 3 is a root of the equation x ax b 0 ,
(a) write down a factor of the equation,
2
(b) Given that x = -2 is the other root, find the values of a and b
x 2 is the other factor
x 3x 2 is the equation factorised
x 2 x 6 0 expanding
a 1, b 6
In FP1 you apply this method to complex roots…
x 3
Problem solving with complex roots
We have seen that complex
roots come in pairs:
Eg x 4 x 13 0
2
x 2 3i
This leads to the logical conclusion that if a complex number
z x iy is a root of an equation, then so is its conjugate z* x iy
We can use this fact to find real quadratic factors of equations:
WB5 Given that 2 – 4i is a root of the equation z2 + pz + q = 0,
where p and q are real constants,
(a) write down the other root of the equation,
2 4i
(b) find the value of p and the value of q.
z 2 4i z 2 4i 0
z 2 2 4i z 2 4i z 2 4i 2 4i 0
z 2 2 z 4iz 2 z 4iz 4 8i 8i 16i 2 0
z 2 4 z 20 0 p 4, q 20
Factor theorem:
If a is a root of f(x)
then ( x a ) is a factor
WB7 Given that 2 and 5 + 2i are roots of the equation
x 3 12 x 2 cx d 0
c, d R
(a) write down the other complex root of the equation.
5 2i
(b) Find the value of c and the value of d.
x 5 2i x 5 2i
x 2 5 2i x 5 2i x 5 2i 5 2i
x 2 5 x 2ix 5 x 2ix 25 10i 10i 10i 2
x 2 10 x 35
x
2
10 x 35 x 2
(c) Show the three roots of this
equation on a single Argand diagram.
Im
x 3 2 x 2 10 x 2 20 x 35 x 70
x 3 12 x 2 55 x 70
c 5, d 70
Re
Problem solving by equating real & imaginary parts
Eg Given that 3 5i a ib 1 i
where a and b are real, find their values
Equating real parts:
a ib 1 i a ai bi b
a b a bi
a b 3 (1)
Equating imaginary parts:
a b 5 (2)
(1) (2) 2a 8 a 4
Sub in (2) 4 b 5 b 1
WB8 Given that z = x + iy, find the value of x and the value of y such that
z + 3iz* = −1 + 13i
where z* is the complex conjugate of z.
z x iy z* x iy
then z 3iz* x iy 3ix iy
x iy 3ix 3 y
x 3 y y 3 xi
Equating real parts: x 3 y 1 (1)
Equating imaginary parts: y 3 x 13
(1) 3 3 x 9 y 3
(2)
(3)
(3) (2) 8 y 16 y 2
Sub in (1) x 6 1 x 5
Eg Find the square roots of 3 – 4i in the form a + ib, where a and b are real
a ib 2 a 2 2abi b2
a 2 b2 2ab i
Equating real parts: a 2 b 2 3 (1)
Equating imaginary parts: 2ab 4 (2)
(2) a
2
b
sub in (1)
4
b2
b2 3
4 b 4 3b 2
b 4 3b 2 4 0
b2 1 b2 4 0
b 1 as b real
sub in (2) a 2
Square roots are -2 + i and 2 - i
Eg Find the roots of x4 + 9 = 0 x 2 3i
a ib 2 a 2 2abi b2
a 2 b2 2ab i
Equating real parts: a 2 b 2 0 (1)
Equating imaginary parts: 2ab 3 (2)
(2) a 23b
sub in (1)
9
4b 2
b2 0
9 4b 4 0
b4
9
4
b
3
2
sub in (2) a
Roots are
3
2
3
2
i
3
2
3
2
,
3
2
3
2
3
2
i
3
2
, 233 i
2
3
2
, 233 i
2
3
2
Modulus-argument form of a complex number
r z x y
2
z x iy and
If
then
2
Im
zx, y
arg z
z r cos ir sin
r
z r cos i sin
known as the modulus-argument
form of a complex number
z 2 i in the form
z r cos i sin
Eg express
From previously,
r 5 and
6
so
z 5 cos 6 i sin 6
y r sin
x r cos
Re
z 2 cos 34 i sin 34
in the form z x iy
Eg express
x r cos
2 cos 34
1
y r sin
2 sin 34
1
z 1 i
The modulus & argument of a product
Eg if
It can be shown that:
It can also be shown that:
z1 z 2 z1 z 2
argz1z2 argz1 argz2
z1 2 i and z 2 1 3i
z1 z 2 z1 z 2 22 12 12 3 2
Eg if
z1 2 i and z 2 1 3i
tan 1 21 0.46...
Im
5 10 5 2
z1
This is easier than
evaluating z1z2 and then
finding the modulus…
z1z2 2 i 1 3i 2 6i i 3 5 5i
z1z2 5 5i 5 5 50 5 2
2
2
z2
Re
arg z1 0.46...
tan 1 31 0.32...
arg z2 2
1.24...
argz1z2 0.46... 1.24...
4
The modulus & argument of a quotient
Eg if
It can be shown that:
It can also be shown that:
z1
z1
z2
z2
z1
arg argz1 arg z 2
z2
z1 2 i and z 2 1 3i
Eg if
From previously,
z1
z1
5
1
z2
z2
10
2
arg z1 0.46...
arg z2 1.24...
This is much easier than
z
evaluating z21 and then
finding the modulus…
arg
z1 2 i 1 3i 2 6i i 3
101 107 i
z2 1 3i 1 3i
1 9
z1
z2
101 107 i
101 2 107 2
z1 2 i and z 2 1 3i
50
100
1
2
0.46... 1.24...
z1
z2
1.71...
WB9
z = – 24 – 7i
(a) Show z on an Argand diagram.
Im
(b) Calculate arg z, giving your answer
in radians to 2 decimal places.
24,7
a ℝ, b ℝ.
It is given that w = a + bi,
Given also that w 4 and arg w
5
6
(c) find the values of a and b
w 4cos 56 i sin
5
6
2
3 2i
(d) find the value of zw
z1 z 2 z1 z 2
z
zw z w 25 4 100
242 72
w 4 given
25
7
tan 0.283...
24
1
Re
arg z
2.86
Modulus-argument form
z r cos i sin
where r z
and arg z
Complex numbers
Manipulation with
complex numbers
z 5 3i 2 2i 10 10i 6i 6 4 16i
21 2i
8
w 2 2i 2 2i
44
Modulus and argument
z 52 32 34
arg z
Complex roots
w 2 2i
z 5 3i
Using:
Im
tan 1 35
Re
Also z1 z 2 z1 z 2
z1
z2
z1
z2
argz1z2 argz1 argz2
arg
z
arg z argz
z1
z2
1
2
w is a root of z az b . Find the values of a and b
2
z 2 2i z 2 2i z 2 z2 2i 2 2i z 2 2i 2 2i
z 2 2 z 2iz 2 z 2iz 4 4i 4i 4
z2 4z 8
Equating real & imaginary parts
wz* p qi
Find the values of p and q
wz* 2 2i 5 3i 4 16i
Equating real parts: p 4
Equating imaginary parts: q 16