complex numbers

REFRESHER COURSE: COMPLEX NUMBERS DEFINITION A complex number z is of the form x + yi where x, y ∈ and i = 1− . x is called the real part of z, denoted by Re(z). y is called the imaginary part of z, denoted by Im(z). Note: Two complex numbers and are equal if and only if their real and imaginary parts are equal. 1z 2z ALGEBRAIC OPERATIONS ON COMPLEX NUMBERS

Addition: + = (a + bi) + (c + di) = (a + c) + (b + d)i , 1z 2z , , ,a b c d ∈

Subtraction: − = (a + bi) − (c + di) = (a − c) + (b − d)i 1z 2z

Multiplication:

(i) k = k(a + bi) = ka + kbi , 1z k∈

(ii) = (a + bi)(c + di) = ac + (ad + bc)i + bdi2 1z 2z

= (ac − bd) + (ad + bc)i Note that i = 1− , so i 2 = 1− Division:

1

1

2

zz

= a bic di++

= ( )(( )(a bi c dic di c di+ −+ −

))

= 2

2 2 2( )( )ac adi bci bdic cdi dci d i− + −− + −

= 2 2

( ) (ac bd bc ad ic d

)+ + −+

= 2 2 2 2

( ) (ac bd bc ad ic d c d

+ −+

+ +)

COMPLEX CONJUGATE If z = x + yi, then the complex number x − yi is called the complex conjugate of z. We denote the complex conjugate of z by z*. Properties of complex conjugates:

S/No. Properties S/No. Properties 1 z = z* ⇔ z is real 6 z z* = x2 + y2 2 (z*)* = z 7 ( )1 2z z± * = z1* ± z2*

3 (kz)* = kz*, k ∈ 8 (z1 z2)* = (z1*)(z2*) 4 z + z* = 2 Re(z) 9 (zn)* = (z*)n, where n ∈ +Z5 z − z* = 2i Im(z) 10

1 1

2 2

* ( *)( *)

z zz z

⎛ ⎞=⎜ ⎟

⎝ ⎠

COMPLEX ROOTS OF A POLYNOMIAL EQUATION WITH REAL COEFFICIENTS If z = a + bi is a root of a equation P(z) = 0 where, P(z) is a polynomial of degree n with real coefficients, then z* = a − bi is also a root of the equation. By the Fundamental Theorem of Algebra, any polynomial P(z) of degree n has exactly n roots, real or complex. Example 1: [NJC Prelim 07/P1/Q1]

Verify that is a root of the equation 2 3i− + 3 25 17 13z z z 0+ + + = and determine the other roots of the equation.

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MODULUS AND ARGUMENT OF A COMPLEX NUMBER Let P represent the complex number . iz x y= +

Modulus of z is the distance between point P representing z and the origin, and is denoted by z where,

2 2z x y= +

Argument of z is the angle measured from the positive real axis to the line segment joining P and the origin and is denoted by ( )arg .z

2

COMLPLEX NUMBER IN ALGEBRAIC, POLAR AND EXPONENTIAL FORM.

Algebraic form: z = x + yi Polar (Trigonometric) Form: ( )cos sinz r iθ θ= + where, r z= and . ( )arg zθ =

Exponential form: iz re θ= where, i cos i sine θ θ θ= + . MUTILPLICATION AND DIVISION OF COMPLEX NUMBERS Let i

1z ae α= and i2z be β= .

1 2z z⋅ i iae beα β= ⋅ ( )iabe α β+= ( ) ( )cos i sinab α β α= + + + β⎡ ⎤⎣ ⎦

From the above result, we observe that:

1) 1 2z z⋅ 1 2.z z= nnz z⇒ =

2) ( )1 2arg z z⋅ 1 2arg argz z= + arg( ) arg( )nz n z⇒ =

In the same way for division of complex numbers,

1

2

zz

i

iaebe

α

β= ( )ia eb

α β−= ( ) ( )cos i sinab

α β α= − + −⎡ ⎤⎣ ⎦β

From the above result, observe that:

3) 1

2

zz

1

2

zz

=

4) 1

2arg z

z⎛ ⎞⎜ ⎟⎝ ⎠

1 2arg argz z= −

If 0 θ π≤ ≤ , the argument of the complex number is measured in the anti-clockwise direction. If 0π θ− < ≤ , the argument of the complex number is measured in the clockwise direction.

π θ π− < ≤ is called the principal range.

Im(z)

Im(z)

( ) 1arg tany

zx

θ − ⎛ ⎞= = − ⎜ ⎟⎜ ⎟

⎝ ⎠

Re(z)

P ( x , y )

θ

( ) 1arg tany

zx

θ π − ⎛ ⎞= = − ⎜ ⎟⎜ ⎟

⎝ ⎠

Re(z)

P ( x , y )

θ

( ) 1arg tany

zx

θ π −⎡ ⎤⎛ ⎞= = − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Re(z)

) Im(z

( ) 1arg tany

zx

θ − ⎛ ⎞= = ⎜ ⎟⎜ ⎟

⎝ ⎠

Im(z)

P ( x , y )

θθ

Re(z) P ( x , y )

Im(z)

P ( x , y )

Re(z)θ

cosr θ

sinr θr


GEOMETRICAL INTERPRETATION

Addition And Subtraction Of Two Complex Numbers:

Geometrically, the addition and subtraction of 2 complex numbers, e.g., and , are in accordance with the vector parallelogram that represents the addition and subtraction of vectors as shown.

1z 2z

3

The vectors representing and 1z z+ 2 21z z− are the diagonals of the parallelogram whose sides are

represented by the line segments and with and representing the complex numbers and respectively.

1OP 2OP 1P 2P 1z 2z

Multiplication And Division Of Two Complex Numbers: Let P be the point representing complex number, iz ae α= and Q be the point representing complex number

iw be β= .

z w⋅ ( )iabe α β+= ( ) ( )cos isinab α β α= + + +⎡ ⎤⎣ ⎦β

i.e., . .z w z w ab= = and ( ) ( ) ( )arg . arg argz w z w α β= + = +

When complex number z is multiplied to complex number w, the line segment OQ is rotated anti-clockwise through α and scaled by a factor of a as shown in the diagram above

Similarly, when complex number w is divided by complex number z, the line segment OQ is rotated

clockwise through α and scaled by a factor of 1a

.

When complex number z is multiplied by i, the line segment OP is rotated

anti-clockwise through 2π

as shown in the diagram on the right. Similarly,

when complex number z is multiplied by i− , the line segment OP is rotated

clockwise through 2π

. Note that: 2i eπ

= and 2i

i eπ

−− =

1 1( )P z0

2 2( )P z

5 1 2( )P z z

4 1 2( )P z z

−

3 2( )P z−

Im(z) +

Re(z)

Im

Re

Im

P ( z )

α

Q ( w )

β

O

Re

Im

P ( z )

α

( iz )

( -iz )

O

Re

P ( z ) α Q ( w )

(z.w)

βα

.z w ab=

O

R


Example 2: [2008 RJC Term 3 Common Test Qn 5a]

The complex number 7 is represented by the point A in an Argand diagram with origin O. Given that OABC is a rectangle described in a clockwise sense with

5i+2OC OA= , find the complex numbers represented

by the points B and C in the form x iy+ , where x and y are real.

DE MOIVRE’S THEOREM AND ITS APPLICATION

(cos sin )n n n inz r n i n r e θθ θ= + =

Example 3: Find the cube root of the complex number 4( 3 )i+ . LOCI IN THE ARGAND DIAGRAM Definition: The locus of a complex variable iz x y= + is a set of points in the complex plane which satisfies a given equation. The set of points form a curve when the condition is an equation and a region if the condition is an inequality. Standard Loci

• ( )z k= , k ∈ is equivalent to the vertical lineRe x k= .

• ( ) is equivalent to the region Re z k≥ x k≥ .

• ( ) is equivalent to the region Re z k< x k< .

• ( ) , h∈ is equivalent to the horizontal lineIm z h= y h= .

• ( ) h≥ is equivalent to the region y h≥ . Im z

• ( ) h< is equivalent to the region Im z y h< .

4

Circle (The locus of a variable from a fixed point)

Perpendicular Bisector (The locus of a variable from 2 fixed points)

r

x

y

O P0 (z0) 0 ,z z r r +− = ∈

The locus of the point P representing z is a circle with centre P0 and radius r.

Half-Line (The locus of a variable whose argument is fixed)

The locus of the point P representing z is the perpendicular bisector of the line segment joining the points P1 and P2, representing and respectively. 1z 2z

1 2z z z z− = −

The locus of the point P representing z is the half-line starting from P0 (excluding P0) and making an angle θ with the positive x-axis.

0arg( )z z θ− =

θ

x

y

O

P0 (z0)

y

x O P2 (z2)

P1 (z1)

P


Example 4: Sketch in separate Argand diagrams the following loci

a) ( )23 7z i− = 5 b) ( )arg(1 ) arg 1iz i− = − + c) 4 2 4iz i z− = − −

Example 5:

Sketch in an Argand diagram the set of points representing all complex numbers z satisfying the following

inequalities: 1 1z iz+ ≥ − , 2 1 3 1z i− < + and 0 arg( 0.5)2

z π≤ − <

EXCERCISE: [JJC MYE 07/P2/ Q10a Modified] 1. If

8

4

( 1 i)(1 3 i)

w − +=

−, find the modulus and argument of w in exact form. Hence, show that the roots of the

equation ( )8

3

4

( 1 i)1(1 3 i)

z − +− =

− can be expressed in the form of 22cos

2i

eθθ⎛ ⎞

⎜ ⎟⎝ ⎠

, where the values of θ are to be

determined.

[JJC MYE 07/P2/ Q10c] 2. a) Write down, in the form eiθ , all the 8th roots of unity, i.e. the roots of the equation . 8 1z =

b) Using the result , express 2( e )( e ) (2cos )i iz z z zθ θ θ−− − = − 1+ 8 1z − as the product of two linear factors and three quadratic factors, where all the coefficients are real and expressed in a non-trigonometric form.

[PJC Prelim 07/P1/Q12 Modified] 3. In a single Argand diagram, sketch the set of points representing all complex numbers z satisfying both of

the following inequalities: 3 3i 1 2iz + + ≤ + and 50 arg( 3 3i)6

z π≤ + + ≤ .

Hence, find a) the maximum and minimum value of arg( 1 3i)z + + b) the maximum and minimum value of 1 3iz + + .

Answers for Examples: 1) 2) 2 3i, 2 3i, and 1z = − + − − − 10 14i, 17 9c b i= − = − 3) ( )18 182 cos sin , 11,1,13k ki kπ π+ = −

4 a) A circle with centre at ( and radius 5 units. )0,1

4 b) A half-line from the point ( )0, 1− making an angle of 34π

− with the positive x-axis.

4 c) A perpendicular bisector of the line segment joining the points ( )0, 4− and ( )2, 4− .

Answers for Exercise:

1) 8

4

( 1 i) 1(1 i 3)− +

=−

, 8

4

( 1 i) 2arg3(1 i 3)π⎡ ⎤− +

= −⎢ ⎥−⎣ ⎦

, 2 4 8, ,9 9 9π π πθ = − − . 2a)

π4eki

z = , where 0, 4, 1, 2, 3k = ± ± ±

2b) 2 2 2( 1)( 1)( 2 1)( 1)( 2 1)z z z z z z z− + − + + + + 3a) π , 23π b) 13 , 2 3−

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complex numbers

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