Complex Numbers

Complex Numbers
Definitions

A number in the form of a + ib, where a, b are real numbers and i =, is called a complex number, may be written as (a, b), where the first number denotes the real part and the

second number denotes the imaginary part. If z = a + ib, then the real part of z is denoted by R_e(z) and the imaginary part by Im(z). A complex number is said to be purely real if Im(z)

= 0, and is said to be purely imaginary if R_e(z) = 0. The complex number 0 = 0 + i0 is both purely real and purely imaginary.

Two complex numbers are said to be equal if and only if their real parts and imaginary parts are separately equal i.e. a + ib = c + id implies a = c and b = d. However, there is no

simple order relation between complex numbers and the symbol a + ib < (or > ) c + id is not defined.

i = is called the imaginary unit. Also i² = -1, i³ = i².i = - i, In general, i⁴ⁿ = 1, i^{4n +1}= i,

i⁴ⁿ⁺² = -1, i^{4n +3} = - i for an integer n. Thus any power of i can be expressed as 1 or i.



Algebraic operations with complex numbers

Addition : (a + ib) + (c + id) = (a + c) + i (b + d)

Subtraction : (a + ib) - (c + id) = (a - c) + i ( b - d)

Multiplication : (a + ib) (c + id) = (ac - bd) + i (ad + bc)

Division : (when at least one of c and d is non-zero)

= =



Modulus and Argument of a complex number

We define modulus of the complex number z = a + ib to be the real number and denote it by |z|. We note that, |z| = 0 iff z = 0. Also - |z| R_e(z) |z|, -|z| I_m(z) |z|.

If z 0 + i0, then is said to be the argument or amplitude of the complex number z = a + ib and is denoted by arg(z). The argument of the complex number 0 is not

defined. The argument of a complex number is not unique. 2n + (n integer) is also the argument of z for various values of n. The value of satisfying the inequality - < is

called the principal value of the argument. The argument of a complex number z = a + ib is any one of the numbers which are solutions of the system of equations



A complex number z = x + iy can be represented by a point P whose Cartesian coordinates are (x, y) referred to the axes OX and OY, usually called the real and the imaginary axes.

The plane of OX and OY is called the Argand diagram or the complex plane. Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real

and purely imaginary.

Now =


If = POM, then cos = tan^-1y/x. Thus |z| is the length of OP and arg.z is the angle which OP makes with the positive direction of the x-axis. If OP = r, then x = r cos cos , and y = r sinq,

and z = r (cos cos + i sin cos ).

The vector can also be used to represent the complex number z = x + iy. The length of the vector, i.e. is the modulus of z.



Also so that

= arg (z) = tan^-1 .
Method of finding the principal value of the argument of a complex number

Step I: Find tanq = and this gives the value of cos in the first quadrant.

Step II: Find the quadrant in which z lies by the sign of x and y co-ordinates.

Step III: Then argument of z will be , - , - , and - according as z lies in the first, second, third or fourth quadrant

.



Conjugate of a complex number

The conjugate of the complex number z = a + ib is defined to be a - ib and is denoted by.

If z = a + ib, z + = 2 a (real),

z -= 2ib (imaginary ) and z= (a + ib) (a - ib) = a² + b² (real) = |z|².

Also R_e(z) = , .



Properties of the Conjugate of a Complex Number:

is the mirror image of z in the real axis.

(i) |z| = ||

(ii) z + = 2 Rez

(iii) z -= 2i Imz

(iv)

(v) = ₁ + ₂





(vi) = ₁ -₂ (vii)

(viii) (ix)

(x)

(xi) If = f(z), then, where w is a function of complex number z.



Properties of the Modulus:

(i) |z| = 0 z = 0 (ii) Re z |z|, Im z |z|

(iii) z= |z|² (iv) |z₁z₂| = |z₁| |z₂|

(v)

(vi) |z₁ + z₂| |z₁| + |z₂| |z₁ + z₂ + ... + z_n| |z₁| + |z₂| + ... + |z_n|

(vii) |z₁ - z₂| ||z₁| - |z₂||

(viii) |z₁ + z₂|² = (z₁ + z₂) (+) = |z₁|² + |z₂|² + z₁+ z₂



Trigonometric or Polar form of a Complex Number

Let z = x + iy. We define x = r cos cos , y = r sin cos so that r = and is the solution of the system cos = sin cos =. Then

z = x + iy = r cos cos + ir sin cos = r (cos cos + i sin cos ) = reⁱ(Euler's formula).

If z₁ = r₁ (cos cos ₁ + i sin cos ₁), and z₂ = r₂ (cos cos ₂ + i sin cos ₂), then z₁z₂ = r₁r₂ and .




Properties of the Argument

arg(z₁z₂) = ₁ + ₂ = argz₁ + argz₂

arg(z₁/z₂) = ₁ - ₂ = argz₁ - argz₂

If arg z = 0 z is real z =


De Moivre's Theorem

If n is an integer, then (cos cos + i sin cos )ⁿ = cos n + i sin n.


The n^th Roots of Unity:

Let x be the n^th root of unity. Then

xⁿ = 1 = cos 2k + i sin 2k (where k is an integer)

x = cos k = 0, 1, 2, ........, n - 1.

Let = cos. Then the n n^th roots of unity are a^t

(t = 0, 1, 2, ......, n - 1), i.e. the n^th roots of unity are 1, , ², ........, ^{n - 1}.



Sum of the Roots:

1 + + ² +...... + ^{n - 1} == 0 the sum of the roots of unity is zero

and .



Product of the Roots:

1..². ....^{n - 1} = =

= cos{(n - 1)} + i sin{(n - 1)} = cos((n - 1)).

If n is even = - 1. If n is odd = 1.


Note :

The points represented by the n nth roots of unity are located at the vertices of a regular polygon of n sides inscribed in a unit circle having centre at the origin, one vertex being on the positive real axis.



Cube Roots of Unity:

For n = 3, we get the cube roots of unity and these are

1, cos+ i sinand cos+ i sini.e. 1, and .



They are generally denoted by 1, and ² and geometrically represented by the vertices of an equilateral triangle whose circumcentre is the origin and circumradius is unity.

Note : ³ = 1 and 1 + + ² = 0.



Square roots of a complex number

In order to find the value of, we write a + ib = R (cos + i sin )

where R = and tan f =. Hence (cos+ i sin )^1/2

= [cos (+ 2k) + i sin ( + 2k)]^1/2

= , k = 0, 1, so that

,

where z₀ = and z₁ = .

If b > 0, x and y take the same sign and if b < 0, x and y take the opposite signs.

For a + ib = 1, a = 1, b = 0 and hence R = 1, with = 0⁰. Hence square roots of unity are

z₀ = [cos 0⁰ + i sin 0⁰] = 1, and z₁ = [cos + i sin ] = - 1.



Geometrical Representation of Complex Numbers

Let P be the point representing the ordered pair (x, y), associated with the complex number z = x + iy.

Draw PM perpendicular to OX and produce it to P' so that |MP| = |MP'|. Then the point P' represents the ordered pair

(x, - y) or it represents the complex number x - iy = ,

= r cos cos - ir sin cos = r (cos (- q) + i sin (- q))

arg = - q = - arg (z).

For z = x + iy, - z = - x - iy arg (- z) = arg (- 1) + arg (z)

i.e. arg (- z) = - p + q = - (p - q) if z lies in the 1^st or the 3^rd quadrant

= p + q if z lies in the 2^nd or the 4^th quadrant.

arg (z) - arg (- z) = p
.

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ be two complex numbers represented by the points P₁(x₁, y₁) and P₂(x₂, y₂) respectively. By definition z₁+ z₂ should be represented by the point (x₁
+ x₂, y₁ + y₂). This point is the vertex, which completes the parallelogram with the line segments joining the origin with z₁ and z₂ as the adjacent sides.




In order to represent z₁ - z₂, we first take point

Q₂ (- x₂, - y₂) and denote z₃ = - x₂ - iy₂. Now complete the parallelogram OP₁RQ₂ with adjacent sides OP₁ and OQ₂. The point R represents the complex number z₁ + z₃ = z₁ - z₂.

In vector notation

z₁ + z₂ = .

Also z₁ - z₂ = .




In the parallelogram OP₁PP₂, the sum of the squares of its sides is equal to the sum of the squares of its diagonals; i.e.



Alternatively, |z₁ + z₂|² + |z₁ - z₂|²

= (z₁ + z₂) =

In any triangle, sum of any two sides is greater than the third side and difference of any two sides is less than the third side; we have

|z₁| + |z₂| |z₁ + z₂|, ||z₁| - |z₂|| |z₁ - z₂|.

These inequalities are called the triangular inequalities. In order to prove these otherwise, we write =

= |z₁|² + |z₂|² + 2Re

|z₁|² + |z₂|² + 2

|z₁ + z₂|² |z₁|² + |z₂|² + 2 |z₁| |z₂|

or |z₁ + z₂| |z₁| + |z₂|.

Further, |z₁ - z₂|² = (z₁ - z₂)

= |z₁|² + |z₂|² - 2Re

or |z₁ - z₂|² |z₁|² + |z₂|² - 2 |z₁| |z₂|

so that |z₁ - z₂|.



Product of two complex numbers z₁ and z₂

Let P₁ and P₂ represent the complex numbers z₁ and z₂ in the complex plane.

Now z = z₁z₂ = r₁ (cos cos ₁ + i sin cos ₁). r₂ (cos cos ₂ + i sin cos ₂)

= r₁r₂ (cos (q₁ + q₂) + i sin (q₁ + q₂))

|z₁z₂| = r₁r₂ and arg (z₁z₂) = q₁ + q₂ = arg z₁ + arg z₂.

If we take a point M on the x-axis so that OM = 1 unit, and construct a triangle OP₂R similar to the triangle OP₁M, we have

OR = OP₁. OP₂ = r₁r₂

and ROX = ROP₂ + P₂OX = ₁ + ₂.


We find that polar representation of R is r₁r₂ cos (₁ + ₂) + i r₁r₂ sin (₁ + ₂)

R represents the complex number z₁z₂.

We know that .

Hence .

First we represent the complex numbers z₁ and in the Argand's diagram and then represent the product .



Rotation

Let z₁, z₂, z₃ be three complex numbers represented by the three vertices of a triangle ABC, taken in the counter clockwise direction. Shift the origin to A so that B and C are now

representing the complex numbers

Z₂ = z₂ - z₁ and Z₃ = z₃ - z₁.

Let Z₂ = , so that







.


Note The arg. (z₃ - z₁) - arg.(z₂ - z₁) = a is the angle through which AB is rotated in the anti-clockwise direction so that it becomes parallel to AC.



Equation of a Straight Line

Writing and re-arranging terms, we find that the equation of the line through z₁ and z₂ is given by



The general equation of a straight line is where b is a real number.

The equation of the line parallel to the line is where c is real. The equation of the line perpendicular to it is where l is purely real.

The equation of the perpendicular bisector of the line segment joining z₁ and z₂ is given by = 0


.
Section Formula

Let C divide AB in the ratio m : n, so that .

Let be z then, z = .

If C divides AB externally in the ratio m : n then


z = .




We may write the section formulas as - lz = mz₂ + nz₁, where m + n = - l, so that

lz + mz₁ + mz₂ = 0 with l + m + n = 0. This is the condition for three points associated with z₁, z₂ and z₃ to be collinear.



Condition for Collinearity

If there are three real numbers (other than 0) l, m and n such that

lz₁ + mz₂ + nz₃= 0 and l + m + n = 0, then complex numbers z₁, z₂ and z₃ will represent collinear points. The complex conjugate of the first equation is. Eliminating

l, m, n from these equations, we get the condition of collinearity as = 0.

Equation of a Circle

Equation of a circle of radius r is |z - z₀| = r whose centre is the point representing the complex number z₀. For z₀ = 0, r = 1, the equation of the circle with unit radius and centre at the

origin is |z| = 1. This is also called the unit circle.

The general equation of a circle is where b is a real number.

The centre of this circle is - a and its radius is . The equation of the circle described on the line segment joining z₁ and z₂ as diameter is


.
Condition for four points to be concyclic

Four complex numbers z₁, z₂, z₃ and z₄ taken in order, will represent concyclic points if, is real.