**Imaginary Quantity**

The square root of a negative real number is called an imaginary quantity or imaginary number. e.g., √-3, √-7/2

The quantity √-1 is an imaginary number, denoted by ‘i’, called iota.

**Integral Powers of Iota (i)**

i=√-1, i^{2} = -1, i^{3} = -i, i^{4}=1

So, i^{4n+1}= i, i^{4n+2} = -1, i^{4n+3} = -i, i^{4n+4} = i^{4n} = 1

In other words,

i^{n} = (-1)^{n/2}, if n is an even integer

i^{n} = (-1)^{(n-1)/2}.i, if is an odd integer

**Complex Number**

A number of the form z = x + iy, where x, y ∈ R, is called a complex number

The numbers x and y are called respectively real and imaginary parts of complex number z.

i.e., x = Re (z) and y = Im (z)

**Purely Real and Purely Imaginary Complex Number**

A complex number z is a purely real if its imaginary part is 0.

i.e., Im (z) = 0. And purely imaginary if its real part is 0 i.e., Re (z)= 0.

**Equality of Complex Numbers**

Two complex numbers z_{1} = a_{1} + ib_{1} and z_{2} = a_{2} + ib_{2} are equal, if a_{2}= a_{2} and b_{1} = b_{2} i.e., Re (z_{1}) = Re (z_{2}) and Im (z_{1}) = Im (z_{2}).

**Algebra of Complex Numbers**

**1. Addition of Complex Numbers**

Let z_{1} = (x_{1} + iy_{i}) and z_{2} = (x_{2} + iy_{2}) be any two complex numbers, then their sum defined as

z_{1} + z_{2} = (x_{1} + iy_{1}) + (x_{2} + iy_{2}) = (x_{1} + x_{2}) + i(y_{1} + y_{2})

**Properties of Addition**

(i) Commutative z_{1} + z_{2} = z_{2} + z_{1}

(ii) Associative (z_{1} + z_{2}) + z_{3} = + (z_{2} + z_{3})

(iii) Additive Identity z + 0 = z = 0 + z

Here, 0 is additive identity.

**2. Subtraction of Complex Numbers**

Let z_{1} = (x_{1} + iy_{1}) and z_{2} = (x_{2} + iy_{2}) be any two complex numbers, then their difference is defined as

z_{1} – z_{2} = (x_{1} + iy_{1}) – (x_{2} + iy_{2})

= (x_{1} – x_{2}) + i(y_{1} – y_{2})

**3. Multiplication of Complex Numbers**

Let z_{1} = (x_{1} + iy_{i}) and z_{2} = (x_{2} + iy_{2}) be any two complex numbers, then their multiplication is defined as

z_{1}z_{2} = (x_{1} + iy_{1})(x_{2} + iy_{2}) = (x_{1}x_{2} – y_{1}y_{2}) + i(x_{1}y_{2} + x_{2}y_{1})

**Properties of Multiplication**

**(i) Commutative** z_{1}z_{2} = z_{2}z_{1}

**(ii) Associative** (z_{1} z_{2}) z_{3} = z_{1}(z_{2} z_{3})

**(iii) Multiplicative Identity** z • 1 = z = 1 • z

Here, 1 is multiplicative identity of an element z.

**(iv) Multiplicative Inverse** Every non-zero complex number z there exists a complex number z_{1} such that z.z_{1} = 1 = z_{1} • z

**(v) Distributive Law**

(a) z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3} (left distribution)

(b) (z_{2} + z_{3})z_{1} = z_{2}z_{1} + z_{3}z_{1} (right distribution)

**4. Division of Complex Numbers**

Let z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} be any two complex numbers, then their division is defined as

where z_{2} # 0.

**Conjugate of a Complex Number**

If z = x + iy is a complex number, then conjugate of z is denoted by z

i.e., z = x – iy

**Properties of Conjugate**

**Modulus of a Complex Number**

If z = x + iy, , then modulus or magnitude of z is denoted by |z| and is given by

|z| = x^{2} + y^{2}.

It represents a distance of z from origin.

In the set of complex number C, the order relation is not defined i.e., z_{1}> z_{2} or z_{i} <z_{2} has no meaning but |z_{1}|>|z_{2}| or |z_{1}|< | z_{2} | has got its meaning, since |z| and |z_{2}| are real numbers.

**Properties of Modulus**

**Reciprocal/Multiplicative Inverse of a Complex Number**

Let z = x + iy be a non-zero complex number, then

Here, z^{-1} is called multiplicative inverse of z.

**Argument of a Complex Number**

Any complex number z=x+iy can be represented geometrically by a point (x, y) in a plane, called Argand plane or Gaussian plane. The angle made by the line joining point z to the origin, with the x-axis is called argument of that complex number. It is denoted by the symbol arg (z) or amp (z).

Argument (z) = θ = tan^{-1}(y/x)

Argument of z is not unique, general value of the argument of z is 2nπ + θ. But arg (0) is not defined.

A purely real number is represented by a point on x-axis.

A purely imaginary number is represented by a point on y-axis.

There exists a one-one correspondence between the points of the plane and the members of the set C of all complex numbers.

The length of the line segment OP is called the modulus of z and is denoted by |z|.

i.e., length of OP = √x^{2} + y^{2}.

Principal Value of Argument

The value of the argument which lies in the interval (- π, π] is called principal value of argument.

(i) If x> 0 and y > 0, then arg (z) = 0

(ii) If x < 0 and y> 0, then arg (z) = π -0

(iii) If x < 0 and y < 0, then arg (z) = – (π – θ)

(iv) If x> 0 and y < 0, then arg (z) = -θ

**Properties of Argument**

**Square Root of a Complex Number**

If z = x + iy, then

**Polar Form**

If z = x + iy is a complex number, then z can be written as

z = |z| (cos θ + i sin θ) where, θ = arg (z)

this is called polar form.

If the general value of the argument is 0, then the polar form of z is

z = |z| [cos (2nπ + θ) + i sin (2nπ + θ)], where n is an integer.

**Eulerian Form of a Complex Number**

If z = x + iy is a complex number, then it can be written as

z = re^{i0}, where

r = |z| and θ = arg (z)

This is called Eulerian form and e^{i0}= cosθ + i sinθ and e^{-i0} = cosθ — i sinθ.

**De-Moivre’s Theorem**

A simplest formula for calculating powers of complex number known as De-Moivre’s theorem.

If n ∈ I (set of integers), then (cosθ + i sinθ)^{n} = cos nθ + i sin nθ and if n ∈ Q (set of rational numbers), then cos nθ + i sin nθ is one of the values of (cos θ + i sin θ)^{n}.

**The nth Roots of Unity**

The nth roots of unity, it means any complex number z, which satisfies the equation z^{n} = 1 or z = (1)^{1/n}

or z = cos(2kπ/n) + isin(2kπ/n) , where k = 0, 1, 2, … , (n — 1)

**Properties of nth Roots of Unity**

- nth roots of unity form a GP with common ratio e
^{(i2π/n)}. - Sum of nth roots of unity is always 0.
- Sum of nth powers of nth roots of unity is zero, if p is a multiple of n
- Sum of pth powers of nth roots of unity is zero, if p is not a multiple of n.
- Sum of pth powers of nth roots of unity is n, ifp is a multiple of n.
- Product of nth roots of unity is (-1)
^{(n – 1)}. - The nth roots of unity lie on the unit circle |z| = 1 and divide its circumference into n equal parts.

**The Cube Roots of Unity**

Cube roots of unity are 1, ω, ω^{2},

where ω = -1/2 + i√3/2 = e^{(i2π/3)} and ω^{2} = (-1 – i√3)/2

ω^{3r + 1} = ω, ω^{3r + 2} = ω^{2}

**Properties of Cube Roots of Unity**

(i) 1 + ω + ω^{2r} =

0, if r is not a multiple of 3.

3, if r is,a multiple of 3.

(ii) ω^{3} = ω^{3r} = 1

(iii) ω^{3r + 1} = ω, ω^{3r + 2} = ω^{2}

(iv) Cube roots of unity lie on the unit circle |z| = 1 and divide its circumference into 3 equal parts.

(v) It always forms an equilateral triangle.

(vi) Cube roots of – 1 are -1, – ω, – ω^{2}.

**Geometrical Representations of Complex Numbers**

**1. Geometrical Representation of Addition**

If two points P and Q represent complex numbers z_{1} and z_{2 }respectively, in the Argand plane, then the sum z_{1} + z_{2} is represented

by the extremity R of the diagonal OR of parallelogram OPRQ having OP and OQ as two adjacent sides.

**2. Geometrical Representation of Subtraction**

Let z_{1} = a_{1} + ib_{1} and z_{2} = a_{2} + ia_{2} be two complex numbers represented by points P (a_{1}, b_{1}) and Q(a_{2}, b_{2}) in the Argand plane. Q’ represents the complex number (—z_{2}). Complete the parallelogram OPRQ’ by taking OP and OQ’ as two adjacent sides.

The sum of z_{1} and —z_{2} is represented by the extremity R of the diagonal OR of parallelogram OPRQ’. R represents the complex number z_{1} — z_{2}.

3. Geometrical Representation of Multiplication of Complex Numbers

R has the polar coordinates (r_{1}r_{2}, θ_{1} + θ_{2}) and it represents the complex numbers z_{1}z_{2}.

**4. Geometrical Representation of the Division of Complex Numbers**

R has the polar coordinates (r_{1/}r_{2}, θ_{1} – θ_{2) }and it represents the complex number z_{1}/z_{2}.

|z|=|z| and arg (z) = – arg (z). The general value of arg (z) is 2nπ – arg (z).

If a point P represents a complex number z, then its conjugate i is represented by the image of P in the real axis.

**Concept of Rotation**

Let z_{1}, z_{2} and z_{3} be the vertices of a ΔABC described in anti-clockwise sense. Draw OP and OQ parallel and equal to AB and AC, respectively. Then, point P is z_{2} – z_{1} and Q is z_{3} – z_{1}. If OP is rotated through angle a in anti-clockwise, sense it coincides with OQ.

**Important Points to be Remembered**

(a) ze^{iα} a is the complex number whose modulus is r and argument θ + α.

(b) Multiplication by e^{-iα} to z rotates the vector OP in clockwise sense through an angle α.

(ii) If z_{1}, z_{2}, z_{3} and z_{4} are the affixes of the points A, B,C and D, respectively in the Argand plane.

(a) AB is inclined to CD at the angle arg [(z_{2} – z_{1})/(z_{4} – z_{3})].

(b) If CD is inclines at 90° to AB, then arg [(z_{2} – z_{1})/(z_{4} – z_{3})] = ±(π/2).

(c) If z_{1} and z_{2} are fixed complex numbers, then the locus of a point z satisfying arg [([(z – z_{1})/(z – z_{2})] = ±(π/2).

**Logarithm of a Complex Number**

Let z = x + iy be a complex number and in polar form of z is re^{iθ} , then

log(x + iy) = log (re^{iθ}) = log (r) + iθ

log(√x^{2} + y^{2}) + itan^{-1} (y/x)

or log(z) = log (|z|)+ iamp (z),

In general,

z = re^{i(θ + 2nπ)}

log z = log|z| + iarg z + 2nπi

**Applications of Complex Numbers in Coordinate Geometry**

Distance between complex Points

(i) Distance between A(z_{1}) and B(_{1}) is given by

AB = |z_{2} — z_{1}| = √(x_{2} + x_{1})^{2} + (y_{2} + y_{1})^{2}

where z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2}

(ii) The point P (z) which divides the join of segment AB in the ratio m : n is given by

z = (mz_{2} + nz_{1})/(m + n)

If P divides the line externally in the ratio m : n, then

z = (mz_{2} – nz_{1})/(m – n)

**Triangle in Complex Plane**

(i) Let ABC be a triangle with vertices A (z_{1}), B(z_{2}) and C(z_{3} ) then

(a) Centroid of the ΔABC is given by

z = 1/3(z_{1} + z_{2} + z_{3})

(b) Incentre of the AABC is given by

z = (az_{1} + bz_{2} + cz_{3})/(a + b + c)

(ii) Area of the triangle with vertices A(z_{1}), B(z_{2}) and C(z_{3}) is given by

For an equilateral triangle,

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{2}z_{3} + z_{3}z_{1} + z_{1}z_{2}

(iii) The triangle whose vertices are the points represented by complex numbers z_{1}, z_{2} and z_{3} is equilateral, if

**Straight Line in Complex Plane**

(i) The general equation of a straight line is az + az + b = 0, where a is a complex number and b is a real number.

(ii) The complex and real slopes of the line az + az are -a/a and – i[(a + a)/(a – a)].

(iii) The equation of straight line through z_{1} and z_{2} is z = tz_{1} + (1 — t)z_{2}, where t is real.

(iv) If z_{1} and z_{2} are two fixed points, then |z — z_{1}| = z — z_{2}| represents perpendicular bisector of the line segment joining z1 and z2.

(v) Three points z_{1}, z_{2} and z_{3} are collinear, if

This is also, the equation of the line passing through _{1}, z_{2} and z_{3} and slope is defined to be (z_{1} – z_{2})/z_{1} – z_{2}

**(vi) Length of Perpendicular** The length of perpendicular from a point z_{1} to az + az + b = 0 is given by |az_{1} + az_{1} + b|/2|a|

(vii) arg (z – z_{1})/(z – z_{2}) = β

Locus is the arc of a circle which the segment joining z_{1} and z_{2} as a chord.

(viii) The equation of a line parallel to the line az + az + b = 0 is az + az + λ = 0, where λ ∈ R.

(ix) The equation of a line parallel to the line az + az + b = 0 is az + az + iλ = 0, where λ ∈ R.

(x) If z_{1} and z_{2} are two fixed points, then I z — z11 =I z z21 represents perpendicular bisector of the segment joining A(z1) and B(z2).

(xi) The equation of a line perpendicular to the plane z(z_{1} – z_{2}) + z(z_{1} – z_{2}) = |z_{1}|^{2} – |z_{2}|^{2}.

(xii) If z_{1}, z_{2} and z_{3} are the affixes of the points A, B and C in the Argand plane, then

(a) ∠BAC = arg[(z_{3} – z_{1}/z_{2} – z_{1})]

(b) [(z_{3} – z_{1})/(z_{2} – z_{1})] = |z_{3} – z_{1}|/|z_{2} – z_{1}| (cos α + isin α), where α = ∠BAC.

(xiii) If z is a variable point in the argand plane such that arg (z) = θ, then locus of z is a straight line through the origin inclined at an angle θ with X-axis.

(xiv) If z is a variable point and z_{1} is fixed point in the argand plane such that (z — z_{1})= θ, then locus of z is a straight line passing through the point z_{1} and inclined at an angle θ with the X-axis.

(xv) If z is a variable point and z_{1}, z_{2} are two fixed points in the Argand plane, then

(a) |z – z_{1}| + |z – z_{2}| = |z_{1}– z_{2}|

Locus of z is the line segment joining z_{1} and z_{2}.

(b) |z – z_{1}| – |z – z_{2}| = |z_{1}– z_{2}|

Locus of z is a straight line joining z_{1} and z_{2} but z does not lie between z1 and z_{2}.

(c) arg[(z – z_{1})/(z – z_{2)] = 0 or π}

Locus z is a straight line passing through z_{1} and z_{2}.

(d) |z – z_{1}|^{2} + |z – z_{2}|^{2} = |z_{1} – z_{2}|^{2}

Locus of z is a circle with z_{1} and z_{2} as the extremities of diameter.

**Circle in Complete Plane**

(i) An equation of the circle with centre at z_{0} and radius r is

|z – z_{0}| = r

or zz – z_{0}z – z_{0}z + z_{0}

- |z — z
_{0}| < r, represents interior of the circle. - |z — z
_{0}| > r, represents exterior of the circle. - |z — z
_{0}| ≤ r is the set of points lying inside and on the circle |z — z_{0}| = r. Similarly, |z — z_{0}| ≥ r is the set of points lying outside and on the circle |z — z_{0}| = r. **General equation of a circle is**

zz – az – az + b = 0

where a is a complex number and b is a real number. Centre of the circle = – a

Radius of the circle = √aa – b or √|a|^{2} – b

(a) Four points z_{1}, z_{2}, z_{3} and z_{4} are concyclic, if

[(z_{4} — z_{1})(z_{2} — z_{3})]/[(z_{4} – z_{3})(z_{2} – z_{1})] is purely real.

(ii) |z — z_{1}|/|z – z_{2}| = k ⇒ Circle, if k ≠ 1 or Perpendicular bisector, if k = 1

(iii) The equation of a circle described on the line segment joining z_{1} and _{1} as diameter is (z – z_{1}) (z – z_{2}) + (z – z_{2}) (z — z_{1}) = 0

(iv) If z_{1}, and z_{2} are the fixed complex numbers, then the locus of a point z satisfying arg [(z – z_{1})/(z – z_{2})] = ± π / 2 is a circle having z_{1} and z_{2 }at the end points of a diameter.

**Conic in Complex plane**

(i) Let z_{1} and z2 be two fixed points, and k be a positive real number.

If k >|z_{1}– z_{2}|, then |z – z_{1}| + |z – z_{2}| = k represents an ellipse with foci at A(z_{1}) and B(z_{2}) and length of the major axis is k.

(ii) Let z_{1} and z2 be two fixed points and k be a positive real number.

If k ≠ |z_{1}– z_{2}| , then |z – z_{1}| – |z – z_{2}| = k represents hyperbola with foci at A(z_{1}) and B(z_{2}).

**Important Points to be Remembered**

- √-a x √-b ≠ √ab

√a x √b = √ab is possible only, if both a and b are non-negative.

So, i^{2} = √-1 x √-1 ≠ √1

- is neither positive, zero nor negative.
- Argument of 0 is not defined.
- Argument of purely imaginary number is π/2
- Argument of purely real number is 0 or π.
- If |z + 1/z| = a then the greatest value of |z| = a + √a
^{2}+ 4/2 and the least value of |z| = -a + √a^{2}+ 4/2 - The value of i
^{i}= e^{-π2} - The complex number do not possess the property of order, i.e., x + iy < (or) > c + id is not defined.
- The area of the triangle on the Argand plane formed by the complex numbers z, iz and z + iz is 1/2|z|
^{2}. - (x) If ω
_{1}and ω_{2}are the complex slope of two lines on the Argand plane, then the lines are

(a) perpendicular, if ω_{1 }+ ω_{2} = 0.

(b) parallel, if ω_{1 }= ω_{2}.