**Imaginary Numbers**

The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc.

The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota.

**Integral Power 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} = 1

Note:

- For any two real numbers a and b, the result √a × √b : √ab is true only, when atleast one of the given numbers i.e. either zero or positive.

√-a × √-b ≠ √ab

So, i^{2}= √-1 × √-1 ≠ 1 - ‘i’ is neither positive, zero nor negative.
- i
^{n}+ i^{n+1}+ i^{n+2}+ i^{n+3}= 0

**Complex Number**

A number of the form x + iy, where x and y are real numbers, is called a complex number, x is called real part and y is called imaginary part of the complex number i.e. Re(Z) = x and Im(Z) = y.

**Purely Real and Purely Imaginary Complex Number**

A complex number Z = x + iy 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 Number**

Two complex numbers z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} are equal, iff x_{1} = x_{2} and y_{1} = y_{2} i.e. Re(z_{1}) = Re(z_{2}) and Im(z_{1}) = Im(z_{2})

Note: Order relation “greater than’’ and “less than” are not defined for complex number.

**Algebra of Complex Numbers**

**Addition of complex numbers**

Let z_{1} = x_{1} + iy_{1} 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**

- Commutative: z
_{1}+ z_{2}= z_{2}+ z_{1} - Associative: z
_{1}+ (z_{2}+ z_{3}) = (z_{1}+ z_{2}) + z_{3} - Additive identity z + 0 = z = 0 + z

Here, 0 is additive identity.

**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})

**Multiplication of complex numbers**

Let z_{1} = (x_{1} + iy_{1}) 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**

- Commutative: z
_{1}z_{2}= z_{2}z_{1} - Associative: z
_{1}(z_{2}z_{3}) = (z_{1}z_{2})z_{3} - Multiplicative identity: z . 1 = z = 1 . z

Here, 1 is multiplicative identity of an element z. - Multiplicative inverse: For every non-zero complex number z, there exists a complex number z
_{1}such that z . z_{1}= 1 = z_{1}. z - Distributive law: z
_{1}(z_{2}+ z_{3}) = z_{1}z_{2}+ z_{1}z_{3}

**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

**Conjugate of Complex Number**

Let z = x + iy, if ‘i’ is replaced by (-i), then said to be conjugate of the complex number z and it is denoted by , i.e. = x – iy

**Properties of Conjugate**

**Modulus of a Complex Number**

Let z = x + iy be a complex number. Then, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e. |z| =

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_{1} < z_{2} has no meaning but |z_{1}| > |z_{2}| or |z_{1}|<|z_{2}| has got its meaning, since |z_{1}| and |z_{2}| are real numbers.

**Properties of Modulus of a Complex number**

**Argand Plane**

Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called argand plane or gaussian plane. A purely number x, i.e. (x + 0i) is represented by the point (x, 0) on X-axis. Therefore, X-axis is called real axis. A purely imaginary number iy i.e. (0 + iy) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is called the imaginary axis.

**Argument of a complex Number**

The angle made by line joining point z to the origin, with the positive direction of X-axis in an anti-clockwise sense is called argument or amplitude of complex number. It is denoted by the symbol arg(z) or amp(z).

arg(z) = θ = tan-1()

Argument of z is not unique, general value of the argument of z is 2nπ + θ, but arg(0) is not defined. The unique value of θ such that -π < θ ≤ π is called the principal value of the amplitude or principal argument.

**Principal Value of Argument**

- if x > 0 and y > 0, then arg(z) = θ
- if x < 0 and y > 0, then arg(z) = π – θ
- if x < 0 and y < 0, then arg(z) = -(π – θ)
- if x > 0 and y < 0, then arg(z) = -θ

**Polar Form of a Complex Number**

If z = x + iy is a complex number, then z can be written as z = |z| (cosθ + isinθ), where θ = arg(z). This is called polar form. If the general value of the argument is θ, then the polar form of z is z = |z| [cos (2nπ + θ) + isin(2nπ + θ)], where n is an integer.

**Square Root of a Complex Number**

**Solution of a Quadratic Equation**

The equation ax^{2} + bx + c = 0, where a, b and c are numbers (real or complex, a ≠ 0) is called the general quadratic equation in variable x. The values of the variable satisfying the given equation are called roots of the equation.

The quadratic equation ax^{2} + bx + c = 0 with real coefficients has two roots given by and , where D = b^{2} – 4ac, called the discriminant of the equation.

Note:

(i) When D = 0, roots ore real and equal. When D > 0 roots are real and unequal. Further If a,b, c ∈ Q and D is perfect square, then the roots of quadratic equation are real and unequal and if a, b, c ∈ Q and D is not perfect square, then the roots are irrational and occur in pair. When D < 0, roots of the equation are non real (or complex).

(ii) Let α, β be the roots of quadratic equation ax^{2} + bx + c = 0, then sum of roots α + β = and the product of roots αβ = .