# Introduction & basic of Complex Numbers

Hey there! This article talks to you about the interesting topic in mathematics i.e.-‘Complex Numbers’ whose other name is ‘Imaginary Numbers’. These are numbers having imaginary content with notation-i. Let us understand deeply what Complex numbers are…

Before going to complex numbers, we first have to know what is i ? Here,

##### ‘i’ is the notation which denotes the square root of (-1) i.e. i=v(-1)

i is imaginary, because, there is no number in the real along with negative square. This originates new set of numbers with the name- Imaginary numbers, which when combined with real part gives a number which is partly real & partly imaginary, such a number we identify as- Complex Number.

# Complex Numbers:

Complex Numbers are of the form **z = a+bi** which is the combination of the real part i.e.(a) & the imaginary part (bi). These complex numbers defined as the order pair of (x,y) of real numbers in the complex plane, with rectangular co-ordinates x & y. The real number x is displayed as- the (x,o) on the real axis (which we consider X-axis) and the imaginary part (o,y) on the imaginary axis (which we consider Y-axis) the point on the behalf of these coordinates is the complex number we want.

The figure below clarifies the above discussion.

(ref: Wikipedia)

# Properties of Complex Numbers:

Let us see about the properties of the Complex Numbers, which they satisfy and also how they behave etc..

Complex numbers, as like as real Numbers satisfy the properties –

- Commutativity
- Associativity
- Identity
- Inverse

Above all properties with respect to both addition & multiplication Complex Numbers do satisfy, As well as

- Associative laws
- Distributive Laws

Both left & right, are also valid for Complex Numbers.

Now we discuss how the arithmetic operations such as addition, subtraction, multiplication, division works on Imaginary Numbers..

# Operations of Complex/ Imaginary Numbers:

### Addition:

Addition of two complex i.e. imaginary Numbers is-

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

### Subtraction:

Subtraction of two complex i.e. imaginary Numbers is-

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

### Multiplication:

Multiplication of two complex i.e. imaginary Numbers is-

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

### Division:

Division of two complex i.e. imaginary Numbers is-

(x_{1}+iy_{1})/(x_{2}+iy_{2}) ={(x_{1}x_{2}+y_{1}y_{2})/(x_{2}^{2}+y_{2}^{2})}+ i{(y_{1}x_{2}-x_{1}y_{2}) / (x_{2}^{2}+y_{2}^{2})}

This is about the introduction & basic operations of the imaginary Numbers. We’ll discuss details about the Singularities in Complex Numbers in further posts…