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.

Imaginary numbers

(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-
(x1+iy1) + (x2+iy2) = (x1+x2) + i(y1+y2)

Subtraction:

Subtraction of two complex i.e. imaginary Numbers is-
(x1+iy1) – (x2+iy2) = (x1+x2) – i(y1+y2)

Multiplication:

Multiplication of two complex i.e. imaginary Numbers is-
(x1+iy1) . (x2+iy2) = (x1.x2-y1.y2) + i(y1x2)

Division:

Division of two complex i.e. imaginary Numbers is-
(x1+iy1)/(x2+iy2) ={(x1x2+y1y2)/(x22+y22)}+ i{(y1x2-x1y2) / (x22+y22)}


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