Basic of Complex Numbers

INTRODUCTION

We know square of any real number is never negative. But in Mathematics and Physics we get many x²+c=0 (c>0) terms. In real number system we haven't any answer on these term. So, solving this problem, we introduced a new system called complex number system, where we solve these equations by using a new term i (√(-1)).

DEFINATION OF COMPLEX NUMBERS

A order pair of real number (a,b) which satisfy the following condition

<1> (a,b)=(c,d) if and only if a=b and c=d

<2> (a,b)±(c,d)={(a±c),(b±d)}
<3> (a,b).(c,d)={(ac-bd),(ad+bc)}
called complex numbers. [(c,d) is a order pair of real number]

NOTATION

It is written by z=a+bi (a,b∈R & i is imaginary unit). Therefore a complex part is multiplication of imaginary unit and real part.

TYPES OF COMPLEX NUMBERS

There are only three types of complex numbers

(1) Pure Real Numbers:- Which Complex Numbers imaginary part is zero and real part may be 0 or not, called Pure Real Numbers .

e.g. 2+0i ,0+0i etc.

• It is located on X axis .

• We know its simply real number. Therefore we learn R⊆ℂ where ℂ is a set of complex numbers.

(2) Pure Complex Number:- Which Complex Numbers real part is zero and always have a non zero imaginary part , called Pure Complex Numbers .

e.g. 2i,4i etc.

• It is located on Y axis.

• 0+0i isn't a complex number.

(3) Mixed Complex Number:-Which Complex Numbers had a non-zero real and complex part called Mixed Complex Numbers .

e.g. 3+2i,4+i etc.

• It is located on argand plane or complex plane.

Location of Argand plane in Number system

Complex Numbers