Complex Number System

Indian mathematician Mahavira (850 A.D.) was first to mention in his work 'Ganitasara sangraha'; 'As in nature of things a negative (quantity) is not a square (quantity), it has, therefore, no square root'. Hence, there is no real number x which satisfies the polynomial equation x²+1=0

A symbol √(-1) , denoted by letter i was introduced by Swiss Mathematician, Leonhard Euler to provide solution of equation x²+1=0.'i'was regarded as a fictions or imaginary number which could be manipulated algebraically like an ordinary real number, except that its square was -1. The letter i was used to denote √(-1),, possibly because i is the first letter of the Latin word, 'imaginarius'.

To permit solutions of such polynomial equations, the set of complex numbers is introduced. We can consider a complex number as having the form a+bi where a and b are real numbers. It is denoted by z, i.e., z=a+bi. 'a' is called as real part of j which is denoted by (Re z) and 'b' is called as imaginary poart of z which is denoted by (Img).

Any complex number is:-

(i) Purely real, if b = 0

(ii) Imaginary, if b 0.

(iii) Purely imaginary, if a =0

Note:

(a) The set R of real numbers is a proper submit of the complex numbers. Hence, the complete number system is

NCWCICQCRCC

(b) Zero is purely real as well as purely imaginary but not imaginary.

(d) √a√b√c = √(abc........)

iff. At least one of a,b,c ........ is non-negative.

(e) If j = a+ib, then a-ib is called complex conjugate of j and written as .

(f) Real numbers satisfy order relations where as imaginary order relation, i.e., i>0, 3+i<2, are meaningless.