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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
(a) The set R of real numbers is a proper submit of the complex numbers. Hence, the complete number system is
(b) Zero is purely real as well as purely imaginary but not imaginary.
(c) i = √(-1), is called the imaginary unit. Also, i2=-1; i3=1; i4=1, etc.
(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.