The history of mathematics shows that man has been developing and enlarging his concept of number according to the saying that “Necessity is the mother of invention”.

 In the remote past they stared with the set of counting numbers and invented, by stages, the rational numbers, negative   numbers,  irrational numbers. Since square of a negative as well as positive number is a positive number   and the square root of a negative number does not exist in the real numbers. So, square roots of negative numbers were given no attention for centuries together.

 The numbers of the form x + iy, where x, y   and i =  ,are called complex numbers, here x is called real part and y is called imaginary part of the complex number. For example, 2+5\imath , 3-\imath etc. are complex numbers.

Important note about complex number

Every real number is a complex number with 0 as its imaginary part (note).

Now , start with considering the equation.

x^{2}+1=0 \newline \Rightarrow x^{2} \, =-1\newline\Rightarrow x = \pm 1

Which does not belong to the set of real numbers.  Therefore   we, for convenience   call it imaginary number and denote it by \imath (read as iota).

The product of a real number and i(iota) is also an imaginary number

Thus 7\imath, -5\imath, 5\imath ,  \imath are all imaginary numbers, \imath which may be written  1.\imath  is also an imaginary number.

Powers of i :

i^{2}= -1\newline i^{3}= i^{2}.i=-1.i=-i \newline i^{4}=i^{2}i^{2}=-1.-1=1

 

 Therefore any power of i  equal to 1, i,-1 or -i.

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