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Complex Numbers a Number of The Form Study Notes

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Complex Numbers Complex Number A number of the form z = x + iy, where x , y ∈ R, is called a complex number. Here, the symbol i is used to denote −1 and it is called iota. The set of complex numbers is denoted by C. Real and Imaginary Parts of a Complex Number Let z = x + iy be a complex number, then x is called the real part and y is called the imaginary part of z and it may be denoted as Re( z ) and Im ( z ), respectively. Purely Real and Purely Imaginary Complex Number A complex number z is a purely real, if its imaginary part is 0. i.e. Im ( z ) = 0. And purely imaginary, if its real part is 0 i.e. Re ( z ) = 0. Zero Complex Number A complex number is said to be zero, if its both real and imaginary parts are zero. Equality of Complex Numbers Two complex numbers z1 = a1 + ib1 and z 2 = a2 + ib2 are equal, iff a1 = a2 and b1 = b2 i.e. Re ( z1 ) = Re ( z 2 ) and Im ( z1 ) = Im ( z 2 ). Iota Mathematician Euler, introduced the symbol i (read as iota) for − 1 with property i 2 + 1 = 0. i.e. i 2 = − 1. He also called this symbol as the imaginary unit. Integral power of iota (i) are given below. i = −1 , i 2 = − 1, i3 = − i , i 4 = 1 So, i 4n + 1 = i , i 4n + 2 = − 1, i 4n +3 = − i , i 4n + 4 = 1  ( −1)n / 2 , if n is an even integer  n −1 In other words, i =  ( −1) 2 ⋅ i , if n is an odd integer n Algebra of Complex Numbers 1. Addition of Complex Numbers Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then their su ...
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