What is Complex Number?
Complex numbers are defined as expressions of the form x = iy
where x, y ∈ R, and i = √-1 . It is denoted by z i.e. z= x + iy
The numbers x and y are called respectively real and imaginary parts of complex number z.
I.e. x = Re (z) and y = Im (z)
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.
- The Set R of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
- Zero is both purely real as well as purely imaginary but not imaginary.
- √x √y = √xy Only if atleast one of either x or y is non negative.
- i = √-1 is called the imaginary unit. Also i2 = -1; i3 = -i; i4 = 1 etc.
Equality of Complex Numbers
Two complex numbers Z1 = a1 + ib1 and z2 = ac2 + ib2 are equal, iff a1 = a2 and b1 = b2 i.e. Re (Z1) = Re (z2) and Im (Z1) = Im (z2).