Complex Number
Let us consider a quadratic equation say, x^{2}
 2x + 5 = 0. Hence the discriminant b^{2 } 4ac, being
negative, it leads to the solution set { 1  2i, 1 + 2i}. The roots
viz x = 1  2i and x = 1 + 2i is that they are partly real and partly
imaginary. We now introduce a number which is a combination of real
and imaginary numbers. This is called a Complex number of the type
a + bi , a, b Î R. We denote this by z then
z = a + ib ...........(ii)
Here a = R (z) ...... real part z and b = I (z) ..... imaginary
part z.
The same may be expressed more elegantly as an ordered pair of
two real numbers a and b as,
z = (a , b) ..........(iii)
For example, z_{1} = (2, 3) = 2 + 3i,
z_{2} = (2, 3) = 2  3i,
z_{3} = (0, 1) = 0 + i = i ,
z_{4} = (0, 0) = 0 + 0i = 0
which is a real number = 0
Conjugate Complex : If z = a + bi is a complex number then
z or z* = a  bi is called conjugate complex of z.
For example, z1_{ }= 3 + 2i then _{1}
= 3  2i, _{2}
=  8  4i then z_{2} =  8 + 4i.
Basic Operations : From the point of view
of an axiomatic foundation, it is advised to treat a complex as
an ordered pair (a,b), a, b Î R; subject to certain operational
rules.
Equality of two complex numbers : Let z_{1}
= (a,b) and z_{2} = (c,d) then z_{1} = z_{2}
iff a = c
and b = d.
Sum : Let z1_{ }= (a,b) and z_{2}
= (c,d) then z_{1} + z_{2} = (a + c, b + d )
Difference : z_{1}  z_{2}_{
}= (a  c, b  d)
Product : z_{1} . _{}
= (a, b) (c,d) = (ac  bd, ad + bc)
Scalar Multiplication : kz = (ka, kb) if
k is a scalar and z = (a,b)
Quotient :
Product of conjugate complexes : z . z = (a + bi)
(a  bi) = a^{2} + b^{2}
Closer Property : From the operations between
any two complex numbers is again a complex number only. As such
we remark that the set 'C' of complex numbers is closed with respect
to these operations.
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