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If an order pair (x, y) of two real numbers x and y is represented by the symbol
x + iy  [where i= √(-1)]

Then the order pair (x, y) is called a complex number (or an imaginary number).
Here x is called the real part of the complex number and y is called its imaginary part.

Addition of two complex numbers is also a complex number.

The sum of two complex numbers can be expressed in the form A + i B
Where A and B are real.

Let z1= a+ib and z2= c+id be two complex numbers (a, b, c, d are real). Then the sum of the complex numbers

=z1+z2
=a+ib+c+id
=a+c + i(b+d)
=A +iB
Where A= a+c and B= b+d and are real.

Therefore addition of two complex numbers will give a complex number.

Example:- Add the following two complex fractions.

(4+3i)/2 and (4-3i)/4

Solution: -

(4+3i)/2 + (4-3i)/4 = [2(4+3i) +(4-3i)]/4
=(2*4 + 2*3i + 4 – 3i) / 4
=(8 + 6i + 4 – 3i) / 4
=(14 +3i)/4
=(14/4) +(3/4)i
=(7/2) +(3/4) i

Therefore after adding complex fractions we got an another complex fraction.

Example 2: - Simplify 1/ (x+iy) + 1/ (x-iy)

Solution: -

1/ (x+iy) + 1/ (x-iy)= [(x+iy) + (x-iy)]/ (x+iy)(x-iy)
=(x+iy+x-iy)/ [(x)^2 –(iy)^2]
=2x / (x^2 – i^2 *y^2)
=2x/ (x^2 +y^2)               (since i^2= -1)