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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.

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.

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

(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)