Adding Complex Fractions

Adding Complex Fractions 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)