Equality: Two complex numbers are equal if and only if their real parts and their imaginary parts are respectively equal.
a+ib =c+id if and only if a=c and b=d
Example 1: If x+4yi+40 = 5x+8i, Find the values x and y.
Solution: Equating the left hand side of the real part = Right hand side of the real part
x +40 = 5x
5x-x = 40
Left hand side of the imaginary part = Right hand side of the imaginary part
4y = 8
y = 2
Therefore the value x is 10 and the value y is 2.
Example 2: Find the values of a and b for the following equations:
Example 3: If z = x + iy and |2z- 1| = |z- 2| then prove that x2 + y2 = 1.
|2z- 1| = |z- 2|
→|2(x+iy)- 1| = |x+iy- 2|
→|(2x-1)+2iy| = |(x-2)+iy|
→√(2x − 1) 2 + 4y2 = √(x − 2)2+ y2
→4x2 - 4x + 1 + 4y2 = x2 − 4x + 4 + y2
(squaring on both sides)
→3x2 + 3y2 - 3 = 0
→3x2 + 3y2 = 3
→x2 + y2 = 1. (Proved).
Example 4: Find the real numbers a and b such that the equation below becomes true.
(a + 6) + 2bi = 4 - 5i
Solution: (a + 6) + 2bi = 4 - 5i
Equating the real and imaginary parts, we get,
a + 6 = 4 and 2b = -5
a = -2 and b = -5/2