# Equality of Complex Numbers

We will discuss about the equality of complex numbers.

Two complex numbers z$$_{1}$$ = a + ib and z$$_{2}$$ = x + iy are equal if and only if a = x and b = y i.e., Re (z$$_{1}$$) = Re (z$$_{2}$$) and Im (z$$_{1}$$) = Im (z$$_{2}$$).

Thus, z$$_{1}$$ = z$$_{2}$$ ⇔ Re (z$$_{1}$$) = Re (z$$_{2}$$) and Im (z$$_{1}$$) = Im (z$$_{2}$$).

For example, if the complex numbers z$$_{1}$$ = x + iy and z$$_{2}$$ = -5 + 7i are equal, then x = -5 and y = 7.

Solved examples on equality of two complex numbers:

1. If z$$_{1}$$ = 5 + 2yi and z$$_{2}$$ = -x + 6i are equal, find the value of x and y.

Solution:

The given two complex numbers are z$$_{1}$$ = 5 + 2yi and z$$_{2}$$ = -x + 6i.

We know that, two complex numbers z$$_{1}$$ = a + ib and z$$_{2}$$ = x + iy are equal if a = x and b = y.

z$$_{1}$$ = z$$_{2}$$

⇒ 5 + 2yi = -x + 6i

⇒ 5 = -x and 2y = 6

⇒ x = -5 and y = 3

Therefore, the value of x = -5 and the value of y = 3.

2. If a, b are real numbers and 7a + i(3a - b) = 14 - 6i, then find the values of a and b.

Solution:

Given, 7a + i(3a - b) = 14 - 6i

⇒ 7a + i(3a - b) = 14 + i(-6)

Now equating real and imaginary parts on both sides, we have

7a = 14 and 3a - b = -6

⇒ a = 2 and 3 2 – b = -6

⇒ a = 2 and 6 – b = -6

⇒ a = 2 and – b = -12

⇒ a = 2 and b = 12

Therefore, the value of a = 2 and the value of b = 12.

3. For what real values of m and n are the complex numbers m$$^{2}$$ – 7m + 9ni and n$$^{2}$$i + 20i -12 are equal.

Solution:

Given complex numbers are m$$^{2}$$ - 7m + 9ni and n$$^{2}$$i + 20i -12

According to the problem,

m$$^{2}$$ - 7m + 9ni = n$$^{2}$$i + 20i -12

⇒ (m$$^{2}$$ - 7m) + i(9n) = (-12) + i(n$$^{2}$$ + 20)

Now equating real and imaginary parts on both sides, we have

m$$^{2}$$ - 7m = - 12 and 9n = n$$^{2}$$ + 20

⇒ m$$^{2}$$ - 7m + 12 = 0 and n$$^{2}$$ - 9n + 20 = 0

⇒ (m - 4)(m - 3) = 0 and (n - 5)(n - 4) = 0

⇒ m = 4, 3 and n = 5, 4

Hence, the required values of m and n are follows:

m = 4, n = 5; m = 4, n = 4; m = 3, n = 5; m = 3, n = 4.