Given that z¹ =4+3i and Z²=2+3i .find z¹÷z²
To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of a complex number a + bi is a - bi.
So, to find z¹ ÷ z², we multiply z¹ and z² by the conjugate of z²:
z¹ = 4 + 3i
z² = 2 + 3i
Conjugate of z² = 2 - 3i
Multiplying z¹ and z² by the conjugate of z²:
z¹ ÷ z² = (4 + 3i)(2 - 3i) / (2 + 3i)(2 - 3i)
Using FOIL (First, Outer, Inner, Last):
z¹ ÷ z² = (4 * 2) + (4 * -3i) + (3i * 2) + (3i * -3i) / (2 * 2) + (2 * -3i) + (3i * 2) + (3i * -3i)
Simplified:
z¹ ÷ z² = (8 - 12i + 6i - 9i²) / (4 - 6i + 6i - 9i²)
Since i² = -1, substitute i² with -1:
z¹ ÷ z² = (8 - 12i + 6i - 9(-1)) / (4 - 6i + 6i - 9(-1))
Simplified further:
z¹ ÷ z² = (8 - 12i + 6i + 9) / (4 - 6i + 6i + 9)
z¹ ÷ z² = (17 - 6i) / 13
Therefore, z¹ ÷ z² = (17 - 6i) / 13