Complex number equation

Question

How do you solve this equation please

Simply the complex number and write in 𝑎 + 𝑏𝑖 form where 𝑎, 𝑏 ∈ R

4−3𝑖 /4+3𝑖

Answer 1

You simplify a complex fraction by multiplying the numerator and denominator by the complex conjugate of the denominator: the complex conjugate of a + bi is a - bi. This removes imaginary terms from the denominator.

(4 - 3i)/(4+3i) * (4 - 3i)/(4 - 3i) = (16 - 24 i + 9 i^2)/(16 - 9 i^2) = (16 - 24i - 9)/(16 + 9) = (7 - 24i)/25

Something fun to notice: in the original fraction, the numerator is the complex conjugate, so they have the same magnitude. In the simplified expression, the magnitude of the numerator is sqrt(7^2 + 24^2) = 25, the magnitude of the denominator.