# What is the conjugate of  7 - 4i?

Dec 3, 2015

$7 + 4 i$

#### Explanation:

The complex conjugate of $a + b i$ is $a - b i$. Complex conjugates have the cool property that $\left(a + b i\right) \left(a - b i\right) = {a}^{2} - a b i + a b i - {b}^{2} {i}^{2} = {a}^{2} + {b}^{2} = | a + b i {|}^{2}$. This makes them useful for rewriting quotients $\frac{a + b i}{c + \mathrm{di}}$ in the standard form $\alpha + \beta i$.

For example,

$\frac{3 + 2 i}{7 - 4 i} = \frac{3 + 2 i}{7 - 4 i} \cdot \frac{7 + 4 i}{7 + 4 i} = \frac{21 + 12 i + 14 i + 8 {i}^{2}}{49 - 16 {i}^{2}}$

$= \frac{13 + 26 i}{49 + 16} = \frac{13}{65} + \frac{26}{65} i = \frac{1}{5} + \frac{2}{5} i$