How do you simplify (2+i) / (3+i)?

Dec 13, 2015

Multiply the numerator and denominator by the conjugate of the denominator to find that
$\frac{2 + i}{3 + i} = \frac{7}{10} + \frac{1}{10} i$

Explanation:

The conjugate of a complex number $a + b i$ is $a - b i$. The product of a complex number and its conjugate is a real number. Using that, we get

$\frac{2 + i}{3 + i} = \frac{2 + i}{3 + i} \cdot \frac{3 - i}{3 - i}$

$= \frac{\left(2 + i\right) \left(3 - i\right)}{\left(3 + i\right) \left(3 - i\right)}$

$= \frac{7 + i}{10}$

$= \frac{7}{10} + \frac{1}{10} i$