## Multiplicative Inverse

If $\alpha=2+i$, then in the form of $a+bi$, what does $\alpha^{-1}$ equal?
Source: NCTM Mathematics Teacher, September 2006

Solution
By definition of multiplication inverse
$\alpha\alpha^{-1}=1$
$\alpha^{-1}=\dfrac{1}{2+i}$
$=\dfrac{1}{2+i}\times\dfrac{2-i}{2-i}$
$=\dfrac{2-i}{2^2-i^2}$
$=\dfrac{2-i}{4+1}$
$=\dfrac{2-i}{5}$
$=\dfrac{2}{5}-\dfrac{1}{5}i$

Answer: $\dfrac{2}{5}-\dfrac{1}{5}i$