2023-01-01

How to simplify $\frac{2+i}{1-i}$ completely?

michellegrosscju

Expert

A complex number a+bi has the conjugate a-bi. A real number is the result of a complex number and its conjugate. This fact will be used to remove the complex number from the given expression's denominator.
$\frac{2+i}{1-i}=\frac{\left(2+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}$
$=\frac{2+2i+i-1}{1+i-i+1}$
$=\frac{1+3i}{2}$
$=\frac{1}{2}+\frac{3}{2}i$

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