Division by rationalising

Problem

Compute $\dfrac{1+i}{2-i}$ by rationalising.

Solution

Multiply numerator and denominator by the conjugate of the denominator, $\overline{2-i}=2+i$: $\frac{1+i}{2-i} = \frac{(1+i)(2+i)}{(2-i)(2+i)} = \frac{2 + i + 2i + i^2}{4 + 1} = \frac{1 + 3i}{5} = \boxed{\frac{1}{5} + \frac{3}{5}i}.$

Links

Topics: Complex numbers
Concepts: Conjugate · Cartesian form · Cartesian plane
Methods: Complex numbers in Cartesian form
Skills: Calculating
Exercise type: Evaluating expressions