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Exercises Equations and inequalities

Example 7 from Algebra and Trigonometry, 2.4 Complex Numbers

Dividing Complex Numbers

Divide: $(2 + 5 i)$ by $(4 - i) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We begin by writing the problem as a fraction.

$\frac{(2 + 5 i)}{(4 - i)}$

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

$\frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)}$

To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).

$\begin{matrix} \frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)} & = & \frac{8 + 2 i + 20 i + 5 i^{2}}{16 + 4 i - 4 i - i^{2}} \\ = & \frac{8 + 2 i + 20 i + 5 (−1)}{16 + 4 i - 4 i - (−1)} & Because i^{2} = −1. \\ = & \frac{3 + 22 i}{17} \\ = & \frac{3}{17} + \frac{22}{17} i & Separate real and imaginary parts . \end{matrix}$
Note that this expresses the quotient in standard form.

How did it go?

PreviousFinding Complex Conjugates Next Simplifying Powers of i

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.