Dividing Complex Numbers

Now that you understand what complex numbers are, it's time to explore what we can do with them. Dividing complex numbers entails multiplying the numerator and denominator by the conjugate of the denominator we're working with. Let's start with a practice problem.

The procedure for dividing complex numbers

Let's say we want to divide $3+5i$ by $-1+2i$ . The first step is rewriting the equation as a fraction:

$\frac{3+5i}{-1+2i}$

Next, we need to figure out what the conjugate of our denominator is. The conjugate of $-1+2i$ is $-1-2i$ , meaning that we need to multiply both our numerator and denominator by $-1-2i$ :

$\frac{\left(3+5i\right)\left(-1-2i\right)}{(-1+2i)\left(-1-2i\right)}$

From here, we have some multiplication to do. Since we're working with binomials, we can use the FOIL method to begin:

$\frac{(-3-6i+5i+10i^2)}{\left(1-4i^2\right)}$

We need to combine like terms to make sense of this, so let's do that next:

$\frac{\left(-3-11i-10i^2\right)}{\left(1-4i^2\right)}$

So far, we've treated the imaginary unit i as any other variable, but remember that i squared $i^2=-1$ . Our next step is substituting -1 for $i^2$ in the equation above:

$\frac{\left(-3-11i+10\right)}{5}$

Almost done! We have to combine the -3 and 10 in the numerator to get $\frac{\left(7-11i\right)}{5}$ , which we then split into separate fractions with like denominators for a final answer of:

$\frac{7}{5}-\left(\frac{11}{5}\right)i$

Dividing complex numbers practice questions

$\frac{\left(4+2i\right)}{\left(-3-2i\right)}$

The first step is multiplying the numerator and denominator by the conjugate of the denominator, which is $-3+2i$ in this case. That gives us:

$\frac{(4+2i)\ast \left(-3+2i\right)}{\left(-3-2i\right)\ast \left(-3+2i\right)}$

Next, we use the FOIL method on both the numerator and denominator. Now we have:

$\frac{\left(-12+8i-6i+4i^2\right)}{\left(9-6i+6i-4i^2\right)}$

Then, we combine like terms noting that $i^2=-1$ :

$\frac{-16+2i}{13}$

$13$ is a prime number in the denominator, so we cannot simplify our answer any further.

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Get help dividing complex numbers

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