Rationalizing Complex Expressions

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Algebra Two $\longrightarrow$
Complex Numbers $\longrightarrow$
  • Rationalize expressions that contain a complex number in the denominator
  • Learn how to divide by a complex number
Lesson Description

Dividing by a complex number, or equivalently, working with a fraction that has a complex number denominator, requires rationalization for the final answer. This lesson covers the nuances of dividing by a complex number via rationalization and simplification.

Practice Problems

Practice problems and worksheet coming soon!


There's No i in.... Denominator?

Because the imaginary number $i$ represents $\sqrt{-1}$, the consensus among teachers is that we treat it like a square root when it comes to agreeing on what it means to "simplify" an expression. As we saw with rationalizing square roots » previously, one of the biggest agreed-upon no-nos for teachers is leaving answers with radicals in the denominator, and because of that, we will also not be allowed to leave $i$ in the denominator.There's another good reason for this, however. As we'll see throughout the lesson, any number fraction with a complex number or pure imaginary denominator can be expressed as a single complex number $a + bi$ where $a$ and $b$ are real numbers (but possibly fractions).

Pure Imaginary

The simplest case for simplifying fractions with imaginary number denominators is to have a purely imaginary number rather than a complex number ». Whether that denominator is just $i$ or a multiple of $i$, we will rationalize it the same way in this case: multiply by $i/i$.
Example 1Rationalize.$$\frac{-6+2i}{i}$$$\blacktriangleright$ Multiply this fraction by $i/i$, which will preserve its value due to the fact that we are multiplying by a form of $1$, and will also create a real number denominator.$$\frac{-6+2i}{i} \cdot \frac{i}{i}$$$$\longrightarrow \frac{-6i + 2i^2}{i^2} \longrightarrow \frac{-2 - 6i}{-1}$$$$\longrightarrow 2+ 6i$$
Pro Tip
All rationalizing happens this way when the denominator of the complex number fraction you are rationalizing is $i$ or a multiple of $i$. Multiply top and bottom by $i/i$ and simplify using the properties of the imaginary number.

Complex Denominators

Recall that purely imaginary numbers are just a subset of complex numbers ». If you have a complex number denominator, the process for rationalizing is similar to the approach for rationalizing mixed radical denominators » which we looked at previously.
Rationalizing a+bi DenominatorsA complex number is of the form $a + bi$ where $a$ and $b$ are each real numbers. The conjugate of $a+bi$ is defined to be $a-bi$.For any fraction with a complex denominator and a real, imaginary, or complex numerator, rationalization is achieved by multiplying by the conjugate of the denominator.$$\frac{c}{a+bi} \cdot \frac{a-bi}{a-bi}$$or$$\frac{di}{a+bi} \cdot \frac{a-bi}{a-bi}$$or$$\frac{c+di}{a+bi} \cdot \frac{a-bi}{a-bi}$$for real numbers $c$ and $d$.
Depending on the numerator, we may be distributing or applying "FOIL". Let's see an example of each.
Example 2Rationalize the following fraction.$$\frac{i}{-1+4i}$$$\blacktriangleright$ The denominator is a complex number, so we will rationalize this fraction by multiplying by the conjugate of the denominator.$$\frac{i}{-1+4i} \cdot \frac{-1 - 4i}{-1 - 4i}$$$$\frac{i(-1-4i)}{(-1+4i)(-1-4i)}$$$$\frac{-i-4i^2}{1 + 4i - 4i -16i^2}$$$$\frac{4 - i}{17}$$
Example 3Rationalize the following fraction.$$\frac{2-5i}{3+i}$$$\blacktriangleright$ Once again, we need to multiply by the conjugate of the denominator.$$\frac{2-5i}{3+i} \cdot \frac{3 - i}{3 - i}$$$$\frac{6 - 2i -15i + 5i^2}{9 - 3i + 3i - i^2}$$$$\frac{1 - 17i}{10}$$
You Should Know
Always read your instructions! Some questions want you to rationalize these types of fractions and turn in answers in the form of a fraction, while other questions ask you to turn in answers in the form $a + bi$. Even though the work process is the same, the answer forms are slightly different.For instance, the answer to Example 3 above would be$$\frac{1}{10} - \frac{17i}{10}$$if the instructions had instead said "Rationalize the following fraction and express your answer in the form $a + bi$". And in that case, if you turned in the answer$$\frac{1 - 17i}{10}$$and your teacher is a mean, they'll dock you points!

This is Division

You may have wondered in the prior lesson about complex number arithmetic » why we looked at addition, subtraction, and multiplication but not division. Remember that fractions are really just shorthand notation for division, and so when you're given a problem such as one we've seen in this lesson, you're being asked to divide.For instance, when you're asked to simplify$$\frac{2-5i}{3+i}$$you're also answering the question "what is $(2-5i) \div (3+i)$?"Rationalizing a complex number fraction isn't just your job for the task of "simplifying", but it's literally how you can divide one complex number by another and get a single $a+bi$ answer! From this example, we can see that$$(2-5i) \div (3+i) = \frac{1}{10} - \frac{17i}{10}$$

Put It To The Test

In each of the following exercises, rationalize or divide. Express your answer as either a fraction with a real number denominator, or a complex number of the form $a + bi$.
Example 4$$\frac{-2}{3i}$$
Show solution
$\blacktriangleright$ Multiply by $i/i$.$$\frac{-2}{3i} \cdot \frac{i}{i}$$$$\frac{-2i}{3i^2}$$$$\frac{2i}{3}$$If you really want to be proper about also reporting the answer in the form $a + bi$, we could say this answer is$$0 + \frac{2}{3} \, i$$Note that Many students are tempted to multiply by $3i/3i$, and while that isn't a wrong or illegal move, it is inefficient.
Example 5$$\frac{60+2i}{7+2i}$$
Show solution
$\blacktriangleright$ We need to multiply the top and bottom of this expression by the conjugate of the denominator.$$\frac{60+2i}{7+2i} \cdot \frac{7-2i}{7-2i}$$$$\frac{420 - 14i + 120i - 4i^2}{49 -14i +14i -4i^2}$$$$\frac{424 + 106i}{53}$$Do be careful to check - this is a tricky one for further simplifying. $53$ goes into each of these numbers!$$\frac{\cancel{53}(8+2i)}{\cancel{53}} = 8 + 2i$$Also, this answer is already in the form $a + bi$.
Example 6$$\frac{9}{6i - 2}$$
Show solution
$\blacktriangleright$ Similar to the last problem, multiply by the conjugate of the denominator.$$\frac{9}{6i - 2} \cdot \frac{6i + 2}{6i + 2}$$$$\frac{54i + 18}{36i^2 + 12i - 12i - 4}$$$$\frac{54i + 18}{-40}$$$$\frac{-9 - 27i}{20}$$Or, in $a + bi$ form,$$-\frac{9}{20} - \frac{27}{20} \, i$$Note that some teachers will try to tell you that you need to rewrite the denominator as $-2 + 6i$, and that the conjugate is really $-2 - 6i$. You will ultimately get an equivalent answer. It happens to be true that the conjugate trick works both ways.
Example 7$$-5 - 10i \div 8 + i$$
Show solution
$\blacktriangleright$ Don't let the division sign fool you - it's equivalent to write this as a fraction, and when we do it will look familiar.$$\frac{-5 - 10i}{8 + i}$$$$\longrightarrow \frac{-5 - 10i}{8 + i} \cdot \frac{8 - i}{8 - i}$$$$\frac{-40 + 5i -80i + 10i^2}{64 - 8i + 8i -i^2}$$$$\frac{-50 - 75i}{65}$$$$\frac{-10 - 15i}{13}$$Or, in $a + bi$ form,$$-\frac{10}{13} - \frac{15}{13} \, i$$
Example 8$$5i + 9 \div 4$$
Show solution
$\blacktriangleright$ We don't actually need to do any rationalizing here - the denominator is not imaginary! Sometimes teachers throw this kind of thing in the mix to trick you.Answer:$$\frac{5i + 9}{4}$$or$$\frac{9}{4} + \frac{5}{4} \, i$$
Lesson Takeaways
  • Understand when teachers expect you to rationalize an imaginary expression
  • Know how to rationalize purely imaginary denominators efficiently (just $i$, ignore the denominator coefficient)
  • Know how to rationalize imaginary expressions with complex number denominators
  • Maintain awareness about the way in which your instructions want you to give answers (often specifically asked for $a + bi$ form)

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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