What Is a Conjugate in Math? Explained for Beginners

When you hear the word conjugate, your mind might go to language class, and you’d be thinking in the right direction. 

The word conjugate comes from the Latin conjugare, meaning “to join together.” In grammar, it refers to the way we change verbs to match the subject or tense.

But in math, the word conjugate takes on a whole new meaning. It isn’t about grammar, it’s about something entirely different that helps us solve and simplify problems in clever ways.

In this guide, we’ll explain what a conjugate is in math, show you how we use it to make tricky expressions easier to handle, give you a chance to practice what you’ve learned, and answer the most common questions we hear from students here at Mathnasium.

What Does "a Conjugate" Mean in Math?

In math, a conjugate is what you get when you take a binomial—an expression with two terms—and flip the sign between them.

So if you have an expression like \( \displaystyle a + b \), the conjugate is \( \displaystyle a - b \). And it works the other way too: the conjugate \( \displaystyle a - b \) is \( \displaystyle a + b \). These can be any two numbers, variables, or a mix of both.

To put this into action:

• If you have \( \displaystyle x + 4 \), the conjugate is \( \displaystyle x - 4 \).

• If your expression is \( \displaystyle 3 - \sqrt{2} \), the conjugate is \( \displaystyle 3 + \sqrt{2} \)

And that’s really all there is to it. A conjugate keeps the same terms in the same order, but the sign in the middle switches from positive to negative, or the other way around.

You May Also Like: What Is a Reciprocal in Math?

How Do We Use Conjugates?

Conjugates help us simplify math problems in ways that would otherwise be much trickier. 

Specifically, we use conjugates to:

• Rationalize radical expressions, especially when we have a square root in the denominator of a fraction.

• Simplify complex numbers, especially when dividing by an expression that includes the imaginary unit i

See how Mathnasium’s proprietary teaching approach, the Mathnasium Method™, helps students learn and master any math topic, including conjugates.

1. Multiplying Conjugates

Since multiplying conjugates is at the core of both simplifying radical expressions and dividing complex numbers, let’s first make sure we know exactly how to do it.

Let’s take a simple pair: \( \displaystyle x + 4 \) and \( \displaystyle x - 4 \). Since these are binomials, you might remember the FOIL method—a way to multiply binomials:

• First: multiply the first terms

• Outer: multiply the outer terms

• Inner: multiply the inner terms

• Last: multiply the last terms

Let’s try it out:

• First: \( \displaystyle x \cdot x = x^2 \)

• Outer: \( \displaystyle x \cdot (-4) = -4x \)

• Inner: \( \displaystyle 4 \cdot x = 4x \)

• Last: \( \displaystyle 4 \cdot (-4) = -16 \)

Put it all together:

\( \displaystyle x^2 - 4x + 4x - 16 \)

Now look at the middle terms:

\( \displaystyle -4x + 4x = 0 \)

They cancel each other out. So the expression simplifies to:

\( \displaystyle x^2 - 16 \)

Now think about this: we did all four FOIL steps, but which parts actually made it into the final answer? Just the first and the last. Why is that?

Because when we multiply two conjugates, the middle terms always cancel out.

So instead of doing all four FOIL steps, we really only need the F and the L, first and last. To put it in math terms, we can use the equation:
\( \displaystyle (a + b)(a - b) = a^2 - b^2 \)

Next, let’s see how conjugates help us rationalize radical expressions and divide complex numbers.

2. Rationalizing Radical Expressions

Sometimes in math, we end up with a square root in the denominator of a fraction, like this:

\( \displaystyle \frac{1}{\sqrt{2} + 3} \)

This is called a radical expression, and having a square root in the denominator isn't ideal. In math, we try to "rationalize" the denominator, meaning we want to remove the radical and make the bottom a regular number.

But how do we do that?

Here’s where conjugates come in.

To get rid of the square root in the denominator, we multiply both the top and bottom of the fraction by the conjugate of the denominator. In this case, the conjugate of\( \displaystyle \sqrt{2} + 3 \) is \( \displaystyle \sqrt{2} - 3 \).

So, we do this:

\( \displaystyle \frac{1}{\sqrt{2} + 3} \times \frac{\sqrt{2} - 3}{\sqrt{2} - 3} \)

Now let’s multiply the denominators using what we learned about conjugates. Remember: when we multiply conjugates, we just use the first and the last term:

• First: \(\displaystyle \sqrt{2} \times \sqrt{2} = \sqrt{4} = 2\)

• Last: \(\displaystyle 3 \times (-3) = -9\)

So the denominator becomes:

\(\displaystyle 2 - 9 = -7\)

Now the numerator:

\( \displaystyle 1 \times (\sqrt{2} - 3) = \sqrt{2} - 3 \)

Putting it all together:

\( \displaystyle \frac{\sqrt{2} - 3}{-7} \)

And that’s it! We’ve eliminated the radical from the denominator.

That’s how conjugates help us rationalize radical expressions: by turning an irrational denominator into a rational one with a simple multiplication step.

3. Multiplying Complex Numbers

Another place where conjugates come in handy is when working with complex numbers, especially when dividing.

Let’s say you’re asked to simplify this:

\( \displaystyle \frac{1}{3 + 2i} \)

This expression has an imaginary number in the denominator, and just like with radicals, we want to get rid of it.

To do that, we multiply both the numerator and the denominator by the conjugate of the denominator.

So what's the conjugate of \( \displaystyle 3 + 2i \)?

It’s \( \displaystyle 3 - 2i \).

Now multiply:

\( \displaystyle \frac{1}{3 + 2i} \times \frac{3 - 2i}{3 - 2i} \)

Let’s simplify the denominator by multiplying the conjugates:

• First: \( \displaystyle 3 \times 3 = 9 \)

• Last: \( \displaystyle 2i \times -2i = -4i^2 \)

But wait—what’s \( \displaystyle i^2 \)?

Remember, in math, i is the imaginary unit, and by definition:

\( \displaystyle i^2 = -1 \)

So when we see \( \displaystyle -4i^2 \), we can replace \( \displaystyle i^2 \) with -1, which gives us:

\( \displaystyle 4 \times (-1) = -4 \)

So the denominator is:

\( \displaystyle 9 + 4 = 13 \)

And the numerator:

\( \displaystyle 1 \times (3 - 2i) = 3 - 2i \)

Finally, we have:

\( \displaystyle \frac{3 - 2i}{13} \)

Specially trained math tutors at Mathnasium tackle complex topics like conjugates in math by breaking them down into manageable parts, helping students build a deep understanding of each concept.  

Practice Time: Conjugates in Action

Ready to put your new skills to the test? Try solving these four tasks using what you’ve learned about conjugates.

Take your time, and when you're finished, you can check your answers at the bottom of the guide.

1. Multiply the conjugates:

\( \displaystyle (7x + 5)(7x - 5) \)

2. Multiply the conjugates:

\( \displaystyle (x + 2\sqrt{3})(x - 2\sqrt{3}) \)

3. Rationalize the expression:

\( \displaystyle \frac{1}{\sqrt{5 + 2}} \)

4. Simplify the complex fraction:

\( \displaystyle \frac{1}{4 - i} \)

FAQs About Conjugates

When students first learn about conjugates, it naturally brings up a lot of questions. 

To help clear things up, we’ve put together a list of the most common questions we hear at Mathnasium of Rolling Hills Estates, along with simple, straightforward answers to help you feel more confident as you practice.

1. When do students learn about conjugates in math?

Most students first encounter conjugates in early high school, usually in Algebra I. That's when they learn to multiply binomials and simplify expressions involving square roots and complex numbers.

Later on, conjugates appear again in Algebra II, Pre-Calculus, and even in college-level courses like Linear Algebra, where students deal with matrix conjugates or complex vector operations. 

2. Do conjugates only work with square roots and complex numbers?

Not at all. While conjugates are most often used with radical expressions and complex numbers, the concept can be applied to any pair of binomials that have the same terms with opposite signs.

3. What’s the difference between a conjugate and a reciprocal?

A conjugate changes the sign in a binomial (like from \( \displaystyle a + b \) to \( \displaystyle a - b \) ). 

A reciprocal, on the other hand, flips the position in a fraction. So the reciprocal of \( \displaystyle \frac{3}{5} \) is \( \displaystyle \frac{5}{3} \) , while the conjugate of \( \displaystyle 3 + \sqrt{2} \) is \( \displaystyle 3 - \sqrt{2} \). They’re both useful, but in different ways.

4. Can you take the conjugate of a single number or variable?

Technically, no. Conjugates apply to binomials, which means expressions with two terms. 

So something like just \( \displaystyle x \) or \( \displaystyle \sqrt{3} \) doesn’t have a conjugate. But if you had \( \displaystyle x + 0 \), you could say its conjugate is \( \displaystyle x - 0 \), which of course just gives you  \( \displaystyle x \) again.

At Mathnasium, we encourage students to ask questions because curiosity leads to clarity. Our instructors provide caring guidance that helps students make sense of math, one question at a time.

Master Conjugates at Mathnasium of Rolling Hills Estates

Mathnasium of Rolling Hills Estates is a math-only learning center for K–12 students in Rolling Hills Estates, CA.

Using the Mathnasium Method™, a proven and proprietary teaching approach, our specially trained math tutors provide face-to-face instruction in a caring and fun group environment to help students truly understand and enjoy any math class and topic, including conjugates typically covered in high school math. 

Each student begins their Mathnasium journey with a diagnostic assessment that helps us identify exactly what they know and where they need support. From there, we create personalized learning plans to guide them toward math success at their own pace.

Whether your student is looking to catch up, keep up, or get ahead, schedule a free assessment and enroll at Mathnasium of Rolling Hills Estates today!

Schedule a Free Assessment at Mathnasium of Rolling Hills Estates

Pssst! Check Your Answers Here

If you’ve given our practice exercises a try, check how you did below.

1. Multiply the conjugates:

\( \displaystyle (7x + 5)(7x - 5) = (7x)^2 - 5^2 = 49x^2 - 25 \)

2. Multiply the conjugates:

\( \displaystyle (x + 2\sqrt{3})(x - 2\sqrt{3}) = x^2 - (2\sqrt{3})^2 = x^2 - 12 \)

3. Rationalize the expression:

\( \displaystyle \frac{1}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{5 - 4} = \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2 \)

4. Simplify the complex fraction:

\( \displaystyle \frac{1}{4 - i} \times \frac{4 + i}{4 + i} = \frac{4 + i}{(4)^2 - (-i)^2} = \frac{4 + i}{16 - (-1)} = \frac{4 + i}{17} \)