SAT Prep: What is a Rational Number?

Stay organized with our free SAT checklist.   Download your copy here.

While students may not need to expressly name rational or irrational numbers on the SAT, they will be working a lot with both those types. Indeed, most students learn to work complex problems that involve both rational and irrational numbers in Algebra class, yet many students soon forget the difference between the two.

 

What is a rational number? What is an irrational number? How is this relevant to the Math section of the SAT test? Read on for a quick yet comprehensive review.

 

What Are Rational Numbers?

 

\(1,2,3,4,5,\) and so on are all integers, as are \(-1,-2,-3,-4\), and so forth. \(1.23\) is a decimal and \(\frac{3}{8}\) is a fraction. All are called different things, yet all are considered rational numbers.

 

At its core, a rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where both the numerator \(p\) and the denominator \(q\) are nonzero integers (e.g. \(\frac{p}{q}=\frac{15}{8}, \frac{1}{2},  \frac{200}{3}\)). Note that 1 is an integer and thus can be the value of q, so all integers are rational numbers (e.g. \(\frac{4}{1}=4\)).

Other examples of rational numbers are decimals with finite decimal numbers (e.g. \(3.125=\frac{25}{8}\)) and decimals with a repeating pattern (e.g. \(1.33333333333…\)). This is because both these types of decimals can be converted to fractions with an integer numerator and denominator (to learn more, read our post: How to Convert a Decimal to a Fraction).

 

It logically follows that any number that does not fit into at least one of these categories is considered an irrational number. What kinds of numbers classify as irrational?

 

For example, \(\pi\) represents \(3.1415926535897932384626433832795…\) and it keeps going. There is no finite end to \(\pi\), so it is considered an irrational number.

 

Similarly, many square roots, cubed roots, and so on are considered irrational numbers because they also solve as decimals with no finite end. \(\sqrt{3}\) is one such example, as it equals \(1.7320508075688772935274463415059…\) and it goes on.

 

However, it is important to note that not all roots are irrational numbers. For example, \(\sqrt{16}\) is \(4\), which is a rational number. Therefore, \(\sqrt{16}\) is considered a rational number as well.

 

So, let’s do a small quiz to make sure we have these concepts right:

 

Is \(3.181818181818\overline{18}\) a rational or irrational number?

 

Answer: A rational number! The decimals are infinite, but there is a pattern to it.

 

Is \(\sqrt{169}\) a rational or irrational number?

 

Answer: a rational number! \(13\:\cdot\:13 = 169\).

 

Last one: is Euler’s number, \(e\), a rational or irrational number?

 

Answer: an irrational number! Euler’s number appears to be a decimal without a discovered end. The first few decimals are \(2.7182818284590452353602874713527…\)

Want to increase your SAT score?

Our students see an average increase of 160+ points in the SAT.

What Do You Need To Know For The SAT?

 

Very few questions on the SAT will ask you directly what is and is not a rational number. Instead, you will see many rational and irrational numbers being used throughout the SAT Math section, and you will need to know how to solve, manipulate, and apply them.

 

Specifically, you will usually be allowed to leave irrational numbers as is on the SAT and not have to simplify or solve them. For example, if you are told to simplify an expression that has \(\pi\) or \(e\) in it, you know that you are not going to be able to simplify those two variables, and your answers will need to be in terms of \(\pi\) or \(e\).

 

Here are some examples of SAT problems that deals with both rational and irrational numbers.

 

Source: SAT Math Practice Problems

 

1. In the complex number system, which of the following is equal to \((14-2i)(7+12i)\)? (\(i = \sqrt{-1}\))?

A. \(74\)

B. \(122\)

C. \(74 + 154i\)

D. \(122 + 154i\)

 

Answer: D

 

In this case, \(i\) is an irrational number because \(\sqrt{-1}\) does not simplify to an integer, a finite decimal, or a fraction with nonzero integers in the numerator and the denominator. Therefore, you can treat \(i\) like a variable, in that you can’t simplify it any further.

 

Once you know this, this problem looks a lot like a FOIL problem using the distributive property. Using the FOIL method:

 

\((14-2i)(7+12i)\)

\(=98 + 168i − 14i − 24i^2\)

Note: \(i^2 = -1\).

Therefore, the expression above can be rewritten as: \(98 + 168i − 14i − (−24)\).

Combining like terms: \(122 + 154i\).

 

2. Simplify: \(\frac{\sqrt[3]{81}}{\sqrt[3]{9}}\)

A. \(9\)

B. \(\sqrt[3]{9}\)

C. \(3\)

D. \(2.08908\)

 

Answer: B.

 

\(\frac{\sqrt[3]{81}}{\sqrt[3]{9}}=\sqrt[3]{\frac{81}{9}}\)

\(=\sqrt[3]{9}\)

 

\(\sqrt[3]{9}\) is an infinite non-repeating decimal, an irrational number. To keep our answer cleaner, we’ll leave it as is. The SAT should not make you write out irrational numbers in decimal form, so you can leave them in their cleanest/simplest form.

 

For More Information

 

Itching for more CollegeVine concept review and testing tips so that you can ace the SAT? Check out our helpful previous blog posts:

 

25 Tips and Tricks for the SAT

30 SAT Math Formulas You Need To Know

15 Hardest SAT Math Questions

 

Lastly, if you feel like you need more of a deep-dive into your SAT Math skills, CollegeVine has a program for that! By signing up for our SAT Tutoring Program, you will get personalized, one-on-one help on whatever SAT sections or concepts you need the most help on, and we’ll give you our tried and true tricks for raising your overall score.

Want more SAT tips sent to you?

Sign up below and we'll send you expert SAT tips and guides.


Sadhvi Mathur
Senior Blogger at CollegeVine
Short bio
Sadhvi is a recent graduate from the University of California, Berkeley, where she double majored in Economics and Media Studies. Having applied to over 8 universities, each with different application platforms and requirements, she is eager to share her knowledge now that her application process is over. Other than writing, Sadhvi's interests include dancing, playing the piano, and trying not to burn her apartment down when she cooks!