SAT Prep: What is a Rational Number?

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. $1.33333333333\dots$). 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 $3.1415926535897932384626433832795\dots$ 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 $1.7320508075688772935274463415059\dots$ 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 $2.7182818284590452353602874713527\dots$