Students typically work with rational numbers in upper elementary and middle school, then learn about irrational numbers in grade 8, when the two concepts get explored together as part of the full system beyond whole numbers and simple fractions. That timing is part of why the two ideas can blur together at first.
The confusion makes sense. Some numbers that look complicated, like repeating decimals, actually turn out to be rational. Meanwhile, some numbers might look familiar when you see them written as decimals, like π, but their digits go on forever without repeating or ending.
We need to know whether a number is rational or irrational because that affects how we work with it later, including how we simplify, estimate, compare, and use it in further calculations.
Today, we're going to define both types clearly, build a simple way to tell them apart, and practice classifying a few tricky examples.
A rational number is any number that can be written as a fraction of two integers, what we'd call a ratio, which is exactly where the name “rational” comes from.
Plenty of numbers, including whole numbers and negative numbers, that don't look like fractions at first glance still count.
Take \(\Large\frac{1}{2}\), 3, and −7:
\(\Large\frac{1}{2}\) is already a fraction,
we can write 3 as \(\Large\frac{3}{1}\),
-7 can be written as \(\Large\frac{-7}{1}\).
All three are rational numbers because we can write them as fractions.
Rational numbers can also show up as decimals. When a decimal terminates, which means it stops, or repeats in a predictable pattern, it's rational.
We can see this with 0.75. The decimal stops after two digits, and we can write it as \(\Large\frac{75}{100}\), or \(\Large\frac{3}{4}\). That makes 0.75 rational.
Repeating decimals can be rational, too. Take 0.333.... The digits keep going, so the number may look confusing at first. But 0.333... equals \(\Large\frac{1}{3}\). Because we can write it as a fraction, 0.333... is also rational.
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An irrational number is any number that cannot be written as a fraction of two integers.
Let’s take \(\sqrt{2}\) as an example. Its decimal, 1.41421..., goes on forever without ever settling into a repeating pattern. No matter which two whole numbers we try to divide, we can never land on \(\sqrt{2}\) exactly. That's what makes it irrational.
Now, we’ll look at π. Its digits, 3.14159..., also continue forever without repeating. That's why we use the symbol π instead of writing out its decimal value in full. We can keep calculating digits, but we never reach an end or a repeating pattern.
For irrational numbers, the decimal keeps going and going, but the digits never repeat in a predictable pattern.
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Rational and irrational numbers can look a lot alike at first, especially in decimal form. So we’ll use these steps to decide which type of number we have.
Let's try it with 0.454545..., which should be rational, and \(\sqrt{5}\), which should be irrational.
We start by asking “Can we write this number as one integer over another?”
0.454545... isn't written as a fraction, so we can't answer this question with confidence yet. That means we need to go further and look at its decimal form.
\(\sqrt{5}\) doesn't obviously reduce to a fraction of two whole numbers, so we can't answer this question with confidence yet. That means we need to go further and look at its decimal form.
If a number's decimal form stops completely or repeats in a predictable pattern, it's rational.
The decimal form of 0.454545... repeats in a predictable pattern (45 repeating), so it's rational.
The decimal form of \(\sqrt{5}\) is 2.23606… It doesn't terminate or settle into a repeating pattern no matter how many digits we calculate. That confirms \(\sqrt{5}\) is irrational.
We can organize the logic in a simple flowchart and use the same questions for any number we need to classify.
Now it's your turn to run a few numbers through the same questions we just covered. Classify each one below as rational or irrational, and check your answers at the bottom of the page.
\(\sqrt{7}\)
0.666...
π
−12
0.32
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Ready to see how you did? Here's the reasoning behind each answer.
\(\sqrt{7}\) → Irrational; its decimal, ≈ 2.6457513… , never terminates or repeats, and it cannot be written as an exact fraction.
0.666... → Rational; it's a repeating decimal equal to the fraction \(\Large\frac{2}{3}\).
π → Irrational; its decimal goes on forever without repeating.
−12 → Rational; it can be written as the fraction \(\Large\frac{-12}{1}\).
0.32 → Rational; it terminates, and it's equal to the fraction \(\Large\frac{32}{100}\) = \(\Large\frac{8}{25}\).
Mathnasium of South Arlington is a math-only learning center for K-12 students in South Arlington, TX. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
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