What Are Rational Numbers?

The word "rational" comes from "ratio" — and that's the key idea. A rational number is any number that can be expressed as a ratio of two integers.

More precisely: a number $r$ is rational if it can be written as

$$r = \frac{p}{q}$$

where $p$ and $q$ are integers and $q \neq 0$.

That covers a lot of ground. Whole numbers are rational ($7 = \frac{7}{1}$). Fractions obviously are. Negative numbers too ($-3 = \frac{-3}{1}$). And as you'll see, most decimals are rational as well — even ones that never end.

Terminating Decimals

A terminating decimal ends after a finite number of digits. For example:

$$\frac{3}{4} = 0.75$$

Once the division is done, it stops. Any terminating decimal is rational — you can always write it as a fraction over the appropriate power of 10.

Converting a terminating decimal to a fraction

Take $1.3456$. The last digit sits in the ten-thousandths place, so place the digits over 10,000:

$$1.3456 = \frac{13456}{10000}$$

Then reduce. Both share a common factor of 16:

$$\frac{13456}{10000} = \frac{841}{625}$$

Done — $1.3456$ is the ratio $\frac{841}{625}$, confirming it's rational.

A simpler example: $0.64$.

$$0.64 = \frac{64}{100} = \frac{16}{25}$$

Repeating Decimals

A repeating decimal goes on forever but cycles through the same pattern of digits. The notation $0.\overline{25}$ means $0.252525\ldots$ — the "25" repeats indefinitely.

Repeating decimals are still rational. The fraction $\frac{1}{3} = 0.\overline{3}$ is the simplest example. But what about something less obvious, like $0.\overline{25}$? There's a clean algebraic method to find the fraction.

Converting a repeating decimal to a fraction

Example: Convert $0.\overline{25}$ to a fraction.

Let $N = 0.252525\ldots$

Since two digits repeat, multiply both sides by 100:

$$100N = 25.252525\ldots$$

Subtract the original equation from this one:

$$100N - N = 25.252525\ldots - 0.252525\ldots$$

$$99N = 25$$

$$N = \frac{25}{99}$$

The repeating part cancels perfectly. The rule for what to multiply by: if one digit repeats, multiply by 10. Two digits → multiply by 100. Three digits → multiply by 1,000.

Example: Convert $4.\overline{5}$ (which is $4.5555\ldots$) to a fraction.

Let $N = 4.5555\ldots$

One digit repeats, so multiply by 10:

$$10N = 45.5555\ldots$$

$$10N - N = 45.5555\ldots - 4.5555\ldots$$

$$9N = 41$$

$$N = \frac{41}{9}$$

Example: Convert $0.\overline{75}$ (which is $0.757575\ldots$) to a fraction.

Two digits repeat → multiply by 100:

$$100N = 75.7575\ldots$$

$$99N = 75$$

$$N = \frac{75}{99} = \frac{25}{33}$$

Irrational Numbers

Not every number is rational. A number is irrational if it cannot be written as $\frac{p}{q}$ for any integers $p$ and $q$. Irrational numbers produce decimals that go on forever without ever settling into a repeating pattern.

Common examples:

$\sqrt{2} = 1.41421356\ldots$
$\pi = 3.14159265\ldots$
$e = 2.71828182\ldots$

None of these ever repeat. No matter how far you compute them, the digits keep changing without cycling back.

A few things worth knowing:

The sum of a rational and an irrational number is always irrational. So $\sqrt{2} + 3.8$ is irrational, because the non-repeating nature of $\sqrt{2}$ survives the addition.
Not every square root is irrational. $\sqrt{9} = 3$, which is rational. Only square roots of non-perfect-squares (like $\sqrt{2}$, $\sqrt{5}$, $\sqrt{7}$) are irrational.
Rational and irrational together make up all real numbers. Every point on the number line is one or the other.

To summarize the distinction:

Type	Decimal behavior	Example
Rational	Terminates or repeats	$0.75$, $0.\overline{3}$, $\frac{5}{8}$
Irrational	Non-terminating, non-repeating	$\sqrt{2}$, $\pi$

Practice Problems

Convert $0.36$ to a fraction in lowest terms.

Show answer$0.36 = \frac{36}{100} = \frac{9}{25}$

Convert $0.\overline{36}$ ($0.363636\ldots$) to a fraction.

Show answerLet $N = 0.363636\ldots$ Two digits repeat → multiply by 100: $100N = 36.3636\ldots$ Subtract: $99N = 36$, so $N = \frac{36}{99} = \frac{4}{11}$

Is $\sqrt{16}$ rational or irrational?

Show answer$\sqrt{16} = 4 = \frac{4}{1}$, so it is rational. It simplifies to a whole number.

Is $0.010010001\ldots$ (where each group of zeros grows by one) rational or irrational?

Show answerIrrational. The decimal never settles into a repeating pattern — the sequence keeps changing. Non-terminating and non-repeating means it can't be expressed as $\frac{p}{q}$.

Convert $2.\overline{142857}$ to a fraction. (Six digits repeat.)

Show answerLet $N = 2.142857142857\ldots$ Six digits repeat → multiply by 1,000,000: $1{,}000{,}000N = 2{,}142{,}857.142857\ldots$ Subtract: $999{,}999N = 2{,}142{,}855$, so $N = \frac{2{,}142{,}855}{999{,}999} = \frac{15}{7}$. Verify: $15 \div 7 = 2.\overline{142857}$ ✓

Type	Decimal behavior	Example
Rational	Terminates or repeats	\(0.75\), \(0.\overline{3}\), \(\frac{5}{8}\)
Irrational	Non-terminating, non-repeating	\(\sqrt{2}\), \(\pi\)