What makes a number a rational number? Are they just whole numbers, or do fractions and decimals count too? These are questions that often puzzle students—and even adults!
In this guide to rational numbers, we’ll clear up any confusion.
Whether you're in middle school just learning about rational numbers or need a quick refresher, you’re in the right place.
Read on to find clear definitions, types, practical examples, a quiz to test your knowledge, and answers to the questions learners commonly ask.
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A rational number is any number that can be written as a fraction where the numerator (top number) and denominator (bottom number) are integers, and the denominator is not zero.
And why can’t the denominator be zero?
Simply because dividing by zero isn’t allowed in math—it’s like trying to share something with zero people!
In mathematical terms, a rational number is often denoted as \(\Large\frac{p}{q}\), where p and q are integers, and q \(\neq\) 0.
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Rational numbers come in different forms, but they all share one key characteristic—they can be written as fractions.
Types of rational numbers are:
Integers are rational because they can be written as fractions with a denominator of 1.
Examples: 8 (\(\Large\frac{8}{1}\)), 0 (\(\Large\frac{0}{1}\)), -5 (\(\Large\frac{-5}{1}\))
Any fraction where the numerator (top number) and denominator (bottom number) are integers, and the denominator isn’t zero, is rational.
Examples: \(\Large\frac{3}{5}\), \(\Large\frac{-4}{7}\)
Terminating Decimals: Decimals that stop are rational because they can be written as fractions.
Examples: 0.75 (\(\Large\frac{3}{4}\)), -2.5 (\(\Large\frac{-5}{2}\))
Repeating Decimals: Decimals with a repeating pattern are rational because they can also be written as fractions.
Examples: 0.333… (\(\Large\frac{1}{3}\)), −1.666… (\(\Large\frac{-5}{3}\))
Numbers whose square roots simplify to whole numbers, also known as perfect squares, are rational because the roots can be written as fractions.
Examples: 4 (\(\sqrt{4}\)=2), 9 (\(\sqrt{9}\)=3)
Not every number is a rational number!
If a number cannot be written as a fraction, it’s called an irrational number. These numbers have decimals that go on forever without repeating a pattern or forming a fraction.
Irrational numbers come in a few shapes and forms, including:
Some decimals go on forever without ever stopping or forming a repeating pattern.
Example: 0.1010010001....
The square roots of numbers that aren’t perfect squares are irrational because they don’t simplify into whole numbers or fractions.
Examples: \(\sqrt{2}\) ≈ 1.414213..., \(\sqrt{7}\) ≈ 2.645751...
A math constant is a fixed value in math that doesn’t change. These constants are irrational because their decimals go on forever without stopping or forming a repeating pattern.
Example: The most well-known mathematical constant is π (pi), approximately equal to 3.14159...
Just like with integers, we can perform the four basic arithmetic operations—addition, subtraction, multiplication, and division—with rational numbers.
Since rational numbers are numbers that can be written as fractions (with integers on top and bottom, and the bottom number not zero), the rules for working with rational numbers are the same as the rules for fractions.
Whichever operation we perform with rational numbers, the result must always be a rational number.
To add rational numbers, the denominators (bottom numbers) must be the same.
Adding rational numbers with the same denominators is super simple!
All we need to do is add the numerators together while keeping the denominator the same like so:
\(\Large\frac{1}{7}\) + \(\Large\frac{3}{7}\) = \(\Large\frac{4}{7}\)
To add rational numbers with different denominators, we need to find a common denominator first.
Once the denominators match, we simply add the numerators (top numbers) and keep the denominator the same.
Let’s see how to add rational numbers with different denominators using this example: \(\Large\frac{2}{3}\) + \(\Large\frac{1}{4}\).
The least common denominator (LCD) is the smallest number that both denominators can divide evenly into. For 3 and 4, the LCD is 12.
We change each fraction so their denominators are both 12:
To turn \(\Large\frac{2}{3}\) into an equivalent fraction with denominator 12, multiply both the numerator and the denominator by 4:
\(\Large\frac{2}{3}\) × \(\Large\frac{4}{4}\) = \(\Large\frac{2×4}{3×4}\) = \(\Large\frac{8}{12}\)
To turn 14 into an equivalent fraction with denominator 12, multiply both the numerator and the denominator by 3:
\(\Large\frac{1}{4}\) × \(\Large\frac{3}{3}\) = \(\Large\frac{1×3}{4×3}\) = \(\Large\frac{3}{12}\)
Now that the denominators are the same, add the numerators:
\(\Large\frac{8}{12}\) + \(\Large\frac{3}{12}\) = \(\Large\frac{11}{12}\)
In this case, \(\Large\frac{11}{12}\) is already in its simplest form, so we’re done.
The final answer: \(\Large\frac{2}{3}\) + \(\Large\frac{1}{4}\) = \(\Large\frac{11}{12}\)
Subtracting rational numbers works just like adding them. The denominators must be the same.
If we are subtracting rational numbers with the same denominators, as in \(\Large\frac{4}{5}\) and \(\Large\frac{2}{5}\), we simply subtract the numerators like so:
\(\Large\frac{4}{5}\) - \(\Large\frac{2}{5}\) = \(\Large\frac{2}{5}\)
If we are subtracting rational numbers with different denominators, we need to find a common denominator first. Once the denominators are like, we simply subtract the numerators (top numbers) and keep the denominator unchanged.
Let’s see how this works in action.
We can subtract \(\Large\frac{5}{6}\) - \(\Large\frac{1}{4}\).
The least common denominator (LCD) for 6 and 4 is 12.
We adjust each fraction so their denominators are both 12:
To turn \(\Large\frac{5}{6}\) into an equivalent fraction with denominator 12, we multiply both the numerator and the denominator by 2:
\(\Large\frac{5}{6}\) × \(\Large\frac{2}{2}\) = \(\Large\frac{5×2}{6×2}\) = \(\Large\frac{10}{12}\)
To turn \(\Large\frac{1}{4}\) into an equivalent fraction with denominator 12, we multiply both the numerator and the denominator by 3:
\(\Large\frac{1}{4}\) × \(\Large\frac{3}{3}\) = \(\Large\frac{1×3}{4×3}\) = \(\Large\frac{3}{12}\)
Now that the denominators are the same, subtract the numerators:
\(\Large\frac{10}{12}\) - \(\Large\frac{3}{12}\) = \(\Large\frac{7}{12}\)
In this case, \(\Large\frac{7}{12}\) is already in its simplest form, so we’re done!
The final answer: \(\Large\frac{5}{6}\) - \(\Large\frac{1}{4}\) = \(\Large\frac{7}{12}\)
Multiplying rational numbers is pretty straightforward—we just have to multiply the numerators (top numbers) and the denominators (bottom numbers).
To show you the process, we will multiply \(\Large\frac{2}{3}\) and \(\Large\frac{4}{5}\).
Multiply the top numbers:
2 × 4 = 8
Step 2: Multiply the Denominators
Multiply the bottom numbers:
3 × 5 = 15
Combine the results from Steps 1 and 2 into a single fraction:
\(\Large\frac{2}{3}\) × \(\Large\frac{4}{5}\) = \(\Large\frac{8}{15}\)
In this case,\(\Large\frac{8}{15}\) is already in its simplest form, so no further simplification is needed.
The final answer: \(\Large\frac{2}{3}\) × \(\Large\frac{4}{5}\) = \(\Large\frac{8}{15}\)
Dividing rational numbers is almost as simple as multiplication! Instead of dividing directly, we flip the second fraction (take its reciprocal) and then multiply.
Let’s see how it works.
We will divide \(\Large\frac{3}{4}\) by \(\Large\frac{2}{5}\).
Take the reciprocal of \(\Large\frac{2}{5}\), which means flipping the numerator and denominator:
\(\Large\frac{2}{5}\) → \(\Large\frac{5}{2}\)
Now, we’re multiplying\(\Large\frac{3}{4}\) × \(\Large\frac{5}{2}\)
Multiply the top numbers (numerators):
3 × 5 = 15
Multiply the bottom numbers (denominators):
4 × 2 = 8
This gives us \(\Large\frac{15}{8}\).
In this case,\(\Large\frac{15}{8}\) is already in its simplest form.
However, since \(\Large\frac{15}{8}\) is an improper fraction (its numerator is bigger than its denominator), we can convert it to a mixed number:
15 ÷ 8 = 1 remainder 7
The whole number (1) is the quotient.
The fraction is the remainder (7).
The denominator (8) stays the same.
So, the final answer is: \(\Large\frac{3}{4}\) ÷ \(\Large\frac{2}{5}\) = 1\(\Large\frac{7}{8}\)
Practice makes perfect! Let’s solve these tasks with rational numbers together.
Determine if \(\sqrt{25}\) is a rational number.
First, we calculate the square root of 25.
\(\sqrt{25}\) = 5
Then, we check if this number can be expressed as a fraction.
5 is a whole that can be expressed as a fraction: 5 = \( \Large \frac{5}{1}\)
Therefore, \(\sqrt{25}\) is a rational number.
Determine if 0.75 is a rational number.
The decimal 0.75 stops, so it’s a terminating decimal. Terminating decimals are rational because they can be written as fractions.
To convert 0.75 to a fraction, we write it as:
\( \Large \frac{75}{100}\)
Next, simplify the fraction by finding the greatest common factor (GCF) of 75 and 100. The GCF of 75 and 100 is 25.
Divide both the numerator and denominator by 25:
\( \Large \frac{75÷25}{100÷25}\) = \( \Large \frac{3}{4}\)
Since 0.75 = \( \Large \frac{3}{4}\), it is a rational number
Add rational numbers \( \Large \frac{1}{3}\) and \( \Large \frac{1}{2}\).
Step 1: Find the Least Common Denominator (LCD)
For 3 and 2, the LCD is 6.
Step 2: Rewrite the Fractions with the Same Denominator
\( \Large \frac{1}{3}\) × \( \Large \frac{2}{2}\) = \( \Large \frac{2}{6}\)
\( \Large \frac{1}{2}\) × \( \Large \frac{3}{3}\) = \( \Large \frac{3}{6}\)
Step 3: Add the Fractions
Since the denominators are the same, add the numerators:
\( \Large \frac{2}{6}\) + \( \Large \frac{3}{6}\) = \( \Large \frac{5}{6}\)
Step 4: Simplify If Needed
\( \Large \frac{5}{6}\) is already in its simplest form, so our job is done.
Multiply \( \Large \frac{4}{7}\) × \( \Large \frac{2}{3}\)
To multiply these fractions, multiply the numerators and denominators:
\( \Large \frac{4}{7}\) × \( \Large \frac{2}{3}\) = \( \Large \frac{8}{21}\)
Since this fraction is already in its simplest form, we are done.
Divide \( \Large \frac{1}{5}\) ÷ \( \Large \frac{1}{3}\)
To divide these fractions, multiply the first fraction by the reciprocal of the second fraction:
\( \Large \frac{1}{5}\) ÷ \( \Large \frac{1}{3}\) = \( \Large \frac{1}{5}\) × \( \Large \frac{3}{1}\) = \( \Large \frac{3}{5}\)
Since \( \Large \frac{3}{5}\) is in its simplest form, our task is done.
Are you ready to put your understanding of rational numbers to the test? Try solving these tasks on your own.
0.123456789101112...
Check your answers at the end of the page to see how you did!
Here are the questions we usually get from students first learning about rational numbers.
Not quite! While most fractions are rational numbers, the denominator of the fraction must not be zero. For example, \( \Large \frac{3}{0}\) is not a rational number because dividing by zero is undefined in math.
No. All whole numbers are rational because they can be written as fractions with a denominator of 1.
For example, 6 = \( \Large \frac{6}{1}\), so whole numbers are always rational.
Yes, percentages are rational numbers because they can always be written as fractions. For example, 25% is the same as \( \Large \frac{25}{100}\), which simplifies to \( \Large \frac{1}{4}\).
No! A fraction like \( \Large \frac{2}{4}\) is still rational, even if it’s not simplified to \( \Large \frac{1}{2}\). Simplifying just makes it easier to read.
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