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What Is a Rational Number?
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. The word "rational" stems from "ratio," indicating that these numbers represent a simple relationship between two whole numbers.
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Definition of a Rational Number
Mathematically, a number is rational if it can be written in the form:
Where:
- p is an integer (positive, negative, or zero).
- q is a non-zero integer (since division by zero is undefined).
This definition means that all integers (like −5, 0, and 42) are rational numbers, because they can be written as fractions with a denominator of 1 (e.g., −5/1, 0/1, 42/1).
Rational vs. Irrational Numbers
Real numbers are divided into two distinct groups: rational and irrational. The difference lies in their decimal representations:
- Rational numbers can always be written as fractions. When written as decimals, they either terminate (end) or repeat in a predictable pattern.
- Irrational numbers cannot be written as fractions. When written as decimals, they go on forever without ever terminating or repeating in a pattern.
| Rational Examples | Fraction Form | Irrational Examples | Decimal Value |
|---|---|---|---|
| Integer (5) | 5 ÷ 1 | Pi (π) | 3.14159265... |
| Fraction (¾) | 3 ÷ 4 | Square Root of 2 (√2) | 1.41421356... |
| Terminating Decimal (0.125) | 1 ÷ 8 | Golden Ratio (φ) | 1.61803398... |
| Repeating Decimal (0.333...) | 1 ÷ 3 | Euler's Number (e) | 2.71828182... |
Key Properties of Rational Numbers
Rational numbers exhibit several important mathematical behaviors under operations:
- Closure Property: The sum, difference, product, and quotient (with a non-zero divisor) of any two rational numbers will always result in another rational number.
- Density Property: Between any two rational numbers, there are infinitely many other rational numbers. For example, between 1.1 and 1.2, you can find 1.11, 1.111, 1.1111, and so on.
- Convertibility: Any repeating or terminating decimal can be systematically converted back into a fraction.
Worked Example: Converting Repeating Decimal to a Fraction
Let's convert the repeating decimal 0.777... into a fraction to prove it is a rational number.
- Let x = 0.777...
- Multiply both sides by 10: 10x = 7.777...
- Subtract the first equation from the second: 10x − x = (7.777...) − (0.777...)
- Simplify: 9x = 7
- Divide by 9: x = 7 ÷ 9
Because we successfully wrote 0.777... as the fraction 7/9, it is proven to be a rational number.
Fraction Conversions
Need to convert decimals or simplify quotients? Use our free converters:
Frequently asked questions
Yes, 0 is a rational number. It can be written as a fraction where the numerator is 0 and the denominator is any non-zero integer, such as 0 ÷ 1, 0 ÷ 5, or 0 ÷ −10. Since it satisfies the p/q definition, 0 is rational.
Pi represents the ratio of a circle's circumference to its diameter, but its decimal representation (3.14159...) goes on forever without repeating or terminating. Because it cannot be written as a simple fraction of two whole integers, it is defined as an irrational number.
No. The square roots of perfect squares are rational numbers. For example, √4 = 2 (which is rational because 2 = 2/1), and √9 = 3. However, the square root of any positive integer that is not a perfect square (like √2, √3, or √5) is irrational.
A terminating decimal has a finite number of digits after the decimal point (e.g., 0.25 = ¼). A repeating decimal has a digit or a block of digits that repeats infinitely (e.g., 0.333... = ⅓). Both types are rational because they can be written as fractions of two integers.