What Is 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).

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 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.

Let's convert the repeating decimal 0.777... into a fraction to prove it is a rational number.

1. Let x = 0.777...
2. Multiply both sides by 10: 10x = 7.777...
3. Subtract the first equation from the second: 10x − x = (7.777...) − (0.777...)
4. Simplify: 9x = 7
5. 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.