Rational Numbers Explained: Properties and Examples

Rational numbers are numbers that can be written as a fraction p ÷ q, where p and q are integers and q is not zero. This single definition covers a huge family: 3, −5, 0, 2⁄3, −7⁄4 and 1.5 are all rational, because each can be written with an integer numerator and a non-zero integer denominator.

Rational numbers matter because measurements in real life are rarely whole. Half a litre of milk, three-quarters of an hour, a discount of 15 percent — all of these are rational numbers, and their arithmetic is the arithmetic of everyday life.

Key Concepts

• Definition: a rational number is any number of the form p/q with p, q integers and q ≠ 0.
• Standard form: the fraction is reduced so that numerator and denominator have no common factor other than 1, and the denominator is positive. For example 4⁄6 reduces to 2⁄3.
• Decimal form: every rational number has a terminating or repeating decimal expansion — 1⁄4 = 0.25, 1⁄3 = 0.333….
• Reciprocal: the reciprocal of a/b is b/a (for a ≠ 0); their product is 1.
• Density property: between any two rational numbers there is always another rational number, for instance their average.
• Closure: rationals are closed under addition, subtraction, multiplication and division (except division by zero).

Formulas

• Addition: a/b + c/d = (ad + bc) ÷ bd.
• Subtraction: a/b − c/d = (ad − bc) ÷ bd.
• Multiplication: (a/b) × (c/d) = ac ÷ bd.
• Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad ÷ bc, provided c ≠ 0.

Worked Example

Compute 2⁄3 + 5⁄6. The common denominator is 6: 2⁄3 = 4⁄6, so 4⁄6 + 5⁄6 = 9⁄6 = 3⁄2 = 1.5. To see the density property, find a rational between 1⁄2 and 2⁄3 by taking their average: (1⁄2 + 2⁄3) ÷ 2 = (7⁄6) ÷ 2 = 7⁄12. Check the order with decimals: 1⁄2 = 0.5, 7⁄12 ≈ 0.583, 2⁄3 ≈ 0.667, and indeed 0.5 < 0.583 < 0.667.

Common Misconceptions

• "Every fraction is rational." Only fractions with integer parts are rational. The number π⁄2 looks like a fraction but is not rational, because π is not an integer.
• "0 cannot be a numerator." It can: 0⁄5 = 0 is a rational number. What is forbidden is a zero denominator.
• "Repeating decimals are not exact numbers." A repeating decimal like 0.666… is exactly equal to 2⁄3 — the repetition is just how the fraction looks in decimal form.

Real-World Applications

• Cooking uses fractions of cups and spoons: half a cup is 1⁄2, a common rational number.
• Shops compute discounts and taxes as rational percentages of prices.
• Engineers read measurements like 3⁄8 of an inch on rulers and calipers.

To sum up, rational numbers are exactly the fractions of integers, closed under all four operations, with decimals that always terminate or repeat. Practise the four operation formulas, and use the averaging trick whenever you need a rational number between two given ones.