Rational Numbers – Numbers That Can Be Written as Fractions
A rational number is any number that can be written as a fraction p/q, where p and q are both integers and q is not zero. This is a much bigger set than it first sounds: it includes every integer (5 = 5/1), every common fraction (3/4), every terminating decimal (0.75 = 3/4), and every repeating decimal (0.333… = 1/3). The symbol for the rational numbers is ℚ, from the Italian word quoziente, meaning “quotient.”
Fractions are one of the oldest ideas in mathematics. The ancient Egyptians, as recorded in the Rhind Mathematical Papyrus (copied around 1650 BCE from an even older document), used an intricate system of unit fractions – fractions with a numerator of 1, like 1/2, 1/3, 1/7 – and expressed almost every other fraction as a sum of distinct unit fractions. The Babylonians, working around the same era, used a base-60 fraction system, fragments of which survive today in how we divide an hour into 60 minutes and a minute into 60 seconds.
What Makes a Number Rational?
Any number is rational if it can be written exactly as a fraction of two integers. This includes whole numbers, fractions, terminating decimals, and repeating decimals.
Yes – 0.75 = 3/4, a fraction of two integers, so 0.75 is rational.
Converting a Terminating Decimal to a Fraction
0.2 = 2/10, which simplifies (dividing top and bottom by 2) to 1/5.
Real-Life Application
- Cooking: 1/2 cup, 3/4 teaspoon – recipe measurements are rational numbers.
- Money: $0.25 = 1/4 of a dollar.
- Probability: a 1-in-4 chance is written as the rational number 1/4.
Key Takeaways
- A rational number can always be written as p/q, where p and q are integers and q ≠ 0.
- Integers, fractions, terminating decimals, and repeating decimals are all rational.
- The ancient Egyptians and Babylonians both developed sophisticated fraction systems thousands of years ago.
- Not every number is rational – the exceptions are covered next.
Practice: Rational Numbers
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