Lesson 2-1: Terminating and Repeating Decimals

Property

A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. All fractions, both positive and negative, are rational numbers.
Since any integer, terminating decimal, or repeating decimal can be written as a ratio of two integers, they are all rational numbers.

Examples

• To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$.
• The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$, which converts to the improper fraction $\frac{937}{100}$.
• The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$.

Explanation

Think of 'rational' as 'ratio-nal.' Any number that can be expressed as a simple fraction or ratio between two integers is a rational number. This includes whole numbers, integers, and decimals that either end or repeat predictably.

Property

A rational number is one that can be expressed as a quotient (or ratio) of two integers, where the denominator is not zero. The decimal representation of a rational number has one of two forms.

1. The decimal representation terminates, or ends.

2. The decimal representation repeats a pattern.

Property

To convert a repeating decimal to a fraction, set the decimal equal to a variable $x$. Multiply the equation by powers of 10 to create two new equations where the infinitely repeating decimal parts are perfectly aligned. Subtracting the smaller equation from the larger one will cancel out the infinite repeating tail, leaving a simple algebraic equation to solve for $x$.

Examples

• Example 1 (Pure Repeating): Convert $0.\overline{5}$ to a fraction.

Let $x = 0.555...$
Multiply by 10 (since 1 digit repeats): $10x = 5.555...$
Subtract the original equation:

• Example 2 (Mixed Repeating): Convert $0.8\overline{3}$ to a fraction.

Let $x = 0.8333...$
Multiply by 10 and 100 to create two equations with aligned repeating parts:

Subtract them: