Lesson 1: Rational Numbers as Decimals

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:

Property

To convert a decimal with a non-repeating part, create two equations by multiplying the original decimal, $x$, by two different powers of 10. The goal is to create two new equations where the repeating decimal parts are perfectly aligned. Subtracting the smaller equation from the larger one will then eliminate the repeating tail, allowing you to solve for $x$.

Examples