Lesson 13-2: Represent Rational Numbers in Decimal Form

Property

To convert a terminating decimal to a fraction, write the digits of the decimal in the numerator. The denominator is a power of 10 with the same number of zeros as there are decimal places. Then, simplify the fraction to its lowest terms.

Examples

• Convert 0.5 to a fraction:

• Convert 0.75 to a fraction:

• Convert 0.125 to a fraction:

Explanation

To change a decimal into a fraction, use its place value to determine the denominator. The number of digits after the decimal point tells you the number of zeros in the denominator (e.g., two places means a denominator of 100). The digits themselves form the numerator. Always remember to simplify the resulting fraction.

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

The repeating decimal $0.\overline{9}$ (or $0.999...$) is mathematically equal to exactly 1.

Examples

• Example 1: Algebraic Proof

Let $x = 0.999...$
Multiply both sides by 10: $10x = 9.999...$
Subtract the original equation: