--- title: "Recurring Decimals as Exact Fractions" description: "A decimal that repeats forever is still an exact fraction, and multiplying by a power of ten before subtracting is what pins it down. This is the step that makes the claim about rational numbers repea" canonical: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd source: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Recurring Decimals as Exact Fractions A decimal that repeats forever is still an exact fraction, and multiplying by a power of ten before subtracting is what pins it down. This is the step that makes the claim about rational numbers repeating into something a learner can act on rather than believe. Subject: Mathematics · Area: Number Representation & Place Value · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/recurring-decimals-as-exact-fractions-mt_Mb-sXIZDWd ## Ready when they can - Converts a recurring decimal such as 0.4545... to a fraction by subtracting a shifted copy of it - Handles a decimal with a non-recurring part before the repeat, such as 0.16333... - States which fractions give terminating decimals and which recur, from the prime factors of the denominator ## Lesson: Endless decimals, exact fractions Terminating decimals stop, like 0.125. Repeating decimals cycle forever, like 0.1666. Both are rational, since each equals some fraction. Test the denominator in lowest terms: only 2s and 5s means it ends, and any other prime means it repeats. So 3/8 ends at 0.375 while 5/11 cycles. **Example.** Name the repeat x and shift by one full repeat, then subtract so the tail cancels. For 0.777, ten x minus x leaves 7, so 9x equals 7 and x is 7/9. For 0.4545, one hundred x minus x leaves 45, so x is 45/99, which simplifies to 5/11. Stray digits before the repeat need two shifts that line the tails up. For 0.16333, one thousand x minus one hundred x gives 900x equals 147, so x is 147/900, which simplifies to 49/300. Always simplify at the end. **Tip.** Shift by exactly one repeat length so the endless tails match digit for digit. The subtraction then leaves plain whole-number arithmetic. That is also why 0.999 reaches exactly 1. **Recap.** Name it x, shift by the repeat, subtract the tail away, and simplify what remains. ## Practice 15 questions on this page, each with its working shown. ## Needs first - [Number Sets & Infinity](https://lightmysky.com/learn/mathematics/number-sets-and-infinity-mt_b4lbTOJYwI) - [Decimals and fractions (age 11+)](https://lightmysky.com/learn/mathematics/decimals-and-fractions-age-11-mt_SBkTGjiZjZ)