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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Name it x, shift by the repeat, subtract the tail away, and simplify what remains.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.