Terminating and recurring decimal conversions · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| recurring decimal/rɪˈkɜːrɪŋ ˈdesɪml/ | 循环小数 | xún huán xiǎo shù |
Does the display tell the whole number?
- A calculator shows 0.333333. Is the display the exact value of one third, or only the digits that fit on the screen?
- This lesson studies recurring decimal 循环小数: A decimal with a digit or block of digits that repeats forever.
Choose the mathematical structure
- A finite decimal uses a power-of-ten denominator before simplifying. For a recurring decimal, multiply by powers of ten so the repeated tails align, then subtract. Use matching decimal places to compare numbers. A displayed rounded decimal need not be the exact fraction.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines recurring decimal?
A decimal with a digit or block of digits that repeats forever.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
0.375=375/1000=3/8. For x=0.272727..., 100x=27.272727..., so 99x=27 and x=27/99=3/11. For y=0.16666..., 100y-10y=16.666...-1.666...=15, so y=15/90=1/6. In a reduced fraction, a denominator containing only factors 2 and 5 gives a terminating decimal.
Terminating and recurring decimal conversions
A finite decimal uses a power-of-ten denominator before simplifying
Compare the model with the worked case and explain one change.
Find the numerator when 0.375 is expressed over 8.
375/1000 reduces to 3/8.
Test a tempting shortcut
- Align the recurring tails before subtracting. 0.333333 is finite and differs from 0.333333... . A non-recurring prefix needs a second power of ten; blindly dividing every digit block by 99 fails.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The finite decimal 0.333333 is exactly one third. This claim is false. Explain which definition or assumption it violates.
Find the denominator of 0.272727... in lowest terms.
100x-x=27 gives 27/99=3/11.
The finite decimal 0.333333 is exactly one third.
Align the recurring tails before subtracting. 0.333333 is finite and differs from 0.333333... . A non-recurring prefix needs a second power of ten; blindly dividing every digit block by 99 fails.
Interpret a new situation
- AQA N10 Higher includes recurring conversion. Both tiers convert and order terminating decimals and fractions. Check a conversion by long division; 3 divided by 8 gives 0.375.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the denominator of 0.16666... in lowest terms.
100y-10y=15 gives 15/90=1/6.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A decimal with a digit or block of digits that repeats forever. Choose the relationship, show the method, check its assumptions and interpret the result.