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| Background: Recurring decimals |
| A recurring decimal is one whose
digits after the decimal point do not end but repeat the same sequence for
ever. These are all recurring decimals: 0.121212... 0.111111... 0.74537435... Recurring decimals are also known as repeating decimals or periodic decimals. The number of digits in the repeating pattern is called the period. So: 0.121212... has a period of 2 0.74537435... has a period of 4 Any repeating decimal can be converted to a fraction. To do this you need to see how the decimal can be written as an infinite geometric series. Here is an example of how you do it. 0.363636...=0.36 + 0.0036 + 0.000036 + ... This infinite sequence is an example of an infinite geometric series. It has a first term, a=0.36=36/100=9/25 and a constant ratio, r=0.01=1/100. In your Advanced Level course, you will meet the following formula for the sum of an infinite geometric series: S=a/(1-r) So, in the above case, 0.363636...=9/25/(1-1/100)=9/25 x 100/99=900/2475=4/11 |
| Exact fractions | ||||
| An exact fraction is
a rational number, especially one written as a quotient of two integers. Here
are some examples: 3/4, 2/3, 100/101, 63/1000 The exact fraction 1/7 can be written in the form of a recurring decimal: 1/7=0.142857142857... which has a period of 6. Similarly 1/6=0.1666... which has a period of 1. The fraction 1/4 can be written as a terminating decimal: 1/4=0.25 All fractions can be written either as recurring decimals or as terminating decimals. All fractions whose denominator only have prime factors of 2 and 5 can be written as terminating decimals. All other fractions can be written as recurring decimals only:
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