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Write the period and length of period in the decimal expansion of 1/7 |
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Answer» ong>ANSWER It is what’s known as a repeating decimal. There are an infinite number of repeating decimals. For instance, 1/3 expressed as a decimal equals 0.33333333··· and so on forever. What’s fascinating about the decimal expansion of 1/7 is that it consists of six UNIQUE digits: 1, 4, 2, 8, 5, and 7, repeated in that ORDER AD infinitum. But here’s the fun part: The same six digits, in the same order, but with a different STARTING digit, also represent the fractions 2/7, 3/7, 4/7, 5/7, and 6/7. In other words… 1/7 = .142857142857142857··· 2/7 = .285714285714285714··· 3/7 = .428571428571428571··· 4/7 = .571428571428571428··· 5/7 = .714285714285714285··· 6/7 = .857142857142857142··· In addition, if you look at just the FIRST two digits of each decimal: .14, .28, etc., you’ll see that they approximate 14 (7x2), 28 (7x4), 42 (7x6), 56 (7x8), 70 (7x10), and 84 (7x12). That’s because 7 x 14 = 98, which is very close to a power of ten (100, or 10 squared). All of this together leads to one of the more fascinating numerical coincidences out there. |
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