write the period and length of period in the decimal expansion of 1/7
Answers
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.
Answer:
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