Recurring decimal - junkanshosou (English spelling)

Japanese: 循環小数 - じゅんかんしょうすう（英語表記）recurring decimal

In decimals such as 0.33..., 0.142857142857..., 0.3212121..., when a sequence of digits (3, 142857, 21 in the above example) continues infinitely while repeating, these decimals are called repeating decimals, and the repeated sequence of digits is called a repeating section. A repeating decimal that has no non-repeating parts is called a pure repeating decimal, and one that has both non-repeating parts and repeating parts is called a mixed repeating decimal. Repeating decimals can be divided into the following parts using repeating sections:

It can be expressed as follows. When the denominator of an irreducible fraction has only the prime factors 2 and 5, it can be expressed as a finite (decimal) decimal, but if it has even one prime factor other than 2 or 5, it becomes an infinite decimal. When it does not have the factors 2 or 5, it can be expressed as a pure repeating decimal, and when it contains at least one of the factors 2 and 5, it can be expressed as a mixed repeating decimal. Conversely, repeating decimals can always be expressed as a fraction. Non-repeating infinite decimals are irrational numbers, and conversely, irrational numbers can be expressed as non-repeating infinite decimals.

Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information

Japanese:

小数 0.33…，0.142857142857…，0.3212121…などにおいて，ある数字の列 (上の例では3，142857，21) が，繰返しながら無限に続く場合に，これらの小数を循環小数といい，繰返される数字の列を循環節という。循環しない部分のないものを，特に純循環小数，循環しない部分と循環する部分をともにもつものを混循環小数という。循環小数は，循環節を用いて，

のように表わされる。既約分数の分母が2および5の素因数だけのときは，(10進の) 有限小数で表わされ，2と5以外の素因数を1つでももてば，無限小数になる。また2と5を因数にもたないときは，純循環小数で表わされ，因数として2と5の少くとも1つを含む場合は，混循環小数で表わされる。逆にまた，循環小数は必ず分数で表わすことができる。循環しない無限小数は無理数であり，逆に無理数は循環しない無限小数で表わされる。

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