Observations on Reciprocals of Leftins

  1. All rationals have reciprocals in all bases.
  2. Rationals have one uniquely determined reciprocal to the same number of significant digits as the original number.
  3. If an integer shares factors with the number base, its reciprocal will require digits to the right of the decimal point.
  4. The number of digits needed to the right of the decimal point for the reciprocal of an integer...
    1. If the number base is prime, it's equal to the number of 0's the number ends with.
    2. If the number base is a prime raised to a power greater than 1, it's equal to the number of 0's the number ends with plus an extra digit if the last non-0 digit is a power of the prime.
    3. If the number base is the product of at least two different prime numbers, it's unrelated to the final digits.
Table of Reciprocals
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© Scott Sherman 1999