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Moreover, in the standard decimal representation of , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if is an integer. Certain procedures for constructing the decimal expansion of x {\displaystyle x} will avoid the problem of trailing 9's.
Example: the decimal number () = (¯) can be rearranged into + ⏟ … Since the 53rd bit to the right of the binary point is a 1 and is followed by other nonzero bits, the round-to-nearest rule requires rounding up, that is, add 1 bit to the 52nd bit.
For example, in duodecimal, 1 / 2 = 0.6, 1 / 3 = 0.4, 1 / 4 = 0.3 and 1 / 6 = 0.2 all terminate; 1 / 5 = 0. 2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1 / 7 = 0. 186A35 has period 6 in duodecimal, just as it does in decimal. If b is an integer base ...
The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number.
A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer as input and produces the -th digit of the real number's decimal expansion as output. (The decimal expansion of a only refers to the digits following the decimal point.)
The value of n is then the period of the decimal expansion of 1/p. [10] At present, more than fifty decimal unique primes or probable primes are known. However, there are only twenty-three unique primes below 10 100. The decimal unique primes are 3, 11, 37, 101, 9091, 9901, 333667, 909091, ... (sequence A040017 in the OEIS).
In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number. Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction.