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Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten. [7] For example, the decimal expressions ,,,, represent the fractions 4 / 5 , 1489 / 100 , 79 / 100000 , + 809 / 500 ...
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [ 9] As with other fractions, the denominator ( b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
In the 3rd century BCE, Archimedes proved the sharp inequalities 223 ⁄ 71 < π < 22 ⁄ 7, by means of regular 96-gons (accuracies of 2·10 −4 and 4·10 −4, respectively). [15] In the 2nd century CE, Ptolemy used the value 377 ⁄ 120, the first known approximation accurate to three decimal places (accuracy 2·10 −5). [16]
5 / 7 = 5 × 0. 142857 = 0. 714285 6 / 7 = 6 × 0. 142857 = 0. 857142; The reason for the cyclic behavior is apparent from an arithmetic exercise of long division of 1 / 7 : the sequential remainders are the cyclic sequence {1, 3, 2, 6, 4, 5}. See also the article 142,857 for more properties of this cyclic number.
The decimal representation represents the infinite sum : Every nonnegative real number has at least one such representation; it has two such representations (with if ) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real ...
This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 85 5 is only slightly bigger than 2 32. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
142857 × 7 4 = 342999657 342 + 999657 = 999999. If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number. [citation needed] 857 2 = 734449 142 2 = 20164 734449 − 20164 = 714285. It is the repeating part in the decimal expansion of the rational number 1 / 7 ...
[4] [5] In many contexts, when a number is spoken, the function of the separator is assumed by the spoken name of the symbol: comma or point in most cases. [6] [2] [7] In some specialized contexts, the word decimal is instead used for this purpose (such as in International Civil Aviation Organization-regulated air traffic control communications).