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Every terminating decimal representation can be written as a decimal fraction, a fraction whose denominator is a power of 10 (e.g. 1.585 = 1585 / 1000 ); it may also be written as a ratio of the form k / 2 n ·5 m (e.g. 1.585 = 317 / 2 3 ·5 2 ).
This form of fraction remained in use for centuries. [27] [30] Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century. [31] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them. [32]
If the integers can be represented by the Kleene plus +, then the set of all signed-digit representations of the decimal fractions, or -adic rationals [], is given by = +, the Cartesian product of the Kleene plus +, the set of all finite concatenated strings of digits … with at least one digit, the singleton consisting of the radix point ...
The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10 −2 power term, also called characteristics, [11] [12] [13] where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic:
In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not. The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then ...
Expanding this form to the six-digit form is as simple as repeating each digit: ... Decimal Fraction 0 00 0 0 3 33 51 0.2 6 66 102 0.4 9 99 153 0.6 C (12) CC 204
A sucrose solution with an apparent specific gravity (20°/20 °C) of 1.040 would be 9.99325 °Bx or 9.99359 °P while the representative sugar body, the International Commission for Uniform Methods of Sugar Analysis (ICUMSA), which favours the use of mass fraction, would report the solution strength as 9.99249%. Because the differences between ...