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The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
Numbers are represented in binary as IEEE 754 floating point doubles. Although this format provides an accuracy of nearly 16 significant digits, it cannot always exactly represent real numbers, including fractions. This becomes an issue when comparing or formatting numbers. For example:
To approximate the greater range and precision of real numbers, we have to abandon signed integers and fixed-point numbers and go to a "floating-point" format. In the decimal system, we are familiar with floating-point numbers of the form (scientific notation): 1.1030402 × 10 5 = 1.1030402 × 100000 = 110304.02. or, more compactly: 1.1030402E5
Like the binary floating-point formats, the number is divided into a sign, an exponent, and a significand. Unlike binary floating-point, numbers are not necessarily normalized; values with few significant digits have multiple possible representations: 1×10 2 =0.1×10 3 =0.01×10 4, etc. When the significand is zero, the exponent can be any ...
The binary encoding, based on binary integer decimal (BID): The significand is encoded as an unsigned integer written in binary. The decimal encoding, based on densely packed decimal (DPD): The significand is encoded as an unsigned integer written in decimal, with groups of 3 digits packed together in a declet and a special rule for the most ...
For other binary formats, the required number of decimal digits is [h] + ⌈ ⌉, where p is the number of significant bits in the binary format, e.g. 237 bits for binary256. When using a decimal floating-point format, the decimal representation will be preserved using:
The full decimal significand is then obtained by concatenating the leading and trailing decimal digits. The 10-bit DPD to 3-digit BCD transcoding for the declets is given by the following table. b 9 … b 0 are the bits of the DPD, and d 2 … d 0 are the three BCD digits.
A decimal data type could be implemented as either a floating-point number or as a fixed-point number. In the fixed-point case, the denominator would be set to a fixed power of ten. In the floating-point case, a variable exponent would represent the power of ten to which the mantissa of the number is multiplied.