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These languages allow the programmer to specify an implicit decimal point in front of one of the digits. For example, a packed decimal value encoded with the bytes 12 34 56 7C represents the fixed-point value +1,234.567 when the implied decimal point is located between the fourth and fifth digits: 12 34 56 7C 12 34.56 7+
Similar binary floating-point formats can be defined for computers. There is a number of such schemes, the most popular has been defined by Institute of Electrical and Electronics Engineers (IEEE). The IEEE 754-2008 standard specification defines a 64 bit floating-point format with: an 11-bit binary exponent, using "excess-1023" format.
The IEEE 754-2008 standard includes decimal floating-point number formats in which the significand and the exponent (and the payloads of NaNs) can be encoded in two ways, referred to as binary encoding and decimal encoding.
The resulting 'raw' exponent is a 8 bit binary integer where the leading bits are not '11', thus values 0 ... 1011 1111 b = 0 ... 191 d, appr. bias is to be subtracted. The significand's leading decimal digit forms from the (0)cde or 100e bits as binary integer. The subsequent digits are encoded in the 10 bit 'declet' fields 'tttttttttt ...
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.
Binary-coded decimal (BCD) is a binary encoded representation of integer values that uses a 4-bit nibble to encode decimal digits. Four binary bits can encode up to 16 distinct values; but, in BCD-encoded numbers, only ten values in each nibble are legal, and encode the decimal digits zero, through nine.
This is called a binary numeral system. However, the x86 processors do have limited support for the decimal numeral system. In addition, the x87 part supports a unique 18-digit (ten-byte) BCD format that can be loaded into and stored from the floating point registers, from where ordinary FP computations can be performed. [1]
It is for this reason that many are surprised to discover that 1/10 + ... + 1/10 (addition of 10 numbers) differs from 1 in binary floating point 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 ...