floatingpoint notation
floatingpoint notation A representation of real numbers that enables both very small and very large numbers to be conveniently expressed. A floatingpoint number has the general form ±m × R^{e} where m is called the mantissa, R is the radix (or base) of the number system, and e is the exponent.
In the context of computing, a more common name for the mantissa is the significand.
IEEE Standard 754 defines the most commonly used representations for real numbers on computers. It defines 32bit (single precision) and 64bit (double precision) as follows.
The first bit is a sign bit, denoting the sign of the significand. This is followed by a fixed number of bits representing the exponent, which is in turn followed by another fixed number of bits representing the magnitude of the significand.
The exponent is often represented using excessn notation. This means that a number, called the characteristic (or biased exponent), is stored instead of the exponent itself. To derive the characteristic for a floatingpoint number from its exponent, the bias (or excess factor) n is added to the exponent. For example, for an 8bit characteristic, exponents in the range −128 to +127 are represented in excess128 notation by characteristics in the range 0 to 255.
IEEE 754 specifies an 8bit singleprecision exponent, with a bias of 127, and an 11bit doubleprecision exponent, with a bias of 1023. A nonzero floatingpoint number is normalized if the leading digit in its significand is nonzero. Since the only possible nonzero digit in base 2 is 1, the leading nonzero digit in the significand need not be explicitly represented. This means that the 23bit significand in the IEEE 754 singleprecision floatingpoint representation effectively provides 24 bits of resolution, and the 52bit doubleprecision significand provides 53 bits of resolution.
Although normalized floatingpoint numbers are most frequently used, unnormalized representations are also needed to represent numbers close to zero.
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