Normal number (computing)

In computing, a normal number is a non-zero number in a floating-point representation which is within the balanced range supported by a given floating-point format: it is a floating point number that can be represented without leading zeros in its significand.

The magnitude of the smallest normal number in a format is given by b^emin, where b is the base (radix) of the format (usually 2 or 10) and emin depends on the size and layout of the format.

Similarly, the magnitude of the largest normal number in a format is given by

b^emax × (b − b^1−p),

where p is the precision of the format in digits and emax is (−emin)+1.

In the IEEE 754 binary and decimal formats, b, p, emin, and emax have the following values:^[1]

Format	b	p	emin	emax
binary16	2	11	−14	15
binary32	2	24	−126	127
binary64	2	53	−1022	1023
binary128	2	113	−16382	16383
decimal32	10	7	−95	96
decimal64	10	16	−383	384
decimal128	10	34	−6143	6144

For example, in the smallest decimal format, the range of positive normal numbers is 10⁻⁹⁵ through 9.999999 × 10⁹⁶.

Non-zero numbers smaller in magnitude than the smallest normal number are called denormal (or subnormal) numbers. Zero is neither normal nor subnormal.

See also

• Normalized number

References

1. IEEE Standard for Floating-Point Arithmetic, 2008-08-29, doi:10.1109/IEEESTD.2008.4610935, retrieved 2015-04-26