Normal number (computing)

For the mathematical meaning, see normal number.

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:

$${\displaystyle b^{E_{\text{min}}}}$$

where b is the base (radix) of the format (like common values 2 or 10, for binary and decimal number systems), and ${\textstyle E_{\text{min}}}$ depends on the size and layout of the format.

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

$${\displaystyle b^{E_{\text{max}}}\cdot \left(b-b^{1-p}\right)}$$

where p is the precision of the format in digits and ${\textstyle E_{\text{min}}}$ is related to ${\textstyle E_{\text{max}}}$ as:

$${\displaystyle E_{\text{min}}\,{\overset {\Delta }{\equiv }}\,1-E_{\text{max}}=\left(-E_{\text{max}}\right)+1}$$

In the IEEE 754 binary and decimal formats, b, p, ${\textstyle E_{\text{min}}}$, and ${\textstyle E_{\text{max}}}$ have the following values:^[1]

Smallest and Largest Normal Numbers for common numerical Formats

Format	${\displaystyle b}$	${\displaystyle p}$	${\displaystyle E_{\text{min}}}$	${\displaystyle E_{\text{max}}}$	Smallest Normal Number	Largest Normal Number
binary16	2	11	−14	15	${\displaystyle 2^{-14}\equiv 0.00006103515625}$	${\displaystyle 2^{15}\cdot \left(2-2^{1-11}\right)\equiv 65504}$
binary32	2	24	−126	127	${\displaystyle 2^{-126}\equiv {\frac {1}{2^{126}}}}$	${\displaystyle 2^{127}\cdot \left(2-2^{1-24}\right)}$
binary64	2	53	−1022	1023	${\displaystyle 2^{-1022}\equiv {\frac {1}{2^{1022}}}}$	${\displaystyle 2^{1023}\cdot \left(2-2^{1-53}\right)}$
binary128	2	113	−16382	16383	${\displaystyle 2^{-16382}\equiv {\frac {1}{2^{16382}}}}$	${\displaystyle 2^{16383}\cdot \left(2-2^{1-113}\right)}$
decimal32	10	7	−95	96	${\displaystyle 10^{-95}\equiv {\frac {1}{10^{95}}}}$	${\displaystyle 10^{96}\cdot \left(10-10^{1-7}\right)\equiv 9.999999\cdot 10^{96}}$
decimal64	10	16	−383	384	${\displaystyle 10^{-383}\equiv {\frac {1}{10^{383}}}}$	${\displaystyle 10^{384}\cdot \left(10-10^{1-16}\right)}$
decimal128	10	34	−6143	6144	${\displaystyle 10^{-6143}\equiv {\frac {1}{10^{6143}}}}$	${\displaystyle 10^{6144}\cdot \left(10-10^{1-34}\right)}$

For example, in the smallest decimal format in the table (decimal32), 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 subnormal numbers (or denormal numbers).

Zero is considered neither normal nor subnormal.

See also

• Normalized number
• Half-precision floating-point format
• Single-precision floating-point format
• Double-precision floating-point format

References

1. IEEE Standard for Floating-Point Arithmetic, 2008-08-29, doi:10.1109/IEEESTD.2008.4610935, ISBN 978-0-7381-5752-8, retrieved 2015-04-26