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:

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

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

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:

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

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