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:

<math display="block">b^{E_{\text{min}}}</math>

where b is the base (radix) of the format (like common values 2 or 10, for binary and decimal number systems), and <math display="inline">E_{\text{min}}</math> depends on the size and layout of the format.

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

<math display="block">b^{E_{\text{max}}}\cdot\left(b - b^{1-p}\right)</math>

where p is the precision of the format in digits and <math display="inline">E_{\text{min}}</math> is related to <math display="inline">E_{\text{max}}</math> as:

<math display="block">E_{\text{min}}\, \overset{\Delta}{\equiv}\, 1 - E_{\text{max}} = \left(-E_{\text{max}}\right) + 1</math>

In the IEEE 754 binary and decimal formats, b, p, <math display="inline">E_{\text{min}}</math>, and <math display="inline">E_{\text{max}}</math> have the following values:<ref>Template:Citation</ref>

Smallest and Largest Normal Numbers for common numerical Formats

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

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

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