Editing Normal number (computing)

{{for|the mathematical meaning|normal number}}
{{refimprove|date=December 2009}}

{{Floating-point}}

In [[computing]], a '''normal number''' is a non-zero number in a [[floating point|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 [[numerical digit|digit]]s 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>{{Citation
 | title = IEEE Standard for Floating-Point Arithmetic
 | date = 2008-08-29
 | url = https://ieeexplore.ieee.org/document/4610935
 | doi =10.1109/IEEESTD.2008.4610935
 | access-date = 2015-04-26| isbn = 978-0-7381-5752-8
 | url-access = subscription
 }}</ref>
{| class="wikitable" style="text-align: right;" |
|+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
|-
|[[Half-precision floating-point format|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>
|-
|[[Single-precision floating-point format|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>
|-
|[[Double-precision floating-point format|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<sup>&minus;95</sup> through 9.999999 × 10<sup>96</sup>.

Non-zero numbers smaller in magnitude than the smallest normal number are called [[subnormal number|'''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 ==
<references />


{{DEFAULTSORT:Normal Number (Computing)}}
[[Category:Computer arithmetic]]