Normal number (computing): Difference between revisions

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:''A different topic is treated in {{for|the article titledmathematical [[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.

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 ''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, theThe magnitude of the largest'''smallest normal number''' in a format is given by:

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

:''b''^''emax'' &times; (''b'' &minus; ''b''^{1&minus;''p''}),

where ''pb'' is the precisionbase (radix) of the format in(like [[numericalcommon digit|digit]]svalues 2 or 10, for binary and ''emax''decimal isnumber systems), and (&minus;''emin<math display="inline">E_{\text{min}}</math>'')+1 depends on the size and layout of the format.

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

In the [[IEEE 754]] binary and proposed decimal formats, ''p'', ''emin'', and ''emax'' have the following values:

{| class="wikitable"

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

!Format!!p!!emin!!emax

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 proposed decimal formats, ''pb'', ''eminp'', <math display="inline">E_{\text{min}}</math>, and ''emax<math display="inline">E_{\text{max}}</math>'' have the following values:<ref>{{Citation

| title = IEEE Standard for Floating-Point Arithmetic

| date = 2008-08-29

| doi =10.1109/IEEESTD.2008.4610935

| isbn = 978-0-7381-5752-8

}}</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

|binary 32-bit||24||−126||127

|<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

|binary 64-bit||53||−1022||1023

|<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

|binary 128-bit||113||−16382||16383

|<math>2^{-1022} \equiv \frac{1}{2^{1022}}</math>

|<math>2^{1023}\cdot\left(2 - 2^{1-53}\right)</math>

|-

|binary 128-bitbinary128||2||113||−16382||16383

|decimal 32-bit||7||−95||96

|<math>2^{-16382} \equiv \frac{1}{2^{16382}}</math>

|<math>2^{16383}\cdot\left(2 - 2^{1-113}\right)</math>

|-

|decimal 32-bitdecimal32||10||7||−95||96

|decimal 64-bit||16||−383||384

|<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>

|-

|decimal 64-bitdecimal64||10||16||−383||384

|decimal 128-bit||34||−6143||6144

|<math>10^{-383} \equiv \frac{1}{10^{383}}

</math>

|<math>10^{384}\cdot\left(10 - 10^{1-16}\right)</math>

|-

|decimal 128-bitdecimal128||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^&minus;95 through 9.999999 &times;× 10⁹⁶.

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

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)}}

Non-zero numbers smaller in magnitude than the smallest normal number are called [[denormal|denormalized numbers]] or [[denormal|subnormal numbers]]. Zero is neither normal nor subnormal.

[[Category:Computer arithmetic]]