Normal number (computing)

{{Short description|Number type in floating-point arithmetic}} {{for|the mathematical meaning|normal number}} {{refimprove|date=December 2009}}

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:

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>{{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 |- |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<sup>−95</sup> through 9.999999 × 10<sup>96</sup>.

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.

== References == <references />

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