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:
{| 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.
See also
- Normalized number
- Half-precision floating-point format
- Single-precision floating-point format
- Double-precision floating-point format