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:

$${\displaystyle b^{E_{\text{min}}}}$$

where b is the base (radix) of the format (like common values 2 or 10, for binary and decimal number systems), and ${\textstyle E_{\text{min}}}$depends on the size and layout of the format.

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

$${\displaystyle b^{E_{\text{max}}}\cdot \left(b-b^{1-p}\right)}$$

where p is the precision of the format in digits and ${\textstyle E_{\text{min}}}$is related to ${\textstyle E_{\text{max}}}$as:

$${\displaystyle E_{\text{min}}\,{\overset {\Delta }{\equiv }}\,1-E_{\text{max}}=\left(-E_{\text{max}}\right)+1}$$

In the IEEE 754 binary and decimal formats, b, p, ${\textstyle E_{\text{min}}}$, and ${\textstyle E_{\text{max}}}$have the following values:

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.

• Normalized number
• Half-precision floating-point format
• Single-precision floating-point format
• Double-precision floating-point format