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:
b E min {\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 E min {\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
b E max ⋅ ( b − b 1 − p ) {\displaystyle b^{E_{\text{max}}}\cdot \left(b-b^{1-p}\right)}
where p is the precision of the format in digits and E min {\textstyle E_{\text{min}}} is related to E max {\textstyle E_{\text{max}}} as:
E min ≡ Δ 1 − E max = ( − E max ) + 1 {\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, E min {\textstyle E_{\text{min}}} , and E max {\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.
See also Normalized number Half-precision floating-point format Single-precision floating-point format Double-precision floating-point format
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