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:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ b^{E_{\text{max}}}\cdot\left(b - b^{1-p}\right) }[/math]

where p is the precision of the format in digits and [math]\displaystyle{ E_{\text{min}} }[/math] is related to [math]\displaystyle{ E_{\text{max}} }[/math] as:

[math]\displaystyle{ 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]\displaystyle{ E_{\text{min}} }[/math], and [math]\displaystyle{ E_{\text{max}} }[/math] have the following values:

Smallest and Largest Normal Numbers for common numerical Formats

Format	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ p }[/math]	[math]\displaystyle{ E_{\text{min}} }[/math]	[math]\displaystyle{ E_{\text{max}} }[/math]	Smallest Normal Number	Largest Normal Number
binary16	2	11	−14	15	[math]\displaystyle{ 2^{-14} \equiv 0.00006103515625 }[/math]	[math]\displaystyle{ 2^{15}\cdot\left(2 - 2^{1-11}\right) \equiv 65504 }[/math]
binary32	2	24	−126	127	[math]\displaystyle{ 2^{-126} \equiv \frac{1}{2^{126}} }[/math]	[math]\displaystyle{ 2^{127}\cdot\left(2 - 2^{1-24}\right) }[/math]
binary64	2	53	−1022	1023	[math]\displaystyle{ 2^{-1022} \equiv \frac{1}{2^{1022}} }[/math]	[math]\displaystyle{ 2^{1023}\cdot\left(2 - 2^{1-53}\right) }[/math]
binary128	2	113	−16382	16383	[math]\displaystyle{ 2^{-16382} \equiv \frac{1}{2^{16382}} }[/math]	[math]\displaystyle{ 2^{16383}\cdot\left(2 - 2^{1-113}\right) }[/math]
decimal32	10	7	−95	96	[math]\displaystyle{ 10^{-95} \equiv \frac{1}{10^{95}} }[/math]	[math]\displaystyle{ 10^{96}\cdot\left(10 - 10^{1-7}\right) \equiv 9.999999 \cdot 10^{96} }[/math]
decimal64	10	16	−383	384	[math]\displaystyle{ 10^{-383} \equiv \frac{1}{10^{383}} }[/math]	[math]\displaystyle{ 10^{384}\cdot\left(10 - 10^{1-16}\right) }[/math]
decimal128	10	34	−6143	6144	[math]\displaystyle{ 10^{-6143} \equiv \frac{1}{10^{6143}} }[/math]	[math]\displaystyle{ 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 10⁻⁹⁵ 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