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:

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

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

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

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

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

$$E_{\text{min}}\stackrel{\mathrm{\Delta }}{\equiv }1-E_{\text{max}}=(-E_{\text{max}})+1$$

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

Smallest and Largest Normal Numbers for common numerical Formats

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

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