Normal number (computing)

{{for|the mathematical meaning|normal number}}

{{refimprove|date=December 2009}}

{{Floating-point}}

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:

$$b^\{E\_\{\backslash text\{min\}\}\}$$

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

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

:$$b^\{E\_\{\backslash text\{max\}\}\}\backslash cdot\backslash left(b\; -\; b^\{1-p\}\backslash right)$$

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

$$E\_\{\backslash text\{min\}\}\backslash ,\; \backslash overset\{\backslash Delta\}\{\backslash equiv\}\backslash ,\; 1\; -\; E\_\{\backslash text\{max\}\}\; =\; \backslash left(-E\_\{\backslash text\{max\}\}\backslash right)\; +\; 1$$

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

| title = IEEE Standard for Floating-Point Arithmetic

| date = 2008-08-29

| url = https://ieeexplore.ieee.org/document/4610935

| doi =10.1109/IEEESTD.2008.4610935

| access-date = 2015-04-26| isbn = 978-0-7381-5752-8

| url-access = subscription

}}

class="wikitable" style="text-align: right;" | |+Smallest and Largest Normal Numbers for common numerical Formats !Format!!$b$!!$p$!!$E\_\{\backslash text\{min\}\}$!!$E\_\{\backslash text\{max\}\}$ !Smallest Normal Number !Largest Normal Number
binary16	2	11	−14	15 |$2^\{-14\}\; \backslash equiv\; 0.00006103515625$ |$2^\{15\}\backslash cdot\backslash left(2\; -\; 2^\{1-11\}\backslash right)\; \backslash equiv\; 65504$
binary32	2	24	−126	127 |$2^\{-126\}\; \backslash equiv\; \backslash frac\{1\}\{2^\{126\}\}$ |$2^\{127\}\backslash cdot\backslash left(2\; -\; 2^\{1-24\}\backslash right)$
binary64	2	53	−1022	1023 |$2^\{-1022\}\; \backslash equiv\; \backslash frac\{1\}\{2^\{1022\}\}$ |$2^\{1023\}\backslash cdot\backslash left(2\; -\; 2^\{1-53\}\backslash right)$
binary128	2	113	−16382	16383 |$2^\{-16382\}\; \backslash equiv\; \backslash frac\{1\}\{2^\{16382\}\}$ |$2^\{16383\}\backslash cdot\backslash left(2\; -\; 2^\{1-113\}\backslash right)$
decimal32	10	7	−95	96 |$10^\{-95\}\; \backslash equiv\; \backslash frac\{1\}\{10^\{95\}\}$ |$10^\{96\}\backslash cdot\backslash left(10\; -\; 10^\{1-7\}\backslash right)\; \backslash equiv\; 9.999999\; \backslash cdot\; 10^\{96\}$
decimal64	10	16	−383	384 |$10^\{-383\}\; \backslash equiv\; \backslash frac\{1\}\{10^\{383\}\}$ |$10^\{384\}\backslash cdot\backslash left(10\; -\; 10^\{1-16\}\backslash right)$
decimal128	10	34	−6143	6144 |$10^\{-6143\}\; \backslash equiv\; \backslash frac\{1\}\{10^\{6143\}\}$ |$10^\{6144\}\backslash cdot\backslash left(10\; -\; 10^\{1-34\}\backslash right)$

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.