Normal number

In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b² pairs of digits are equally likely with density b⁻², all b³ triplets of digits equally likely with density b⁻³, etc.

Intuitively this means that no digit, or (finite) combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored".

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^emin, where b is the base (radix) of the format (usually 2 or 10) and emin depends on the size and layout of the format.

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

where p is the precision of the format in digits and emax is (−emin)+1.

In the IEEE 754 binary and decimal formats, p, emin, and emax have the following values:

For example, in the smallest decimal format, 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 denormal (or subnormal) numbers. Zero is neither normal nor subnormal.

See also