Arithmetic coding actually refers to half of an Arithmetic Coding data compression system.

It has two parts:

- An arithmetic coder
- A data model (e.g., Markovian model)

- There is a new data character (a fixed number of bits per character)
- There is a set of probabilities for each possible character

The larger the range, the less bits it takes to code the character. The smaller the range, the more bits it takes to code the character.

Typically, the model used to code the data changes based on the data input stream contents. This is known as adaptive coding.

*See also :*Data compression, Range encoder

An arithmetic encoder takes a string of symbols as input and produces a rational number in the interval [0, 1) as output. As each symbol is processed, the encoder will restrict the output to a smaller interval.

Let N be the number of distinct symbols in the input; let x_{1}, x_{2} ... x_{N} represent the symbols, and let P_{1}, P_{2} ... P_{N} represent the probability of each symbol appearing.
At each step in the process, the output is restricted to the current interval [y, y+R).
Partition this interval into N disjoint subintervals:

- I
_{1}= [y, y + P_{1}R) - I
_{2}= [y + P_{1}R, y + P_{1}R + P_{2}R) - .
- .
- .

Note that at each stage, all the possible intervals are pairwise disjoint. Therefore a specific sequence of symbols produces exactly one unique output range, and the process can be reversed.

Since arithmetic encoders are typically implemented on binary computers, the actual output of the encoder is generally the shortest sequence of bits representing the fractional part of a rational number in the final interval.

Suppose our entire input string contains M symbols: then x_{i} appears exactly P_{i}M times in the input. Therefore, the size of the final interval will be

However, IBM and other companies own patents in the United States and other countries on algorithms essential for implementing an arithmetic encoder.
*But are those patent holders willing to license the patents royalty-free for use in open-source software?*

An earlier version of the above article was posted on PlanetMath. This article is open content