English: An arithmetic coding example assuming a fixed probability distribution of three Symbols "A","B" and "C".

An arithmetic coding example assuming a fixed probability distribution of three Symbols "A","B" and "C". Probability of "A" is 50%, probability of "B" is 33% and probability of "C" is 17%. Furthermore we assume that the recursion depth is known in each step. In step one we code "B" which is inside of the interval [0.5,0.83): The binary number "0.10x" is the shortest code that represents an Interval that is entirely inside of [0.5,0.83). "x" represents here an arbitrary bit sequence. There are two extreme cases: the smallest x stands for an infinite number of zeros which represents the left side of the represented Interval. Then the left side of the Interval is dec(0.10)=0.5. The greatest x stands for an infinite number of ones which gives a number that converges towards dec(0.11)=0.75 Therefore "0.10x" represents the Interval [0.5,0.75) which is inside of [0.5,0.83)

Now we can leave out the "0." part since all Intervals begin with "0." and we can ignore the x because no matter what Bit-sequence it represents, we will stay inside of [0.5,0.75)