 Denote by the floor function of "x", that is, the greatest integer less than or equal to "x", and let {"x"} = "x" − ⌊"x"⌋ be the fractional part of "x".There exists an integer "k" such that .Set
and
For , put

In other words, the canonical β-expansion of "x" is defined by choosing the largest "d""k" such that , then choosing the largest "d""k"−1 such that , etc.