digitexpansion

Digit expansion is the representation of numbers using a positional numeral system, in which the value is obtained by summing digits multiplied by powers of a chosen base. In base b (b > 1), a nonnegative integer has a finite expansion with digits 0 through b−1, for example 1234 in base 10 equals 1×10^3 + 2×10^2 + 3×10^1 + 4×10^0. A fractional part can be written as 0.d1 d2 d3…, where each di is a digit and the value is d1×b^−1 + d2×b^−2 + d3×b^−3, and so on.

In base 10, every real number has a digit expansion consisting of an integer part and a fractional part. Terminating expansions occur when only finitely many digits are nonzero; nonterminating expansions may be infinite. For rational numbers in base b, the expansion is eventually periodic, meaning it repeats a finite block of digits. For example, 1/3 in base 10 is 0.333…, and 1/7 in base 10 is 0.142857142857…, repeating every six digits. Irrational numbers have nonterminating expansions that do not repeat.

Digit expansions use a digit set consisting of the integers 0 through b−1. When b exceeds 10, additional symbols such as letters are commonly used to represent digits beyond 9. Variants and extensions include negative bases (such as base −2), balanced digit sets, and complex or nonstandard bases that enable alternative representations or compact encodings. These systems underpin arithmetic algorithms, base conversion, and data representation in computing.