Positional numeral system in the context of Quipu

An abacus (pl. abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times, in the ancient Near East, Europe, China, and Russia, until largely replaced by handheld electronic calculators, during the 1980s, with some ongoing attempts to revive their use. An abacus consists of a two-dimensional array of slidable beads (or similar objects). In their earliest designs, the beads could be loose on a flat surface or sliding in grooves. Later the beads were made to slide on rods and built into a frame, allowing faster manipulation.

Each rod typically represents one digit of a multi-digit number laid out using a positional numeral system such as base ten (though some cultures used different numerical bases). Roman and East Asian abacuses use a system resembling bi-quinary coded decimal, with a top deck (containing one or two beads) representing fives and a bottom deck (containing four or five beads) representing ones. Natural numbers are normally used, but some allow simple fractional components (e.g. 1⁄2, 1⁄4, and 1⁄12 in Roman abacus), and a decimal point can be imagined for fixed-point arithmetic.

The decimal numeral system (also called the base-ten positional numeral system and denary /diːnəri/ or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or as in 25.9703 or 3,1415).Decimal may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals".

In a positional numeral system, the radix (pl. radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (x)_y with x as the string of digits and y as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosing parentheses), as it is the most common way to express value. For example, (100)₁₀ is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)₂ (in the binary system with base 2) represents the number four.