Numeral

Definition

A numeral is a symbolic representation of a natural number or, more generally, an integer. The number itself is an abstract mathematical object, while a numeral is the written expression used to denote it.

For example, the numerals 7, VII, and 111 in base $2$ can all denote the same number.

Unary numerals

A unary numeral system represents a number by repeating a single mark. This is the simplest numeral system and is closely related to tally marks.

If the alphabet is $\{\mid \}$, we may define the decoding function ${\mathcal{N}}_1$ by

${\mathcal{N}}_1(ϵ)=0$

${\mathcal{N}}_1(\mid ::xs)=\mathrm{suc}({\mathcal{N}}_1(xs))$

Thus:

• ${\mathcal{N}}_1(ϵ)=0$
• ${\mathcal{N}}_1(\mid )=1$
• ${\mathcal{N}}_1(\mid \mid \mid )=3$

Unary notation is conceptually simple, but inefficient for large numbers.

Positional numerals

Most familiar numeral systems are positional: the value of a digit depends both on the digit itself and on its position.

Fix a base $n\ge 2$ with digit alphabet $[n]=\{0,1,\dots ,n-1\}$. A numeral is then a finite string of digits from $[n]$. The associated decoding function ${\mathcal{N}}_n$ is given recursively by

${\mathcal{N}}_n(ϵ)=0$
${\mathcal{N}}_n(xs:{:}^{-}i)=n\cdot {\mathcal{N}}_n(xs)+i$

Where $:{:}^{-}$ is a ‘reverse’ cons. This expresses the usual rule for positional notation: appending a digit shifts the previous value one place to the left and then adds the new digit.

Examples

• In base $2$, the numeral 101 denotes $5$.
• In base $10$, the numeral 203 denotes $2\cdot 10^2+0\cdot 10+3$.
• In base $16$, the numeral A denotes $10$.

Numerals and numbers

The distinction between numerals and numbers is analogous to the distinction between a word and its meaning. Different numeral systems can represent the same number in different ways.

• Decimal 4
• Binary 100
• Roman IV
• Unary ||||

Each of these is a different numeral for the same number.

Related Concepts

• Natural Number
• Integer