Numeral
ArithmeticA numeral is a symbol or combination of symbols used to represent a number, such as "7," "VII," or "seven."
Definition
A numeral is the written symbol we use to show a number. The number itself is an idea; the numeral is how we write that idea down.
Example
The numeral "$5$," the Roman numeral "V," and the word "five" all represent the same number. They are different numerals for the same value.
Key Insight
A number is an abstract idea (the concept of five-ness), while a numeral is the symbol we use to write it. One number can have many different numerals.
Definition
A numeral is a written representation of a number. Numeral systems include Hindu-Arabic (our standard $0$-$9$ digits), Roman (I, V, X, L, C, D, M), and many others. The Hindu-Arabic system is positional; Roman numerals are additive/subtractive and non-positional.
Example
The number forty-seven can be written as the numeral $47$ (Hindu-Arabic), XLVII (Roman), $101111$ (binary), or $2F$ (hexadecimal). All represent the same abstract quantity.
Key Insight
The positional Hindu-Arabic numeral system, adopted in Europe through Arabic mathematicians, replaced Roman numerals precisely because arithmetic (like long multiplication) is far easier with a positional system.
Definition
A numeral is a formal expression in a numeral system, which is a writing system for expressing numbers. Positional systems are defined by a base $b$ and represent $n$ as a finite or infinite string of digits using the polynomial value $\sum d_k \cdot b^k$. The distinction between a number (abstract object) and its numeral (syntactic representation) is important in mathematical logic and computer science.
Example
In Peano arithmetic, each natural number has a canonical numeral: $0$, $S(0)$, $S(S(0))$, etc. The standard numerals $0, 1, 2, \ldots$ are abbreviations of these.
Key Insight
Godel numbering assigns a unique numeral to every formula in a formal system, enabling a formal system to "talk about" its own syntax. This technique is central to the proof of Godel's incompleteness theorems.