Numeral system

Numeral systems by culture
Hindu–Arabic numerals
Western Arabic; Eastern Arabic; Khmer	Indian family; Brahmi; Thai
East Asian numerals
Chinese; Suzhou; Counting rods	Japanese; Korean
Alphabetic numerals
Abjad; Armenian; Cyrillic; Ge'ez	Hebrew; Greek (Ionian); Āryabhaṭa;
Other systems
Attic; Babylonian; Coptic; Egyptian; Etruscan	Mayan; Roman; Urnfield
	List of numeral system topics
Positional systems by base
	Decimal (10)
	2, 4, 8, 16, 32, 64
	1, 3, 9, 12, 20, 24, 30, 36, 60, more…
	v; t; e;

A numeral system (also called a number system or system of numeration) is a way to write numbers. Roman numerals and tally marks are examples. "11" usually means eleven, but if the numeral system is binary, then "11" means three.

A numeral is a way to represent a number. It may be a symbol or a word in a natural language, or a group of them. Numerals differ from numbers just as the word "rock" differs from a real rock. The symbols "11", "eleven" and "XI" are all numerals that represent the same number. Babylonian numerals, Greek numerals and Roman numerals are among the systems that were long used, before the Hindu–Arabic numeral system largely replaced them.

Bases

Various symbols are used as numerals to make numbers. A system with base 10 (the common decimal system), normally uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of the numbers 0 to 9 can be written as one symbol, 0 ... 9. To count past 9, symbols have to be put together. 10 can be seen as 1 in the tens' place and 0 in the ones' place, or as 1 times 10¹ plus 0 times 10⁰. With a base of 2, only the symbols 0 and 1 are used. 10_{base 2} is therefore 1 times 2¹ plus 0 times 2⁰. This is the same as 2, in the base 10 notation.

For bases bigger than 10, capital letters are used as symbols. For example, the hexadecimal numeral system (base 16) uses the numerical digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Today, base 10 is the most commonly used system. Computers use binary and people who study computers often use octal and hexadecimal numeral systems. Ancient Sumer used sexagesimal (base 60). New world used base 20.

Most electronic calculations are done in binary (base 2), but most people do calculations in decimal (base ten) or duodecimal.

Hypercomplex numbers

Mathematician Robert P. C. de Marrais lists different hypercomplex number systems belonging to different dimensions.^[1]

Name	Dimension	Symbol
real numbers	1 = 2⁰	$\mathbb {R}$
complex numbers	2 = 2¹	$\mathbb {C}$
quaternions	4 = 2²	$\mathbb {H}$
octonions	8 = 2³	$\mathbb {O}$
sedenions	16 = 2⁴	$\mathbb {S}$
pathions	32 = 2⁵	$\mathbb {P}$
chingons	64 = 2⁶	$\mathbb {X}$
routons	128 = 2⁷	$\mathbb {U}$
voudons	256 = 2⁸	$\mathbb {V}$
2ⁿ-ions	2ⁿ

References

↑ de Marrais, Robert P. C. (2002). "Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2ⁿ-ions Beyond the Sedenions". arXiv:math/0207003. doi:10.48550/arXiv.math/0207003.

Other websites

History of Counting and Numeral Systems-PlainMath.Net Archived 2007-07-15 at the Wayback Machine
Online Numeral Base Converter for Different Numeral Systems (Base 2-36)
Online Converter Archived 2007-01-04 at the Wayback Machine for Decimal/Roman Numerals (JavaScript, GPL)
Number Sense & Numeration Lessons
Counting Systems of Papua New Guinea

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[Box-Kite-1] Marrais, Robert P. C. (2002). "Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2ⁿ-ions Beyond the Sedenions". arXiv:math/0207003. doi:10.48550/arXiv.math/0207003.

[1]

v t e Number systems
Commonly used systems	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ )
Hypercomplex numbers	Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ ) Sedenions ( $\mathbb {S}$ ) Pathions ( $\mathbb {P}$ ) Chingons ( $\mathbb {X}$ ) Routons ( $\mathbb {U}$ ) Voudons ( $\mathbb {V}$ )