List of numeral systems#Standard positional numeral systems

"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system."{{cite journal |last1=Chrisomalis |first1=Stephen |date=2004 |title=A cognitive typology for numerical notation |journal=Cambridge Archaeological Journal |volume=14 |issue=1 |pages=37–52 |doi=10.1017/S0959774304000034 }}{{rp|38}} The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base).

{{anchor|Positional numeral systems}}

Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base.

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File:Binary clock.svg might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.]]

The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name.For the mixed roots of the word "hexadecimal", see {{citation|title=Discrete Mathematics with Applications|first=Susanna|last=Epp|edition=4th|publisher=Cengage Learning|year=2010|isbn=9781133168669|page=91|url=https://books.google.com/books?id=HUAIAAAAQBAJ&pg=PA91}}. There have been some proposals for standardisation.[http://www.dozenal.org/drupal/sites_bck/default/files/MultiplicationSynopsis.pdf Multiplication Tables of Various Bases], p. 45, Michael Thomas de Vlieger, Dozenal Society of America

• Factorial number system {1, 2, 3, 4, 5, 6, ...}
• Even double factorial number system {2, 4, 6, 8, 10, 12, ...}
• Odd double factorial number system {1, 3, 5, 7, 9, 11, ...}
• Primorial number system {2, 3, 5, 7, 11, 13, ...}
• Fibonorial number system {1, 2, 3, 5, 8, 13, ...}
• {60, 60, 24, 7} in timekeeping
• {60, 60, 24, 30 (or 31 or 28 or 29), 12, 10, 10, 10} in timekeeping
• (12, 20) traditional English monetary system (£sd)
• (20, 18, 13) Maya timekeeping

• Quote notation
• Redundant binary representation
• Hereditary base-n notation
• Asymmetric numeral systems optimized for non-uniform probability distribution of symbols
• Combinatorial number system

All known numeral systems developed before the Babylonian numerals are non-positional,{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=kXZhBAAAQBAJ&pg=PA254 p. 254:] Chrisomalis calls the Babylonian system "the first positional system ever"}} as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

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• {{annotated link|History of ancient numeral systems}}
• {{annotated link|History of the Hindu–Arabic numeral system}}
• {{annotated link|List of numeral system topics}}
• {{annotated link|Numeral prefix}}
• {{annotated link|Radix}}
• {{annotated link|Radix economy}}
• {{annotated link|Timeline of numerals and arithmetic}}
• {{annotated link|List of books on history of number systems}}

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Systems

Numeral