Numeral systems by culture
Hindu-Arabic numerals
Western Arabic (Hindu numerals) Eastern Arabic Indian family Tamil	Burmese Khmer Lao Mongolian Thai
East Asian numerals
Chinese Japanese Suzhou	Korean Vietnamese Counting rods
Alphabetic numerals
Abjad Armenian Āryabhaṭa Cyrillic	Ge'ez Greek Georgian Hebrew
other historical systems
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit	Kharosthi Mayan Quipu Roman
Positional systems by base
Decimal (10)
2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 27, 30, 36, 60, 64
Balanced ternary
Non-positional system
Unary numeral system (Base 1)
List of numeral systems
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This list is incomplete; you can help by expanding it.

This is a list of numeral systems.

[edit] Positional notation

These numeral systems use place-value notation.

Standard [link]

A binary clock 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.

For the definition of standard positional numeral systems, see Non-standard positional numeral systems.

The common names are derived somewhat arbitrarily from Latin and Greek. For more information, see Hexadecimal#Etymology.

Base	Name	Usage
2	Binary	All modern digital computations.
3	Ternary	Cantor set (all points in [0,1] that can be represented in ternary with no 1s.)
4	Quaternary	Data transmission and Hilbert curves.
5	Quinary
6	Senary	Diceware
7	Septenary
8	Octal	Charles XII of Sweden, Unix-like permissions
9	Nonary
10	Decimal	Most widely used by modern civilizations.^[1]^[2]
11	Undecimal
12	Duodecimal
13	Tridecimal	The Maya calendar.
14	Tetradecimal	Programming for the HP 9100A/B calculator^[3] and image processing applications^[4].
15	Pentadecimal	Telephony routing over IP and the Huli language.
16	Hexadecimal	Human-friendly representation (hex dump) of binary data and Base16 encoding.
20	Vigesimal	Celtic numerals, Maya numerals
24	Tetravigesimal
26	Hexavigesimal
27	Septemvigesimal	Telefol and Oksapmin languages.
30	Trigesimal
32	Duotrigesimal	Base32 encoding and the Ngiti language.
36	Hexatridecimal	Base36 encoding.
60	Sexagesimal	The Babylonian numerals positional numeral system.
64	Tetrasexagesimal	Base64 encoding.
85		Ascii85 encoding.

[edit] Non-standard

[edit] Bijective numeration

Base	Name	Usage
1	Unary	Tally marks.
10	Decimal without a zero

[edit] Signed-digit representation

Base	Name	Usage
2	Non-adjacent form
3	Balanced ternary	Ternary computers.

[edit] Negative bases

The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:

Base	Name	Usage
−2	Negabinary
−3	Negaternary
−10	Negadecimal

[edit] Complex bases

Base	Name	Usage
2i	Quater-imaginary base
−1 ± i	Twindragon base	Twindragon fractal shape.

[edit] Non-integer bases

Base	Name	Usage
φ	Golden ratio base	Early Beta encoder.^[5]
e	Base e
π	Base π
√2	Base √2

Non-positional notation [link]

All known numeral systems developed before the Babylonian numerals are non-positional.

Numerals [link]

Name	Base	Sample	Approx. first appearance
Babylonian numerals	60		3100 B.C.
Greek numerals	10	α β γ δ ε ϝ ζ η θ ι
Roman numerals	10	I II III IV V VI VII VIII IX X	1000 B.C.
Chinese rod numerals	10		1st century
Arabic numerals	10	0 1 2 3 4 5 6 7 8 9 10	9th century

References [link]

^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994 (Also: The Universal History of Numbers: From prehistory to the invention of the computer, Georges Ifrah, ISBN 0-471-39340-1, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk)
^ https://www.hpmuseum.org/prog/hp9100pr.htm
^ See a patent at Free Patents Online
^ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory 54 (9): 4324–4334, DOI:10.1109/TIT.2008.928235

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Helena Blavatsky

Helena Petrovna Blavatsky (Russian: Еле́на Петро́вна Блава́тская, Yelena Petrovna Blavatskaya; 12 August [O.S. 31 July] 1831 – 8 May 1891) was an occultist, spirit medium, and author who co-founded the Theosophical Society in 1875. She gained an international following as the leading theoretician of Theosophy, the esoteric movement that the Society promoted.

Born into an aristocratic Russian-German family in Yekaterinoslav, Blavatsky traveled widely around the Russian Empire as a child. Largely self-educated, she developed an interest in Western esotericism during her teenage years. According to her later claims, in 1849 she embarked on a series of world travels, visiting Europe, the Americas, and India. She alleged that during this period she encountered a group of spiritual adepts, the "Masters of the Ancient Wisdom", who sent her to Shigatse, Tibet, where they trained her to develop her own psychic powers. Both contemporary critics and later biographers have argued that some or all of these foreign visits were fictitious, and that she spent this period in Europe. By the early 1870s, Blavatsky was involved in the Spiritualist movement; although defending the genuine existence of Spiritualist phenomena, she argued against the mainstream Spiritualist idea that the entities contacted were the spirits of the dead. Relocating to the United States in 1873, she befriended Henry Steel Olcott and rose to public attention as a spirit medium, attention that included public accusations of fraudulence. One of her sources of inspiration was a book by Emil Schlagintweit, Le Bouddhisme Au Tibet, published in 1881.

Septenary (Theosophy)

The Septenary in Helena Blavatsky's teachings refers to the seven principles of man. In The Key to Theosophy she presents a synthesis of Eastern (Advaita Vedanta, Samkhya) and Western (Platonism, 19th century Occultism) ideas, according to which human nature consists of seven principles. These are:

Atma - Spirit or Self - one with The Absolute as Its Radiation.

Buddhi - Spiritual Soul - vehicle of pure universal spirit.

Manas - consisting of Higher Manas, the spiritual, inner, or higher Ego; and Lower Manas, the ordinary mind.

Kamarupa - the "desire body", seat of animal desires and passions.

Prana - the vital principle.

Linga Sharira - the double, or astral body.

Sthula Sharira - the physical body.

Each of these principles are embodied as such:

The first body is called sthula-sarira (Sanskrit, from sthula meaning coarse, gross, not refined, heavy, bulky, fat in the sense of bigness, conditioned and differentiated matter + sarira to moulder, waste away). A gross body, impermanent because of its wholly compound character. The physical body is usually considered as the lowest substance-principle. The physical form is the result of the harmonious co-working on the physical plane of forces and faculties streaming through their astral vehicle or linga-sarira, the pattern or model of the physical body.

Contents

[edit] Positional notation

Standard [link]

[edit] Non-standard

[edit] Bijective numeration

[edit] Signed-digit representation

[edit] Negative bases

[edit] Complex bases

[edit] Non-integer bases

Other [link]

Non-positional notation [link]

Numerals [link]

See also [link]

References [link]

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Helena Blavatsky

Septenary (Theosophy)

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