# List of numeral systems

(Redirected from Base 50 (numeral system))

There are many different numeral systems, that is, writing systems for expressing numbers.

## By culture / time period

Name Base Sample Approx. First Appearance
Proto-cuneiform numerals 10+60 c. 3500–2000 BCE
Indus numerals c. 3500–1900 BCE
Proto-Elamite numerals 10+60 3,100 BCE
Sumerian numerals 10+60 3,100 BCE
Egyptian numerals 10
3,000 BCE
Babylonian numerals 10+60 2,000 BCE
Chinese numerals
Japanese numerals
Korean numerals (Sino-Korean)
Vietnamese numerals (Sino-Vietnamese)
10

〇一二三四五六七八九十百千万亿 (Default, Simplified Chinese)

1,600 BCE
Aegean numerals 10 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  ( )
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘  ( )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡  ( )
𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪  ( )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳  ( )
1,500 BCE
Roman numerals I V X L C D M 1,000 BCE
Hebrew numerals 10 א ב ג ד ה ו ז ח ט
י כ ל מ נ ס ע פ צ
ק ר ש ת ך ם ן ף ץ
800 BCE
Indian numerals 10 Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९
Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩

750–690 BCE
Bengali numerals 10 ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ 500 BCE
Greek numerals 10 ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ
<400 BCE
Phoenician numerals 10 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [1] <250 BCE[2]
Chinese rod numerals 10 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 1st Century
Coptic numerals 10 Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ 2nd Century
Ge'ez numerals 10 ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱
፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻
3rd–4th Century
15th Century (Modern Style)[3]
Armenian numerals 10 Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ Early 5th Century
Khmer numerals 10 ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ Early 7th Century
Thai numerals 10 ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ 7th Century[4]
Abjad numerals 10 غ ظ ض ذ خ ث ت ش ر ق ص ف ع س ن م ل ك ي ط ح ز و هـ د ج ب ا <8th Century
Eastern Arabic numerals 10 ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ 8th Century
Vietnamese numerals (Chữ Nôm) 10 𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩 <9th Century
Western Arabic numerals 10 0 1 2 3 4 5 6 7 8 9 9th Century
Glagolitic numerals 10 Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... 9th Century
Cyrillic numerals 10 а в г д е ѕ з и ѳ і ... 10th Century
Rumi numerals 10 10th Century
Burmese numerals 10 ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ 11th Century[5]
Tangut numerals 10 𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗 11th Century (1036)
Cistercian numerals 10 13th Century
Maya numerals 5+20 <15th Century
Muisca numerals 20 <15th Century
Korean numerals (Hangul) 10 영 일 이 삼 사 오 육 칠 팔 구 15th Century (1443)
Aztec numerals 20 16th Century
Sinhala numerals 10 ෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣
𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴
<18th Century
Cherokee numerals 10 19th Century (1820s)
Osmanya numerals 10 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 20th Century (1920s)
Kaktovik numerals 5+20 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 20th Century (1994)

## By type of notation

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.

### Standard positional numeral systems

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.

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.[6] There have been some proposals for standardisation.[7]

Base Name Usage
2 Binary Digital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon)
3 Ternary Cantor set (all points in [0,1] that can be represented in ternary with no 1s); counting Tasbih in Islam; hand-foot-yard and teaspoon-tablespoon-shot measurement systems; most economical integer base
4 Quaternary Data transmission, DNA bases and Hilbert curves; Chumashan languages, and Kharosthi numerals
5 Quinary Gumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6 Senary Diceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7 Septenary Weeks timekeeping, Western music letter notation
8 Octal Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China)
9 Nonary Base9 encoding; compact notation for ternary
10 Decimal (also known as denary) Most widely used by modern civilizations[8][9][10]
11 Undecimal A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[11] and the Pangwa (Tanzania) in the 20th century.[12] Briefly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal. Used as a check digit in ISBN for 10-digit ISBNs.
12 Duodecimal Languages in the Nigerian Middle Belt Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozen-gross-great gross counting; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions; penny and shilling
13 Tridecimal Base13 encoding; Conway base 13 function.
14 Tetradecimal Programming for the HP 9100A/B calculator[13] and image processing applications;[14] pound and stone.
15 Pentadecimal Telephony routing over IP, and the Huli language.

Base16 encoding; compact notation for binary data; tonal system; ounce and pound.
18 Octodecimal Base18 encoding; a base such that 7n is palindromic for n = 3, 4, 6, 9.
20 Vigesimal Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
21 Unvigesimal Base21 encoding; also the smallest base where all of 1/2 to 1/18 have periods of 4 or shorter.
22 Duovigesimal Base22 encoding.
23 Trivigesimal Kalam language,[15] Kobon language[citation needed]
24 Tetravigesimal 24-hour clock timekeeping; Kaugel language.
25 Pentavigesimal Base25 encoding; sometimes used as compact notation for quinary.
26 Hexavigesimal Base26 encoding; sometimes used for encryption or ciphering,[16] using all letters in the English alphabet
27 Heptavigesimal Septemvigesimal Telefol[17] and Oksapmin[18] languages. Mapping the nonzero digits to the alphabet and zero to the space is occasionally used to provide checksums for alphabetic data such as personal names,[19] to provide a concise encoding of alphabetic strings,[20] or as the basis for a form of gematria.[21] Compact notation for ternary.
28 Octovigesimal Base28 encoding; months timekeeping.
29 Enneavigesimal Base29 encoding.
30 Trigesimal The Natural Area Code, this is the smallest base such that all of 1/2 to 1/6 terminate, a number n is a regular number if and only if 1/n terminates in base 30.
31 Untrigesimal Base31 encoding.
32 Duotrigesimal Base32 encoding; the Ngiti language.
33 Tritrigesimal Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong.
34 Tetratrigesimal Using all numbers and all letters except I and O; the smallest base where 1/2 terminates and all of 1/2 to 1/18 have periods of 4 or shorter.
35 Pentatrigesimal Using all numbers and all letters except O.
36 Hexatrigesimal Base36 encoding; use of letters with digits.
37 Heptatrigesimal Base37 encoding; using all numbers and all letters of the Spanish alphabet.
38 Octotrigesimal Base38 encoding; use all duodecimal digits and all letters.
39 Enneatrigesimal Base39 encoding.
40 Quadragesimal DEC RADIX 50/MOD40 encoding used to compactly represent file names and other symbols on Digital Equipment Corporation computers. The character set is a subset of ASCII consisting of space, upper case letters, the punctuation marks "\$", ".", and "%", and the numerals.
42 Duoquadragesimal Base42 encoding; largest base for which all minimal primes are known.
47 Septaquadragesimal Smallest base for which no generalized Wieferich primes are known.
49 Enneaquadragesimal Compact notation for septenary.
50 Quinquagesimal Base50 encoding; SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits.
52 Duoquinquagesimal Base52 encoding, a variant of Base62 without vowels except Y and y[22] or a variant of Base26 using all lower and upper case letters.
54 Tetraquinquagesimal Base54 encoding.
56 Hexaquinquagesimal Base56 encoding, a variant of Base58.[23]
57 Heptaquinquagesimal Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[24] or I, 1, l, 0, and O.[25]
58 Octoquinquagesimal Base58 encoding, a variant of Base62 excluding 0 (zero), I (capital i), O (capital o) and l (lower case L).[26]
60 Sexagesimal Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[27] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian
62 Duosexagesimal Base62 encoding, using 0–9, A–Z, and a–z.
64 Tetrasexagesimal Base64 encoding; I Ching in China.
This system is conveniently coded into ASCII by using the 26 letters of the Latin alphabet in both upper and lower case (52 total) plus 10 numerals (62 total) and then adding two special characters (+ and /).
72 Duoseptuagesimal Base72 encoding; the smallest base >2 such that no three-digit narcissistic number exists.
80 Octogesimal Base80 encoding; Supyire as a sub-base.
81 Unoctogesimal Base81 encoding, using as 81=34 is related to ternary.
85 Pentoctogesimal Ascii85 encoding. This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME-64 encoding, since 855 is only slightly bigger than 232. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
89 Enneaoctogesimal Largest base for which all left-truncatable primes are known.
90 Nonagesimal Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
91 Unnonagesimal Base91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92 Duononagesimal Base92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[28]
93 Trinonagesimal Base93 encoding, using all of ASCII printable characters except for "," (0x27) and "-" (0x3D) as well as the Space character. "," is reserved for delimiter and "-" is reserved for negation.[29]
94 Tetranonagesimal Base94 encoding, using all of ASCII printable characters.[30]
95 Pentanonagesimal Base95 encoding, a variant of Base94 with the addition of the Space character.[31]
96 Hexanonagesimal Base96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits.
97 Septanonagesimal Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff primes are known.
100 Centesimal As 100=102, these are two decimal digits.
120 Centevigesimal Base120 encoding.
121 Centeunvigesimal Related to base 11.
125 Centepentavigesimal Related to base 5.
128 Centeoctovigesimal Using as 128=27.
169 Centenovemsexagesimal Two Tridecimal digits.
185 Centepentoctogesimal Smallest base which is not perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
200 Duocentesimal Base200 encoding.
210 Duocentedecimal Smallest base such that all of 1/2 to 1/10 terminate.
216 Duocentehexidecimal related to base 6.
256 Duocentehexaquinquagesimal Base256 encoding, as 256=28.
300 Trecentesimal Base300 encoding.
360 Trecentosexagesimal Degrees for angle.

### Non-standard positional numeral systems

#### Bijective numeration

Base Name Usage
1 Unary (Bijective base‑1) Tally marks, Counting
10 Bijective base-10 To avoid zero
26 Bijective base-26 Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.[32]

#### Signed-digit representation

Base Name Usage
3 Balanced ternary Ternary computers
4 Balanced quaternary
5 Balanced quinary
6 Balanced senary
7 Balanced septenary
8 Balanced octal
9 Balanced nonary
10 Balanced decimal John Colson
Augustin Cauchy
11 Balanced undecimal
12 Balanced duodecimal

#### 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
−4 Negaquaternary
−5 Negaquinary
−6 Negasenary
−8 Negaoctal

#### Complex bases

Base Name Usage
2i Quater-imaginary base related to base −4 and base 16
${\displaystyle {\sqrt {2}}i}$ Base ${\displaystyle {\sqrt {2}}i}$ related to base −2 and base 4
${\displaystyle {\sqrt[{4}]{2}}i}$ Base ${\displaystyle {\sqrt[{4}]{2}}i}$ related to base 2
${\displaystyle 2\omega }$ Base ${\displaystyle 2\omega }$ related to base 8
${\displaystyle {\sqrt[{3}]{2}}\omega }$ Base ${\displaystyle {\sqrt[{3}]{2}}\omega }$ related to base 2
−1 ± i Twindragon base Twindragon fractal shape, related to base −4 and base 16
1 ± i Negatwindragon base related to base −4 and base 16

#### Non-integer bases

Base Name Usage
${\displaystyle {\frac {3}{2}}}$ Base ${\displaystyle {\frac {3}{2}}}$ a rational non-integer base
${\displaystyle {\frac {4}{3}}}$ Base ${\displaystyle {\frac {4}{3}}}$ related to duodecimal
${\displaystyle {\frac {5}{2}}}$ Base ${\displaystyle {\frac {5}{2}}}$ related to decimal
${\displaystyle {\sqrt {2}}}$ Base ${\displaystyle {\sqrt {2}}}$ related to base 2
${\displaystyle {\sqrt {3}}}$ Base ${\displaystyle {\sqrt {3}}}$ related to base 3
${\displaystyle {\sqrt[{3}]{2}}}$ Base ${\displaystyle {\sqrt[{3}]{2}}}$
${\displaystyle {\sqrt[{4}]{2}}}$ Base ${\displaystyle {\sqrt[{4}]{2}}}$
${\displaystyle {\sqrt[{12}]{2}}}$ Base ${\displaystyle {\sqrt[{12}]{2}}}$ usage in 12-tone equal temperament musical system
${\displaystyle 2{\sqrt {2}}}$ Base ${\displaystyle 2{\sqrt {2}}}$
${\displaystyle -{\frac {3}{2}}}$ Base ${\displaystyle -{\frac {3}{2}}}$ a negative rational non-integer base
${\displaystyle -{\sqrt {2}}}$ Base ${\displaystyle -{\sqrt {2}}}$ a negative non-integer base, related to base 2
${\displaystyle {\sqrt {10}}}$ Base ${\displaystyle {\sqrt {10}}}$ related to decimal
${\displaystyle 2{\sqrt {3}}}$ Base ${\displaystyle 2{\sqrt {3}}}$ related to duodecimal
φ Golden ratio base Early Beta encoder[33]
ρ Plastic number base
ψ Supergolden ratio base
${\displaystyle 1+{\sqrt {2}}}$ Silver ratio base
e Base ${\displaystyle e}$ Lowest radix economy
π Base ${\displaystyle \pi }$
eπ Base ${\displaystyle e\pi }$
${\displaystyle e^{\pi }}$ Base ${\displaystyle e^{\pi }}$

Base Name Usage
6 Hexadic number not a field
10 Decadic number not a field
12 Dodecadic number not a field

• 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

### Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[34] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

## References

1. ^ Everson, Michael (2007-07-25). "Proposal to add two numbers for the Phoenician script" (PDF). UTC Document Register. Unicode Consortium. L2/07-206 (WG2 N3284).
2. ^ Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
3. ^ Chrisomalis, Stephen (2010-01-18). Numerical Notation: A Comparative History. ISBN 9781139485333.
4. ^ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
5. ^ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 2017.
6. ^ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
7. ^ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
8. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
9. ^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
10. ^ 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
11. ^ Overmann, Karenleigh A (2020). "The curious idea that Māori once counted by elevens, and the insights it still holds for cross-cultural numerical research". Journal of the Polynesian Society. 129 (1): 59–84. doi:10.15286/jps.129.1.59-84. Retrieved 24 July 2020.
12. ^ Thomas, N.W (1920). "Duodecimal base of numeration". Man. 20 (1): 56–60. doi:10.2307/2840036. JSTOR 2840036. Retrieved 25 July 2020.
13. ^ HP 9100A/B programming, HP Museum
14. ^ Free Patents Online
15. ^ Laycock, Donald (1975). "Observations on Number Systems and Semantics". In Wurm, Stephen (ed.). New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene. Pacific Linguistics C-38. Canberra: Research School of Pacific Studies, Australian National University. pp. 219–233.
16. ^
17. ^ Laycock, Donald (1975). "Observations on Number Systems and Semantics". In Wurm, Stephen (ed.). New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene. Pacific Linguistics C-38. Canberra: Research School of Pacific Studies, Australian National University. pp. 219–233.
18. ^ Saxe, Geoffrey B.; Moylan, Thomas (1982). "The development of measurement operations among the Oksapmin of Papua New Guinea". Child Development. 53 (5): 1242–1248. doi:10.1111/j.1467-8624.1982.tb04161.x. JSTOR 1129012..
19. ^ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proceedings. AMIA Symposium: 305–309, PMC 2244404, PMID 12463836.
20. ^ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
21. ^ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
22. ^ "Base52". GitHub. Retrieved 2016-01-03.
23. ^ "Base56". Retrieved 2016-01-03.
24. ^ "Base57". GitHub. Retrieved 2016-01-03.
25. ^ "Base57". GitHub. Retrieved 2019-01-22.
26. ^ "The Base58 Encoding Scheme". Internet Engineering Task Force. November 27, 2019. Archived from the original on August 12, 2020. Retrieved August 12, 2020. Thanks to Satoshi Nakamoto for inventing the Base58 encoding format
27. ^ "NewBase60". Retrieved 2016-01-03.
28. ^ "Base92". GitHub. Retrieved 2016-01-03.
29. ^ "Base93". 26 September 2013. Retrieved 2017-02-13.
30. ^ "Base94". Retrieved 2016-01-03.
31. ^ "base95 Numeric System". Archived from the original on 2016-02-07. Retrieved 2016-01-03.
32. ^ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
33. ^ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, arXiv:0806.1083, Bibcode:2008arXiv0806.1083W, doi:10.1109/TIT.2008.928235, S2CID 12926540
34. ^ Chrisomalis calls the Babylonian system "the first positional system ever" in Chrisomalis, Stephen (2010), Numerical Notation: A Comparative History, Cambridge University Press, p. 254, ISBN 9781139485333.