# Tilde

(Redirected from ̴)
~
Tilde
˜  ∼  ◌̃
Small tilde  Tilde operator Combining tilde
Punctuation
apostrophe   '
brackets [ ]  ( )  { }  ⟨ ⟩
colon :
comma ,  ،
dash   –  —  ―
ellipsis   ...  . . .
exclamation mark !
full stop, period .
hyphen
hyphen-minus -
question mark ?
quotation marks ‘ ’  “ ”  ' '  " "
semicolon ;
slash, stroke, solidus /
Word dividers
interpunct ·
space
General typography
ampersand &
asterisk *
at sign @
backslash \
bullet
caret ^
dagger † ‡
degree °
ditto mark
inverted exclamation mark ¡
inverted question mark ¿
number sign, pound, hash, octothorpe #
numero sign
obelus ÷
ordinal indicator º ª
percent, per mil % ‰
plus and minus + −
basis point
pilcrow
prime
section sign §
tilde ~
underscore, understrike _
vertical bar, pipe, broken bar |    ¦
Intellectual property
service mark
Uncommon typography
asterism
hedera
index, fist
interrobang
irony punctuation
lozenge
reference mark
tie
Related
In other scripts

The tilde (, ; ˜ or ~; also referred to informally as squiggly or squiggle(s)) is a grapheme with several uses. The name of the character comes from Portuguese and Spanish, from the Latin titulus meaning "title" or "superscription", though the term "tilde" has evolved and now has a different meaning in linguistics. Some may refer to it as a "flourish".

It was originally written over a letter as a scribal abbreviation, as a "mark of suspension", shown as a straight line when used with capitals. Thus the commonly used words Anno Domini were frequently abbreviated to Ao Dñi an elevated terminal with a suspension mark placed above the "n". Such mark could denote the omission of one letter or several letters. This saved on the expense of the scribe's labour and the cost of vellum and ink. Mediaeval European charters written in Latin are largely made up of such abbreviated words with suspension marks, with few being given in full, generally only uncommon words. It has since acquired a number of other uses as a diacritic mark or a character in its own right. These are encoded in Unicode at U+0303 ◌̃ combining tilde and U+007E ~ tilde (as a spacing character), and there are additional similar characters for different roles. In lexicography, the latter kind of tilde and the swung dash () are used in dictionaries to indicate the omission of the entry word.[1]

## Common use

This symbol (in English) informally[2] means "approximately", such as: "~30 minutes ago" meaning "approximately 30 minutes ago".[3] It can mean "similar to",[4] including "of the same order of magnitude as",[2] such as: "x ~ y" meaning that x and y are of the same order of magnitude. Another approximation symbol is , meaning "approximately equal to"[3][4][5] the critical difference being the subjective level of accuracy: ≈ indicates a value which can be considered functionally equivalent for a calculation within an acceptable degree of error, whereas ~ is usually used to indicate a larger, possibly significant, degree of error. The tilde is also used to indicate equal to, or approximately equal to by placing it over the "=" symbol.

## Diacritical use

In some languages, the tilde is used as a diacritical mark ( ˜ ) placed over a letter to indicate a change in pronunciation, such as nasalization.

### Pitch

It was first used in the polytonic orthography of Ancient Greek, as a variant of the circumflex, representing a rise in pitch followed by a return to standard pitch.

### Abbreviation

Carta marina showing Finnish economy, with the captions Hic fabricantur naves and Hic fabricantur bombarde abbreviated

Later, it was used to make abbreviations in medieval Latin documents. When an n or m followed a vowel, it was often omitted, and a tilde (i.e., a small n) was placed over the preceding vowel to indicate the missing letter; this is the origin of the use of tilde to indicate nasalization (compare the development of the umlaut as an abbreviation of e.) The practice of using the tilde over a vowel to indicate omission of an n or m continued in printed books in French as a means of reducing text length until the 17th century. It was also used in Portuguese, Catalan and Spanish.

The tilde was also used occasionally to make other abbreviations, such as over the letter q ("") to signify the word que ("that").

### Nasalization

It is also as a small n that the tilde originated when written above other letters, marking a Latin n which had been elided in old Galician-Portuguese. In modern Portuguese it indicates nasalization of the base vowel: mão "hand", from Lat. manu-; razões "reasons", from Lat. rationes. This usage has been adopted in the orthographies of several native languages of South America, such as Guarani and Nheengatu, as well as in the International Phonetic Alphabet (IPA) and many other phonetic alphabets. For example, [ljɔ̃] is the IPA transcription of the pronunciation of the French place-name Lyon.

In Breton, the symbol ñ after a vowel means that the letter n serves only to give the vowel a nasalised pronunciation, without being itself pronounced, as it normally is. For example an gives the pronunciation [ãn] whereas gives [ã].

### Palatal n

Main article: Ñ

The tilded n (ñ, Ñ) developed from the digraph nn in Spanish. In this language, ñ is considered a separate letter called eñe (IPA: [ˈeɲe]), rather than a letter-diacritic combination; it is placed in Spanish dictionaries between the letters n and o. In addition, the word tilde can refer to any diacritic in this language; for example, the acute accent in José is also called a tilde in Spanish.[6] Current languages in which the tilded n (ñ) is used for the palatal nasal consonant /ɲ/ include:

### Tone

In Vietnamese, a tilde over a vowel represents a creaky rising tone (ngã).

### International Phonetic Alphabet

In phonetics, a tilde is used as a diacritic either placed above a letter, below it or superimposed onto the middle of it (see International Phonetic Alphabet → Diacritics):

• A tilde above a letter indicates nasalization, e.g. [ã], [ṽ].
• A tilde superimposed onto the middle of a letter indicates velarization or pharyngealization, e.g. [ɫ], [z̴]. If no precomposed unicode character exists, the unicode character U+0334 ◌̴ combining tilde overlay can be used to generate one.
• A tilde below a letter indicates laryngealisation, e.g. [d̰]. If no precomposed unicode character exists, the unicode character U+0330 ◌̰ combining tilde below can be used to generate one.

### Letter extension

In Estonian, the symbol õ stands for the close-mid back unrounded vowel, and it is considered an independent letter.

### Other uses

Some languages and alphabets use the tilde for other purposes:

• Arabic script: A symbol resembling the tilde (maddah U+0653 ـٓ arabic maddah above) is used over the letter ا (/a/) to become آ, denoting a long /aː/ sound ([ʔæː]).
• Guaraní: The tilded (note that G/g with tilde is not available as a precomposed glyph in Unicode) stands for the velar nasal consonant. Also, the tilded y () stands for the nasalized upper central rounded vowel [ɨ̃]. A small number of other alphabets also use .
• Unicode has a combining vertical tilde character,  ̾  (U+033E). It is used to indicate middle tone in linguistic transcription of certain dialects of the Lithuanian language[7] and for transliteration of the Cyrillic palatalization sign,  ҄  (U+0484).[citation needed]

### Precomposed Unicode characters

The following characters using the tilde as a diacritic exist as precomposed Unicode characters:

Character Code point Name
U+00C3 Ã LATIN CAPITAL LETTER A WITH TILDE
U+00D1 Ñ LATIN CAPITAL LETTER N WITH TILDE
U+00D5 Õ LATIN CAPITAL LETTER O WITH TILDE
U+00E3 ã LATIN SMALL LETTER A WITH TILDE
U+00F1 ñ LATIN SMALL LETTER N WITH TILDE
U+00F5 õ LATIN SMALL LETTER O WITH TILDE
U+0128 Ĩ LATIN CAPITAL LETTER I WITH TILDE
U+0129 ĩ LATIN SMALL LETTER I WITH TILDE
U+0168 Ũ LATIN CAPITAL LETTER U WITH TILDE
U+0169 ũ LATIN SMALL LETTER U WITH TILDE
U+019F Ɵ LATIN CAPITAL LETTER O WITH MIDDLE TILDE
U+022C Ȭ LATIN CAPITAL LETTER O WITH TILDE AND MACRON
U+022D ȭ LATIN SMALL LETTER O WITH TILDE AND MACRON
U+026B ɫ LATIN SMALL LETTER L WITH MIDDLE TILDE
U+1D6C LATIN SMALL LETTER B WITH MIDDLE TILDE
U+1D6D LATIN SMALL LETTER D WITH MIDDLE TILDE
U+1D6E LATIN SMALL LETTER F WITH MIDDLE TILDE
U+1D6F LATIN SMALL LETTER M WITH MIDDLE TILDE
U+1D70 LATIN SMALL LETTER N WITH MIDDLE TILDE
U+1D71 LATIN SMALL LETTER P WITH MIDDLE TILDE
U+1D72 LATIN SMALL LETTER R WITH MIDDLE TILDE
U+1D73 LATIN SMALL LETTER R WITH FISHHOOK AND MIDDLE TILDE
U+1D74 LATIN SMALL LETTER S WITH MIDDLE TILDE
U+1D75 LATIN SMALL LETTER T WITH MIDDLE TILDE
U+1D76 LATIN SMALL LETTER Z WITH MIDDLE TILDE
U+1E1A LATIN CAPITAL LETTER E WITH TILDE BELOW
U+1E1B LATIN SMALL LETTER E WITH TILDE BELOW
U+1E2C LATIN CAPITAL LETTER I WITH TILDE BELOW
U+1E2D LATIN SMALL LETTER I WITH TILDE BELOW
U+1E4C LATIN CAPITAL LETTER O WITH TILDE AND ACUTE
U+1E4D LATIN SMALL LETTER O WITH TILDE AND ACUTE
U+1E4E LATIN CAPITAL LETTER O WITH TILDE AND DIAERESIS
U+1E4F LATIN SMALL LETTER O WITH TILDE AND DIAERESIS
U+1E74 LATIN CAPITAL LETTER U WITH TILDE BELOW
U+1E75 LATIN SMALL LETTER U WITH TILDE BELOW
U+1E78 LATIN CAPITAL LETTER U WITH TILDE AND ACUTE
U+1E79 LATIN SMALL LETTER U WITH TILDE AND ACUTE
U+1E7C LATIN CAPITAL LETTER V WITH TILDE
U+1E7D LATIN SMALL LETTER V WITH TILDE
U+1EAA LATIN CAPITAL LETTER A WITH CIRCUMFLEX AND TILDE
U+1EAB LATIN SMALL LETTER A WITH CIRCUMFLEX AND TILDE
U+1EB4 LATIN CAPITAL LETTER A WITH BREVE AND TILDE
U+1EB5 LATIN SMALL LETTER A WITH BREVE AND TILDE
U+1EBC LATIN CAPITAL LETTER E WITH TILDE
U+1EBD LATIN SMALL LETTER E WITH TILDE
U+1EC4 LATIN CAPITAL LETTER E WITH CIRCUMFLEX AND TILDE
U+1EC5 LATIN SMALL LETTER E WITH CIRCUMFLEX AND TILDE
U+1ED6 LATIN CAPITAL LETTER O WITH CIRCUMFLEX AND TILDE
U+1ED7 LATIN SMALL LETTER O WITH CIRCUMFLEX AND TILDE
U+1EE0 LATIN CAPITAL LETTER O WITH HORN AND TILDE
U+1EE1 LATIN SMALL LETTER O WITH HORN AND TILDE
U+1EEE LATIN CAPITAL LETTER U WITH HORN AND TILDE
U+1EEF LATIN SMALL LETTER U WITH HORN AND TILDE
U+1EF8 LATIN CAPITAL LETTER Y WITH TILDE
U+1EF9 LATIN SMALL LETTER Y WITH TILDE
U+2C62 LATIN CAPITAL LETTER L WITH MIDDLE TILDE

## Similar characters

There are a number of Unicode characters similar to the tilde.

~ U+007E TILDE Same as keyboard tilde. In-line.
˜ U+02DC SMALL TILDE Raised but quite small.
◌̃ U+0303 COMBINING TILDE
͊ U+034A COMBINING NOT TILDE ABOVE Raised, small, with slash through.
◌̰ U+0330 COMBINING TILDE BELOW Used in IPA to indicate creaky voice
◌̴ U+0334 COMBINING TILDE OVERLAY Used in IPA to indicate velarization or pharyngealization
ס֘ U+0598 HEBREW ACCENT ZARQA Hebrew cantillation mark
ס֮ U+05AE HEBREW ACCENT ZINOR Hebrew cantillation mark
◌᷉ U+1DC9 COMBINING ACUTE-GRAVE-ACUTE Used in IPA as a tone mark
U+2053 SWUNG DASH
U+223C TILDE OPERATOR Used in mathematics. In-line. Ends not curved as much.
U+223D REVERSED TILDE In some fonts it is the tilde's simple mirror image; others extend the tips to resemble a
U+223F SINE WAVE
U+2248 ALMOST EQUAL TO
U+301C WAVE DASH Used in Japanese punctuation
U+3030 WAVY DASH
U+FE4B WAVY OVERLINE
U+FE4F WAVY LOW LINE
U+FF5E FULLWIDTH TILDE 50% wider. In-line. Ends not curved much.

## ASCII tilde (U+007E)

 Serif: —~— Sans-serif: —~— Monospace: —~— A tilde between two em dashes in three font families
Raised tilde from a dot matrix printer

Most modern proportional fonts align plain spacing tilde at the same level as dashes, or only slightly upper. This distinguish it from small tilde ˜, which is always raised. But in some monospace fonts, especially used in text user interfaces, ASCII tilde character is raised too. This apparently is a legacy of typewriters, where pairs of similar spacing and combining characters relied on one glyph. Even in line printers' age character repertoires were often not large enough to distinguish between plain tilde, small tilde and combining tilde. Overprinting of a letter by the tilde was a working method of combining a letter.

## Punctuation

The swung dash (~) is used in various ways in punctuation:

### Range

In some languages (though not English), a tilde-like wavy dash may be used as punctuation (instead of an unspaced hyphen or en-dash) between two numbers, to indicate a range rather than subtraction or a hyphenated number (such as a part number or model number).

Before a number the tilde is used to mean "approximately"; "~42" means "approximately 42".[8] Japanese and other East Asian languages almost always use this convention, but it is often done for clarity in some other languages as well.

Chinese uses the wavy dash and full-width em dash interchangeably for this purpose. In English, the tilde is often used to express ranges and model numbers in electronics but rarely in formal grammar or type-set documents, as a wavy dash preceding a number sometimes represents an approximation (see the Mathematics section, below).

### Japanese

Further information: Japanese punctuation

The wave dash (波ダッシュ nami dasshu?) is used for various purposes in Japanese, including to denote ranges of numbers, in place of dashes or brackets, and to indicate origin. The wave dash is also used to separate a title and a subtitle in the same line, as a colon is used in English.

When used in conversations via email or instant messenger it may be used as a sarcasm mark.

The sign is used as a replacement for the chouon, katakana character, in Japanese, extending the final syllable.

In informal messaging in China the tilde is sometimes used at the end of sentences to indicate a semi-excited but not alarmed tone; somewhere between a "." and a "!".[9][citation needed]

#### Unicode and Shift JIS encoding of wave dash

Correct JIS wave dash.
Incorrect Unicode wave dash.

In practice the full-width tilde (全角チルダ zenkaku chiruda?), Unicode U+FF5E, is often used instead of the wave dash (波ダッシュ nami dasshu?), Unicode U+301C, because the Shift JIS code for the wave dash, 0x8160, which is supposed to be mapped to U+301C,[10][11] is not mapped to U+301C but mapped to U+FF5E[12] in code page 932 (Microsoft's code page for Japanese), a widely used extension of Shift JIS, in order to avoid the shape definition error in Unicode: the wave dash glyph in JIS/Shift JIS[13] is identical to the Unicode reference glyph for U+FF5E,[14] while the reference glyph for U+301C[15] was incorrectly turned upside down when Unicode imported the JIS wave dash. In other platforms such as Mac OS and Mac OS X, 0x8160 is correctly mapped to U+301C. It is generally difficult, if not impossible, for users of Japanese Windows to type U+301C, especially in legacy, non-Unicode applications.

Nevertheless, the Japanese wave dash is still formally mapped to U+301C as of JIS X 0213. Those two code points have the identical or very similar glyph in several fonts, reducing the confusion and incompatibility.

## Mathematics

### As an unary operator

A tilde in front of a single quantity can mean "approximately", "about" or "of the same order of magnitude as".

In written mathematical logic, the tilde represents negation: "~p" means "not p", where "p" is a proposition. Modern use has been replacing the tilde with the negation symbol (¬) for this purpose, to avoid confusion with equivalence relations.

### As a binary operator

In the 1800s x ~ y was the 'difference' operator, and thus meant | xy |, (the absolute value of xy).[citation needed]

### As an equivalence operator

In mathematics, the tilde operator (Unicode U+223C), sometimes called "twiddle", is often used to denote an equivalence relation between two objects. Thus "x ~ y" means "x is equivalent to y". It is a weaker statement than stating that x equals y. The expression "x ~ y" is sometimes read aloud as "x twiddles y", perhaps as an analogue to the verbal expression of "x = y".[16]

The tilde can indicate approximate equality in a variety of ways. It can be used to denote the asymptotic equality of two functions. For example, f (x) ~ g(x) means that limx → ∞ f( x) ∕ g(x) = 1.[2]

A tilde is also used to indicate "approximately equal to" (e.g. 1.902 ~= 2). This usage probably developed as a typed alternative to the libra symbol used for the same purpose in written mathematics, which is an equal sign with the upper bar replaced by a bar with an upward hump, bump, or loop in the middle (♎) or, sometimes, a tilde (≃). The symbol "≈" is also used for this purpose.

In physics and astronomy, a tilde can be used between two expressions (e.g. h ~ 10−34 J s) to state that the two are of the same order of magnitude.[2]

In statistics and probability theory, the tilde means "is distributed as";[2] see random variable.

A tilde can also be used to represent geometric similarity (e.g. ABC ~ ∆DEF, meaning triangle ABC is similar to DEF). A triple tilde () is often used to show congruence, an equivalence relation in geometry.

### As an accent

The symbol "$\tilde{f}$" is often pronounced "eff twiddle" or, particularly in American English, "eff wiggle".[17] This can be used to denote the Fourier transform of f, or a lift of f, and can have a variety of other meanings depending on the context.

A tilde placed below a letter in mathematics can represent a vector quantity (e.g. $(x_1, x_2, x_3, \ldots, x_n) = \underset{^\sim}{\mathbf x}$).

In statistics and probability theory, a tilde placed on top of a variable is sometimes used to represent the median of that variable; thus $\tilde{\mathbf y}$ would indicate the median of the variable $\mathbf y$. A tilde over the letter n ($\tilde{n}$) is sometimes used to indicate the harmonic mean.

## Physics

Often in physics, one can consider an equilibrium solution to an equation, and then a perturbation to that equilibrium. For the variables in the original equation (for instance $X$) a substitution $X\to x+\tilde{x}$ can be made, where $x$ is the equilibrium part and $\tilde{x}$ is the perturbed part.

## Economics

For relations involving preference, economists sometimes use the tilde to represent indifference between two or more bundles of goods. For example, to say that a consumer is indifferent between bundles x and y, an economist would write x ~ y.

## Electronics

It can approximate the sine wave symbol (, U+223F), which is used in electronics to indicate alternating current, in place of +, −, or for direct current.

## Computing

### Games

In many games, the tilde key (on U.S. English keyboards) is used to open the console. This is true for games such as Battlefield 3, Half-Life, Halo CE, Quake, Half-Life 2, Soldier of Fortune II: Double Helix, Unreal, Counter-Strike, Crysis, Morrowind, Oblivion, Skyrim, Fallout: New Vegas, Fallout 3, RuneScape, and others based on the Quake engine or Source engine.

It is sometimes used in Rogue-like games to represent water or snakes.

### Other uses

Computer programmers use the tilde in various ways and sometimes call the symbol (as opposed to the diacritic) a squiggle, squiggly, or twiddle. According to the Jargon File, other synonyms sometimes used in programming include not, approx, wiggle, enyay (after eñe) and (humorously) sqiggle . It is used in many languages as a binary inversion operator, swapping a number's binary 1's and 0's for example ~18 (binary ~1010) is equal to 9 (binary 0101).

In Perl 6, "~~" is used instead of "=~".

## Juggling notation

In the juggling notation system Beatmap, tilde can be added to either "hand" in a pair of fields to say "cross the arms with this hand on top". Mills Mess is thus represented as (~2x,1)(1,2x)(2x,~1)*.[27]

## Keyboards

Where a tilde is on the keyboard depends on the computer's language settings according to the following chart. On many keyboards it is primarily available through a dead key that makes it possible to produce a variety of precomposed characters with the diacritic.[citation needed] In that case, a single tilde can typically be inserted with the dead key followed by the space bar, or alternatively by striking the dead key twice in a row.

To insert a tilde with the dead key, it is often necessary to simultaneously hold down the Alt Gr key. On the keyboard layouts that include an Alt Gr key, it typically takes the place of the right-hand Alt key. With a Macintosh either of the Alt/Option keys function similarly.

In the US and European Windows systems, the Alt code for a single tilde is 126.

Keyboard Insert a single tilde (~) Insert a precomposed character with tilde (e.g. ã)
Arabic (Saudi) Shift+ذّ
Croatian Alt Gr+1
Danish Alt Gr+¨ followed by Space Alt Gr+¨ followed by the relevant letter
Dvorak Alt Gr+= followed by Space, or

Alt Gr+ Shift+' followed by Space

Alt Gr+= followed by the relevant letter, or

Alt Gr+ Shift+' followed by the relevant letter

English (Australia) Shift+
English (Canada) Shift+
English (UK) Shift+#
English (US) Shift+ Ctrl+~ followed by the relevant letter
Faroese Alt Gr+ð followed by Space Alt Gr+ð followed by the relevant letter
Finnish Alt Gr+¨ followed by Space, or

Alt Gr+¨¨

Alt Gr+¨ followed by the relevant letter
French (Canada) Alt Gr+ç followed by Space, or

Alt Gr+çç

Alt Gr+ç followed by the relevant letter
French (France) Alt Gr+é followed by Space, or

Alt Gr+éé
Alt+n (on Mac OS X)

Alt Gr+é followed by the relevant letter
French (Switzerland) Alt Gr+^ followed by Space, or

Alt Gr+^^

Alt Gr+^ followed by the relevant letter
German (Germany) Alt Gr++
German (Switzerland) Alt Gr+^ followed by Space, or

Alt Gr+^^

Alt Gr+^ followed by the relevant letter
Hebrew (Israel) Shift+~ Ctrl+ Shift+~ followed by the relevant letter
Hindi (India) Alt Gr+ Shift+ the key to the left of 1
Hungarian Alt Gr+1
Icelandic Alt Gr+' (the same key as ?)
Italian Alt+5 (on Mac OS X)

Alt Gr+ì (on Linux)

Norwegian Alt Gr+¨ followed by Space, or

Alt Gr+¨¨.

On Mac: Option+ Command+¨ followed by Space.

Alt Gr+¨ followed by the relevant letter.

On Mac: Option+ Command+¨ followed by the relevant letter.

Polish Shift+ followed by Space,

or Shift+`

The dead key is not generally used for inserting characters with tilde; when followed by {a|c|e|l|n|o|s|x|z}, it results in {ą|ć|ę|ł|ń|ó|ś|ź|ż} instead.
Portuguese ~ followed by Space ~ followed by the relevant letter
Slovak Alt Gr+1
Spanish (Spain) Alt Gr+4 followed by Space, or

Alt Gr+44

Alt Gr+4 followed by the relevant letter
Spanish (Latin America) Alt Gr++
Swedish Alt Gr+¨ followed by Space, or

Alt Gr+¨¨

Alt Gr+¨ followed by the relevant letter
Turkish Alt Gr+ü followed by Space, or

Alt Gr+üü

Alt Gr+ü followed by the relevant letter

## References

1. ^ "Swung dash", WordNet (search) (3.0 ed.)
2. "Tilde". MathWorld. Wolfram. 3 November 2011. Retrieved 11 November 2011.
3. ^ a b "All Elementary Mathematics – Mathematical symbols dictionary". Bymath. Retrieved 11 November 2011.
4. ^ a b Quinn, Liam. "HTML 4.0 Entities for Symbols and Greek Letters". HTML help. Retrieved 11 November 2011.
5. ^ "Math Symbols... Those Most Valuable and Important: Approximately Equal Symbol". Solving Math problems. 20 September 2010. Retrieved 11 November 2011.
6. ^ Ortografía de la lengua española. Madrid: Real Academia Española. 2010. p. 279. ISBN 978-84-670-3426-4.
7. ^ Lithuanian Standards Board (LST), proposal for a zigazag diacritic.
8. ^ "Other symbols", Abstract Math.
9. ^ Chinese Friends
10. ^ "Appendix 1: Shift_JIS-2004 vs Unicode mapping table", JIS X 0213:2004, X 0213.
11. ^ Shift-JIS to Unicode, Unicode.
12. ^ "Windows 932_81". Microsoft. Retrieved 2010-07-30.
13. ^ "Microsoft Word – 233cover_rev.doc" (PDF). JP: IPSJ. Retrieved 2010-07-30.
14. ^ UFF00 (chart), Unicode.
15. ^ U3000 (chart), Unicode.
16. ^ Derbyshire, J (2004), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, New York: Penguin.
17. ^ Choy, Stephen TL; Jesudason, Judith Packer; Lee, Peng Yee (1988). Proceedings of the Analysis Conference, Singapore 1986. Elsevier. Retrieved 11 November 2011.
18. ^ "Tilde expansion", C Library Manual, The GNU project, retrieved 4 July 2010.
19. ^ "Module mod_userdir", HTTP Server Documentation (version 2.0 ed.), The Apache foundation, retrieved 4 July 2010.
20. ^ RFC 3986, IETF.