The hyphen ‐ is a punctuation mark used to join words and to separate syllables of a single word. The use of hyphens is called hyphenation.^[1] Son-in-law is an example of a hyphenated word.

The hyphen is sometimes confused with dashes (en dash – and em dash — and others), which are longer, or with the minus sign −, which is also longer and usually higher up to match the crossbar in the plus sign +.

As an orthographic concept, the hyphen is a single entity. In character encoding it is represented by any of several characters and glyphs, including the Unicode hyphen (shown at the top of the infobox on this page), the hyphen-minus, the soft hyphen, and the nonbreaking hyphen. The character most often used to represent a hyphen (and the one produced by the key on a keyboard) is called the "hyphen-minus" by Unicode, deriving from the original ASCII standard, where it was called "hyphen (minus)".^[2]

Etymology[edit]

The word is derived from Ancient Greek ὑφ' ἕν (huph' hén), contracted from ὑπό ἕν (hypó hén), "in one" (literally "under one").^[3]^[4] An (ἡ) ὑφέν ((he) hyphén) was an undertie-like ‿ sign written below two adjacent letters to indicate that they belong to the same word when it was necessary to avoid ambiguity, before word spacing was practiced.

History[edit]

First page of the first volume: the epistle of St Jerome to Paulinus from the University of Texas copy. The page has 40 lines.

The first known documentation of the hyphen is in the grammatical works of Dionysius Thrax. At the time hyphenation was joining two words that would otherwise be read separately by a low tie mark between the two words.^[5] In Greek these marks were known as enotikon, officially romanized as a hyphen.^[6]

With the introduction of letter spacing in the Middle Ages, the hyphen, still written beneath the text, reversed its meaning. Scribes used the mark to connect two words that had been incorrectly separated by a space. This era also saw the introduction of the marginal hyphen, for words broken across lines.^[7]

The modern format of the hyphen originated with Johannes Gutenberg of Mainz, Germany, c. 1455 with the publication of his 42-line Bible. His tools did not allow for a sublinear hyphen, and he thus moved it to the middle of the line.^[8] Examination of an original copy on vellum (Hubay index #35) in the U. S. Library of Congress shows that Gutenberg's movable type was set justified in a uniform style, 42 equal lines per page. The Gutenberg printing press required words made up of individual letters of type to be held in place by a surrounding nonprinting rigid frame. Gutenberg solved the problem of making each line the same length to fit the frame by inserting a hyphen as the last element at the right-side margin. This interrupted the letters in the last word, requiring the remaining letters be carried over to the start of the line below. His double hyphen appears throughout the Bible as a short, double line inclined to the right at a 60-degree angle: ⸗^{[citation needed]}

Use in English[edit]

The English language does not have definitive hyphenation rules,^[9] though various style guides provide detailed usage recommendations and have a significant amount of overlap in what they advise. Hyphens are mostly used to break single words into parts or to join ordinarily separate words into single words. Spaces are not placed between a hyphen and either of the elements it connects except when using a suspended or "hanging" hyphen that stands in for a repeated word (e.g., nineteenth- and twentieth-century writers). Style conventions that apply to hyphens (and dashes) have evolved to support ease of reading in complex constructions; editors often accept deviations if they aid rather than hinder easy comprehension.

The use of the hyphen in English compound nouns and verbs has, in general, been steadily declining. Compounds that might once have been hyphenated are increasingly left with spaces or are combined into one word. Reflecting this changing usage, in 2007, the sixth edition of the Shorter Oxford English Dictionary removed the hyphens from 16,000 entries, such as fig-leaf (now fig leaf), pot-belly (now pot belly), and pigeon-hole (now pigeonhole).^[10] The increasing prevalence of computer technology and the advent of the Internet have given rise to a subset of common nouns that might have been hyphenated in the past (e.g., toolbar, hyperlink, and pastebin).

Despite decreased use, hyphenation remains the norm in certain compound-modifier constructions and, among some authors, with certain prefixes (see below). Hyphenation is also routinely used as part of syllabification in justified texts to avoid unsightly spacing (especially in columns with narrow line lengths, as when used with newspapers).

Separating[edit]

Justification and line-wrapping[edit]

When flowing text, it is sometimes preferable to break a word into two so that it continues on another line rather than moving the entire word to the next line. The word may be divided at the nearest break point between syllables (syllabification) and a hyphen inserted to indicate that the letters form a word fragment, rather than a full word. This allows more efficient use of paper, allows flush appearance of right-side margins (justification) without oddly large word spaces, and decreases the problem of rivers. This kind of hyphenation is most useful when the width of the column (called the "line length" in typography) is very narrow. For example:

Justified text without hyphenation		Justified text with hyphenation
We, therefore, the representatives of the United States of America ...		We, therefore, the represen- tatives of the United States of America ...

Rules (or guidelines) for correct hyphenation vary between languages, and may be complex, and they can interact with other orthographic and typesettingpractices. Hyphenation algorithms, when employed in concert^{[clarification needed]} with dictionaries, are sufficient for all but the most formal texts.

It may be necessary to distinguish an incidental line-break hyphen from one integral to a word being mentioned (as when used in a dictionary) or present in an original text being quoted (when in a critical edition), not only to control its word wrap behavior (which encoding handles with hard and soft hyphenshaving the same glyph) but also to differentiate appearance (with a different glyph). Webster's Third New International Dictionary^[11] and the Chambers Dictionary^[12] use a double hyphen for integral hyphens and a single hyphen for line-breaks, whereas Kromhout's Afrikaans–English dictionary uses the opposite convention.^[13] The Concise Oxford Dictionary (fifth edition) suggested repeating an integral hyphen at the start of the following line.^[14]

Prefixes and suffixes[edit]

Prefixes (such as de-, pre-, re-, and non-^[15]) and suffixes (such as -less, -like, -ness, and -hood) are sometimes hyphenated, especially when the unhyphenated spelling resembles another word or when the affixation is deemed misinterpretable, ambiguous, or somehow "odd-looking" (for example, having two consecutive monographs that look like the digraphs of English, like e+a, e+e, or e+i). However, the unhyphenated style, which is also called closed up or solid, is usually preferred, particularly when the derivative has been relatively familiarized or popularized through extensive use in various contexts. As a rule of thumb, affixes are not hyphenated unless the lack of a hyphen would hurt clarity.

The hyphen may be used between vowel letters (e.g., ee, ea, ei) to indicate that they do not form a digraph. Some words have both hyphenated and unhyphenated variants: de-escalate/deescalate, co-operation/cooperation, re-examine/reexamine, de-emphasize/deemphasize, and so on. Words often lose their hyphen as they become more common, such as email instead of e-mail. When there are tripled letters, the hyphenated variant of these words is often more common (as in shell-like instead of shelllike).

Closed-up style is avoided in some cases: possible homographs, such as recreation (fun or sport) versus re-creation (the act of creating again), retreat(turn back) versus re-treat (give therapy again), and un-ionized (not in ion form) versus unionized (organized into trade unions); combinations with propernouns or adjectives (un-American, de-Stalinisation);^[16]^[17] acronyms (anti-TNF antibody, non-SI units); or numbers (pre-1949 diplomacy, pre-1492 cartography). Although proto-oncogene is still hyphenated by both Dorland's and Merriam-Webster's Medical, the solid (that is, unhyphenated) styling (protooncogene) is a common variant, particularly among oncologists and geneticists.^{[citation needed]}

A diaeresis may also be used in a like fashion, either to separate and mark off monographs (as in coöperation) or to signalize a vocalic terminal e (for example, Brontë). This use of the diaeresis peaked in the late 19th and early 20th centuries, but it was never applied extensively across the language: only a handful of diaereses, including coöperation and Brontë, are encountered with any appreciable frequency in English; thus reëxamine, reïterate, deëmphasize, etc. are seldom encountered. In borrowings from Modern French, whose orthography utilizes the diaeresis as a means to differentiate graphemes, various English dictionaries list the dieresis as optional (as in naive and naïve) despite the juxtaposition of a and i.^{[citation needed]}

Syllabification and spelling[edit]

Hyphens are occasionally used to denote syllabification, as in syl-la-bi-fi-ca-tion. Various British and North American dictionaries use an interpunct, sometimes called a "middle dot" or "hyphenation point", for this purpose, as in syl·la·bi·fi·ca·tion. This allows the hyphen to be reserved only for places where a hard hyphen is intended (for example, self-con·scious, un·self-con·scious, long-stand·ing). Similarly, hyphens may be used to indicate how a word is being or should be spelled. For example, W-O-R-D spells "word".

In nineteenth-century American literature, hyphens were also used irregularly to divide syllables in words from indigenous North American languages, without regard for etymology or pronunciation,^[18] such as "Shuh-shuh-gah" (from Ojibwe zhashagi, "blue heron") in The Song of Hiawatha.^[19] This usage is now rare and proscribed, except in some place names such as Ah-gwah-ching.

Joining[edit]

Compound modifiers[edit]

Compound modifiers are groups of two or more words that jointly modify the meaning of another word. When a compound modifier other than an adverb–adjective combination appears before a term, the compound modifier is often hyphenated to prevent misunderstanding, such as in American-football player or little-celebrated paintings. Without the hyphen, there is potential confusion about whether the writer means a "player of American football" or an "American player of football" and whether the writer means paintings that are "little celebrated" or "celebrated paintings" that are little.^[20] Compound modifiers can extend to three or more words, as in ice-cream-flavored candy, and can be adverbial as well as adjectival (spine-tinglingly frightening). However, if the compound is a familiar one, it is usually unhyphenated. For example, some style guides prefer the construction high school students, to high-school students.^[21]^[22] Although the expression is technically ambiguous ("students of a high school"/"school students who are high"), it would normally be formulated differently if other than the first meaning were intended. Noun–noun compound modifiers may also be written without a hyphen when no confusion is likely: grade point average and department store manager.^[22]

When a compound modifier follows the term to which it applies, a hyphen is typically not used if the compound is a temporary compound. For example, "that gentleman is well respected", not "that gentleman is well-respected"; or "a patient-centered approach was used" but "the approach was patient centered."^[23] But permanent compounds, found as headwords in dictionaries, are treated as invariable, so if they are hyphenated in the cited dictionary, the hyphenation will be used in both attributive and predicative positions. For example, "A cost-effective method was used" and "The method was cost-effective" (cost-effective is a permanent compound that is hyphenated as a headword in various dictionaries). When one of the parts of the modifier is a proper noun or a proper adjective, there is no hyphen (e.g., "a South American actor").^[24]

When the first modifier in a compound is an adverb ending in -ly (e.g., "a poorly written novel"), various style guides advise no hyphen.^[24]^{[additional citation(s) needed]} However, some do allow for this use. For example, The Economist Style Guide advises: "Adverbs do not need to be linked to participles or adjectives by hyphens in simple constructions ... Less common adverbs, including all those that end -ly, are less likely to need hyphens."^[25] In the 19th century, it was common to hyphenate adverb–adjective modifiers with the adverb ending in -ly (e.g., "a craftily-constructed chair"). However, this has become rare. For example, wholly owned subsidiary and quickly moving vehicle are unambiguous, because the adverbs clearly modify the adjectives: "quickly" cannot modify "vehicle".

However, if an adverb can also function as an adjective, then a hyphen may be or should be used for clarity, depending on the style guide.^[17] For example, the phrase more-important reasons ("reasons that are more important") is distinguished from more important reasons ("additional important reasons"), where more is an adjective. Similarly, more-beautiful scenery (with a mass-noun) is distinct from more beautiful scenery. (In contrast, the hyphen in "a more-important reason" is not necessary, because the syntax cannot be misinterpreted.) A few short and common words—such as well, ill, little, and much—attract special attention in this category.^[25] The hyphen in "well-[past_participled] noun", such as in "well-differentiated cells", might reasonably be judged superfluous (the syntax is unlikely to be misinterpreted), yet plenty of style guides call for it. Because early has both adverbial and adjectival senses, its hyphenation can attract attention; some editors, due to comparison with advanced-stage disease and adult-onset disease, like the parallelism of early-stage disease and early-onset disease. Similarly, the hyphen in little-celebrated paintings clarifies that one is not speaking of little paintings.

Hyphens are usually used to connect numbers and words in modifying phrases. Such is the case when used to describe dimensional measurements of weight, size, and time, under the rationale that, like other compound modifiers, they take hyphens in attributive position (before the modified noun),^[26]although not in predicative position (after the modified noun). This is applied whether numerals or words are used for the numbers. Thus 28-year-oldwoman and twenty-eight-year-old woman or 32-foot wingspan and thirty-two-foot wingspan, but the woman is 28 years old and a wingspan of 32 feet.^[a]However, with symbols for SI units (such as m or kg)—in contrast to the names of these units (such as metre or kilogram)—the numerical value is always separated from it with a space: a 25 kg sphere. When the unit names are spelled out, this recommendation does not apply: a 25-kilogram sphere, a roll of 35-millimetre film.^[27]

In spelled-out fractions, hyphens are usually used when the fraction is used as an adjective but not when it is used as a noun: thus two-thirds majority^[a]and one-eighth portion but I drank two thirds of the bottle or I kept three quarters of it for myself.^[28] However, at least one major style guide^[26] hyphenates spelled-out fractions invariably (whether adjective or noun).

In English, an en dash, –, sometimes replaces the hyphen in hyphenated compounds if either of its constituent parts is already hyphenated or contains a space (for example, San Francisco–area residents, hormone receptor–positive cells, cell cycle–related factors, and public-school–private-school rivalries).^[29] A commonly used alternative style is the hyphenated string (hormone-receptor-positive cells, cell-cycle-related factors). (For other aspects of en dash–versus–hyphen use, see Dash § En dash.)

Object–verbal-noun compounds[edit]

When an object is compounded with a verbal noun, such as egg-beater (a tool that beats eggs), the result is sometimes hyphenated. Some authors do this consistently, others only for disambiguation; in this case, egg-beater, egg beater, and eggbeater are all common.

An example of an ambiguous phrase appears in they stood near a group of alien lovers, which without a hyphen implies that they stood near a group of lovers who were aliens; they stood near a group of alien-lovers clarifies that they stood near a group of people who loved aliens, as "alien" can be either an adjective or a noun. On the other hand, in the phrase a hungry pizza-lover, the hyphen will often be omitted (a hungry pizza lover), as "pizza" cannot be an adjective and the phrase is therefore unambiguous.

Similarly, a man-eating shark is nearly the opposite of a man eating shark; the first refers to a shark that eats people, and the second to a man who eats shark meat. A government-monitoring program is a program that monitors the government, whereas a government monitoring program is a government program that monitors something else.

Personal names[edit]

Some married couples compose a new surname (sometimes referred to as a double-barrelled name) for their new family by combining their two surnames with a hyphen. Jane Doe and John Smith might become Jane and John Smith-Doe, or Doe-Smith, for instance. In some countries only the woman hyphenates her birth surname, appending her husband's surname.

With already-hyphenated names, some parts are typically dropped. For example, Aaron Johnson and Samantha Taylor-Wood became Aaron Taylor-Johnson and Sam Taylor-Johnson. Not all hyphenated surnames are the result of marriage. For example Julia Louis-Dreyfus is a descendant of Louis Lemlé Dreyfus whose son was Léopold Louis-Dreyfus.

Other compounds[edit]

Connecting hyphens are used in a large number of miscellaneous compounds, other than modifiers, such as in lily-of-the-valley, cock-a-hoop, clever-clever, tittle-tattle and orang-utan. Use is often dictated by convention rather than fixed rules, and hyphenation styles may vary between authors; for example, orang-utan is also written as orangutan or orang utan, and lily-of-the-valley may be hyphenated or not.

Suspended hyphens[edit]

A suspended hyphen (also called a suspensive hyphen or hanging hyphen, or less commonly a dangling or floating hyphen) may be used when a single base word is used with separate, consecutive, hyphenated words that are connected by "and", "or", or "to". For example, short-term and long-term plans may be written as short- and long-term plans. This usage is now common and specifically recommended in some style guides.^[22] Suspended hyphens are also used, though less commonly, when the base word comes first, such as in "investor-owned and -operated". Uses such as "applied and sociolinguistics" (instead of "applied linguistics and sociolinguistics") are frowned upon; the Indiana University style guide uses this example and says "Do not 'take a shortcut' when the first expression is ordinarily open" (i.e., ordinarily two separate words).^[22] This is different, however, from instances where prefixes that are normally closed up (styled solidly) are used suspensively. For example, preoperative and postoperative becomes pre- and postoperative(not pre- and post-operative) when suspended. Some editors prefer to avoid suspending such pairs, choosing instead to write out both words in full.^[26]

Other uses[edit]

A hyphen may be used to connect groups of numbers, such as in dates (see § Usage in date notation), telephone numbers or sports scores.

It can also be used to indicate a range of values, although many styles prefer an en dash (see Dash § En dash §§ Ranges of values).

It is sometimes used to hide letters in words (filleting for redaction or censoring), as in "G-d", although an en dash can be used as well ("G–d").^[30]

It is often used in reduplication.^[31]

Due to their similar appearances, hyphens are sometimes mistakenly used where an en dash or em dash would be more appropriate.^[32]

Varied meanings[edit]

Some stark examples of semantic changes caused by the placement of hyphens to mark attributive phrases:

Disease-causing poor nutrition is poor nutrition that causes disease.
- Disease causing poor nutrition is a disease that causes poor nutrition.
A hard-working man is a man who works hard.
- A hard working man is a working man who is tough.
A man-eating shark is a shark that eats humans.
- A man eating shark is a man who is eating shark meat.
Three-hundred-year-old trees are an indeterminate number of trees that are each 300 years old.
- Three hundred-year-old trees are three trees that are each 100 years old.
- Three hundred year-old trees are 300 trees that are each a year old.

Use in computing[edit]

Hyphen-minuses[edit]

In the ASCII character encoding, the hyphen (or minus) is character 45₁₀.^[33] As Unicode is identical to ASCII (the 1967 version) for all encodings up to 127₁₀, the number 45₁₀ (2D₁₆) is also assigned to this character in Unicode, where it is denoted as U+002D - HYPHEN-MINUS.^[34] Unicode has, in addition, other encodings for minus and hyphen characters: U+2212 − MINUS SIGN and U+2010 ‐ HYPHEN, respectively. The unambiguous hyphen at U+2010 is generally inconvenient to enter on most keyboards and the glyphs for this hyphen and the hyphen-minus are identical in most fonts (Lucida Sans Unicodeis one of the few exceptions). Consequently, use of the hyphen-minus as the hyphen character is very common. Even the Unicode Standard regularly uses the hyphen-minus rather than this hyphen.

The hyphen-minus has limited use in indicating subtraction; for example, compare 4+3−2=5 (minus) and 4+3-2=5 (hyphen-minus) — in most fonts, the hyphen-minus will not have the optimal width, thickness, or vertical position, whereas the minus character will. Nevertheless, in many spreadsheet and programming applications the hyphen-minus must be typed to indicate subtraction, as use of the Unicode minus sign will produce an error.

The hyphen-minus is often used instead of dashes or minus signs in situations where the latter characters are unavailable (such as type-written or ASCII-only text), where they take effort to enter (via dialog boxes or multi-key keyboard shortcuts), or when the writer is unaware of the distinction. Consequently, some writers use two or three hyphen-minuses (-- or ---) to represent an em dash.^[35] In the TeX typesetting languages, a single hyphen-minus (-) renders a hyphen, a single hyphen-minus in math mode ( $-$ ) renders a minus sign, two hyphen-minuses (--) renders an en dash, and three hyphen-minuses (---) renders an em dash.

The hyphen-minus character is also often used when specifying command-line options. The character is usually followed by one or more letters that indicate specific actions. Typically it is called a dash or switch in this context. Various implementations of the getopt function to parse command-line options additionally allow the use of two hyphen-minus characters, --, to specify long option names that are more descriptive than their single-letter equivalents. Another use of hyphens is that employed by programs written with pipelining in mind: a single hyphen may be recognized in lieu of a filename, with the hyphen then serving as an indicator that a standard stream, instead of a file, is to be worked with.

Soft and hard hyphens[edit]

Although software (hyphenation algorithms) can often automatically make decisions on when to hyphenate a word at a line break, it is also sometimes useful for the user to be able to insert cues for those decisions (which are dynamic in the online medium, given that text can be reflowed). For this purpose, the concept of a soft hyphen (discretionary hyphen, optional hyphen) was introduced, allowing such manual specification of a place where a hyphenated break is allowed but not forced. That is, it does not force a line break in an inconvenient place when the text is later reflowed.

Soft hyphens are inserted into the text at the positions where hyphenation may occur. It can be a tedious task to insert the soft hyphens by hand, and tools using hyphenation algorithms are available that do this automatically. Current modules^[which?] of the Cascading Style Sheets (CSS) standard provide language-specific hyphenation dictionaries.

Some (OpenType) fonts will change the character at the end of a word. An example is a font that places a long s, 'ſ ', everywhere except at the end of a word, where a round s, 's', is used. A soft hyphen can be used to change the previous letter to a round s in the middle of a word. For example, 'prinſeſſen' can be corrected by inserting a soft hyphen between the 'ſ 's: 'prinſeſ-ſen' becomes 'prinſesſen' (which is correct in Norwegian).

In contrast, a hyphen that is always displayed and printed is called a "hard hyphen". This can be a Unicode hyphen, a hyphen-minus, or a nonbreaking hyphen (see below). Confusingly, the term is sometimes limited to nonbreaking hyphens.^{[citation needed]}

Nonbreaking hyphens[edit]

The non-breaking hyphen, nonbreaking hyphen, or no-break hyphen looks identical to the regular hyphen, but word processors treat it as a letter so that the hyphenated word will not be divided at the hyphen should this fall at what would be the end of a line of text; instead, either the whole hyphenated word will remain in full at the end of the line or it will go in full to the beginning of the next line. The nonbreaking space exists for similar reasons.

The word segmentation rules of most text systems consider a hyphen to be a word boundary and a valid point at which to break a line when flowing text. However, this is not always desirable behavior, especially when it could lead to ambiguity (e.g. retreat and re‑treat would be indistinguishable with a line break after re), or in languages other than English (e.g., a line break at the hyphen in Irish an t‑athair or Romanian s‑a would be undesirable). The nonbreaking hyphen addresses this need.

Use in date notation[edit]

Use of hyphens to delineate the parts of a written date (rather than the slashes used conventionally in Anglophone countries) is specified in the international standard ISO 8601. Thus, for example, 1789-07-14 is the standard way of writing the date of Bastille Day. This standard has been transposed as European Standard EN 28601 and has been incorporated into various national typographic style guides (e.g., DIN 5008 in Germany). Now all official European Union (and many member state) documents use this style. This is also the typical date format used in large parts of Europe and Asia, although sometimes with other separators than the hyphen.

This method has gained influence within North America, as most common computer file systems make the use of slashes in file names difficult or impossible. DOS, OS/2 and Windows use / to introduce and separate switches to shell commands, and on both Windows and Unix-like systems slashes in a filename introduce subdirectories which may not be desirable. Besides encouraging use of dashes, the Y-M-D order and zero-padding of numbers less than 10 are also copied from ISO 8601 to make the filenames sort by date order.

Unicode[edit]

Unicode has multiple hyphen characters:^[36]

U+002D - HYPHEN-MINUS, a character of multiple uses
U+00AD SOFT HYPHEN (HTML )^[b]
U+2010 ‐ UNICODE HYPHEN (HTML &dash; or &hyphen;)^[c]
U+2011 ‑ NON-BREAKING HYPHEN
U+2E5D ⹝ OBLIQUE HYPHEN for medieval texts^[38]

And in non-Latin scripts:^[36]

U+058A ֊ ARMENIAN HYPHEN
U+05BE ־ HEBREW PUNCTUATION MAQAF
U+1806 ᠆ MONGOLIAN TODO SOFT HYPHEN
U+1B60 ᭠ BALINESE PAMENENG (used only as a line-breaking hyphen)
U+2E17 ⸗ DOUBLE OBLIQUE HYPHEN (used in ancient Near-Eastern linguistics and in blackletter typefaces)
U+30FB ・ KATAKANA MIDDLE DOT (has the Unicode property of "Hyphen" despite its name)
U+FE63 ﹣ SMALL HYPHEN-MINUS (compatibility character for a small hyphen-minus, used in East Asian typography)
U+FF0D － FULLWIDTH HYPHEN-MINUS (compatibility character for a wide hyphen-minus, used in East Asian typography)
U+FF65 ･ HALFWIDTH KATAKANA MIDDLE DOT (compatibility character for a wide katakana middle dot, has the Unicode property of "Hyphen" despite its name)

Unicode distinguishes the hyphen from the general interpunct. The characters below do not have the Unicode property of "Hyphen" despite their names:^[36]

U+1400 ᐀ CANADIAN SYLLABICS HYPHEN
U+2027 ‧ HYPHENATION POINT
U+2043 ⁃ HYPHEN BULLET (HTML &hybull;)
U+2E1A ⸚ HYPHEN WITH DIAERESIS
U+2E40 ⹀ DOUBLE HYPHEN
U+30A0 ゠ KATAKANA-HIRAGANA DOUBLE HYPHEN
U+10EAD 𐺭 YEZIDI HYPHENATION MARK

(See interpunct and bullet (typography) for more round characters.)

Arithmetic

‐
‐
Unicode hyphen

From Wikipedia, the free encyclopedia

Arithmetic (from Ancient Greek ἀριθμός (arithmós) 'number', and τική [τέχνη] (tikḗ [tékhnē]) 'art, craft') is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms,^{[disputed – discuss]} which are highly important to the field of mathematical logic today.

History[edit]

The prehistory of arithmetic is limited to a small number of artifacts that may indicate the conception of addition and subtraction; the best-known is the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed.^[1]

The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations: addition, subtraction, multiplication, and division, as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board (or the Roman abacus) to obtain the results.

Early number systems that included positional notation were not decimal; these include the sexagesimal (base 60) system for Babylonian numerals and the vigesimal (base 20) system that defined Maya numerals. Because of the place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.

The continuous historical development of modern arithmetic starts with the Hellenistic period of ancient Greece; it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. Nicomachus is an example of this viewpoint, using the earlier Pythagorean approach to numbers and their relationships to each other in his work, Introduction to Arithmetic.

Greek numerals were used by Archimedes, Diophantus, and others in a positional notation not very different from modern notation. The ancient Greeks lacked a symbol for zero until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the units place, one for the tens place, and one for the hundreds. For the thousands place, they would reuse the symbols for the units place, and so on. Their addition algorithm was identical to the modern method, and their multiplication algorithm was only slightly different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method of successive approximation because, once computed, a digit does not change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60 instead of negative powers of 10 for the fractional part 0.934.^[2]

The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. Since they also lacked a symbol for zero, they had one set of symbols for the units place and a second set for the tens place. For the hundreds place, they then reused the symbols for the units place, and so on. Their symbols were based on ancient counting rods. The exact time when the Chinese started calculating with positional representation is unknown, though it is known that the adoption started before 400 BC.^[3] The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui and dates back to the 2nd century BC.

The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing 0. This allowed the system to consistently represent both large and small integers—an approach that eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system into his work and experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number and determined the results for multiplication, division, addition, and subtraction of zero and all other numbers—except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht (650 AD) said, "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols."^[4] The Arabs also learned this new method and called it hesab.

Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa (Fibonacci) was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He wrote, "The method of the Indians (Latin Modus Indorum) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero".^[5]

In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

The flourishing of algebra in the medieval Islamic world, and also in Renaissance Europe, was an outgrowth of the enormous simplification of computation through decimal notation.

Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of abaci. More recent examples include slide rules, nomograms and mechanical calculators, such as Pascal's calculator. At present, they have been supplanted by electronic calculators and computers.

Arithmetic operations[edit]

The basic arithmetic operations are addition, subtraction, multiplication and division, although arithmetic also includes more advanced operations, such as manipulations of percentages,^[6] square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.^[7]

Addition[edit]

Addition, denoted by the symbol $+$ , is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers (such as $2 + 3 = 5$ or $3 + 5 = 8$ ).

Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as summation, a term also used to denote the definition for "adding infinitely many numbers" in an infinite series. Repeated addition of the number 1 is the most basic form of counting; the result of adding $1$ is usually called the successor of the original number.

Addition is commutative and associative, so the order in which finitely many terms are added does not matter.

The number $0$ has the property that, when added to any number, it yields that same number; so, it is the identity element of addition, or the additive identity.

For every number $x$ , there is a number denoted $- x$ , called the opposite of $x$ , such that $x + (- x) = 0$ and $(- x) + x = 0$ . So, the opposite of $x$ is the inverseof $x$ with respect to addition, or the additive inverse of $x$ . For example, the opposite of $7$ is $-7$ , since $7 + (-7) = 0$ .

Addition can also be interpreted geometrically, as in the following example. If we have two sticks of lengths 2 and 5, then, if the sticks are aligned one after the other, the length of the combined stick becomes 7, since $2 + 5 = 7$ .

Subtraction[edit]

Subtraction, denoted by the symbol $-$ , is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: $D = M - S .$ Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: $D + S = M .$ ^[8]

For positive arguments $M$ and $S$ holds:

If the minuend is larger than the subtrahend, the difference

D

is positive.

If the minuend is smaller than the subtrahend, the difference

D

is negative.

In any case, if minuend and subtrahend are equal, the difference $D = 0.$

Subtraction is neither commutative nor associative. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under § Addition), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, $a - b = a + (- b)$ . The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved.

For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation). The trade-off is the halving of the number range for a fixed word length.

A formerly widespread method to achieve a correct change amount, knowing the due and given amounts, is the counting up method, which does not explicitly generate the value of the difference. Suppose an amount P is given in order to pay the required amount Q, with P greater than Q. Rather than explicitly performing the subtraction P − Q = C and counting out that amount C in change, money is counted out starting with the successor of Q, and continuing in the steps of the currency, until P is reached. Although the amount counted out must equal the result of the subtraction P − Q, the subtraction was never really done and the value of P − Q is not supplied by this method.

Multiplication[edit]

Multiplication, denoted by the symbols $\times$ or $\cdot$ , is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are called factors.

Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say x, is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0, in such a way that 1 goes to the multiplicand.

Another view on multiplication of integer numbers (extendable to rationals but not very accessible for real numbers) is by considering it as repeated addition. For example. $3 \times 4$ corresponds to either adding $3$ times a $4$ , or $4$ times a $3$ , giving the same result. There are different opinions on the advantageousness of these paradigmata in math education.

Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except $0$ is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity $1$ . $0$ is the only number without a multiplicative inverse, and the result of multiplying any number and $0$ is again $0.$ One says that $0$ is not contained in the multiplicative group of the numbers.

The product of a and b is written as $a \times b$ or $a \cdot b$ . It can also written by simple juxtaposition: ab. In computer programming languages and software packages (in which one can only use characters normally found on a keyboard), it is often written with an asterisk: a * b.

Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and applying repeated addition, or on employing tables or slide rules, thereby mapping multiplication to addition and vice versa. These methods are outdated and are gradually replaced by mobile devices. Computers use diverse sophisticated and highly optimized algorithms, to implement multiplication and division for the various number formats supported in their system.

Division[edit]

Division, denoted by the symbols $\div$ or $/$ , is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividenddivided by the divisor. Under common rules, dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than or equal to 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. So as explained in § Subtraction, the construction of the division in modern algebra is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced in § Multiplication. Hence division is the multiplication of the dividend with the reciprocal of the divisor as factors, that is, $a \div b = a \times 1 / b .$

Within the natural numbers, there is also a different but related notion called Euclidean division, which outputs two numbers after "dividing" a natural $N$ (numerator) by a natural $D$ (denominator): first a natural $Q$ (quotient), and second a natural $R$ (remainder) such that $N = D \times Q + R$ and $0 \leq R < Q .$

In some contexts, including computer programming and advanced arithmetic, division is extended with another output for the remainder. This is often treated as a separate operation, the Modulo operation, denoted by the symbol $\%$ or the word $mod$ , though sometimes a second output for one "divmod" operation.^[9] In either case, Modular arithmetic has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc.) correspond with different implementations of modulus.

Fundamental theorem of arithmetic[edit]

The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization:

252 = 2² × 3² × 7¹

Euclid's Elements first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.

The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number. Other reasons include the sieve of Eratosthenes, and the definition of a prime number itself (a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.).

Decimal arithmetic[edit]

Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal")positional notation; however, any numeral system based on powers of 10, e.g., Greek, Cyrillic, Roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indorum" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (10²), plus 0 tens (10¹), plus 7 units (10⁰), plus 3 tenths (10⁻¹) plus 6 hundredths (10⁻²).

The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jaintext from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci^[10] using the Hindu–Arabic numeral system.

Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero). This adjustment is termed a carry of the value 1.

The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from 1 to 8 ( $9 \times 9 = 81$ ). Additional steps define the final result.

Similar techniques exist for subtraction and division.

The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10ⁿ with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 10², 10, 1, 10⁻¹, 10⁻², ...}.

Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.

Compound unit arithmetic[edit]

Compound^[11] unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches; gallons and pints; pounds, shillings and pence; and so on. Before decimal-based systems of money and units of measure, compound unit arithmetic was widely used in commerce and industry.

Basic arithmetic operations[edit]

The techniques used in compound unit arithmetic were developed over many centuries and are well documented in many textbooks in many different languages.^[12]^[13]^[14]^[15] In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:

Reduction, in which a compound quantity is reduced to a single quantity—for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches.^[16]
Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to 1 lb 8 oz.
Normalization is the conversion of a set of compound units to a standard form—for example, rewriting "1 ft 13 in" as "2 ft 1 in".

Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.

Principles of compound unit arithmetic[edit]

There are two basic approaches to compound unit arithmetic:

Reduction–expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by Microsoft Excelwhere all time intervals are processed internally as days and decimal fractions of a day.
On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the ongoing normalization method as applied to addition is shown below.

The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results.

The total in the pence column is 25. Since there are 12 pennies in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column.

For the sake of simplicity, the example chosen did not have farthings.

Operations in practice[edit]

During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pence and farthings, and ready reckoners, which are books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet^[17] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".

The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth")^[18] in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (e.g., "2.5 ft" is displayed rather than "2 ft 6 in").

Number theory[edit]

Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's Last Theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.

Arithmetic in education[edit]

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism.

The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.^[19]

Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati.^[20] The book begins with the foundations of mathematics and proceeds to its application in the later chapters.

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President of the United States
Presidential seal
Presidential flag
Incumbent Joe Biden since January 20, 2021
Executive branch of the U.S. government Executive Office of the President
Style	Mr. President^[1]^[2] (informal) The Honorable^[3] (formal) His Excellency^[4]^[5] (diplomatic)
Type	Head of state Head of government Commander-in-chief
Abbreviation	POTUS
Member of	Cabinet Domestic Policy Council National Economic Council National Security Council
Residence	White House
Seat	Washington, D.C.
Appointer	Electoral College or via succession from vice presidency
Term length	Four years, renewable once
Constituting instrument	Constitution of the United States
Formation	March 4, 1789 (234 years ago)^[6]^[7]^[8]
First holder	George Washington^[9]
Salary	400,000 United States dollars per year
Website	www.whitehouse.gov

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History[edit]

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Separating[edit]

Justification and line-wrapping[edit]

Prefixes and suffixes[edit]

Syllabification and spelling[edit]

Joining[edit]

Compound modifiers[edit]

Object–verbal-noun compounds[edit]

Personal names[edit]

Other compounds[edit]

Suspended hyphens[edit]

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Varied meanings[edit]

Use in computing[edit]

Hyphen-minuses[edit]

Soft and hard hyphens[edit]

Nonbreaking hyphens[edit]

Use in date notation[edit]

Unicode[edit]

History[edit]

Arithmetic operations[edit]

Addition[edit]

Subtraction[edit]

Multiplication[edit]

Division[edit]

Fundamental theorem of arithmetic[edit]

Decimal arithmetic[edit]

Compound unit arithmetic[edit]

Basic arithmetic operations[edit]

Principles of compound unit arithmetic[edit]

Operations in practice[edit]

Number theory[edit]

Arithmetic in education[edit]

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An Independent Mind, Knot Logic

An Independent Mind, Knot Logic

Cantore Arithmetic: Page 1 Equated Word street

Karen A. Placek, aka Karen Placek, K.A.P., KAP

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