Overline

An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In mathematical notation, an overline has been used for a long time as a vinculum, a way of showing that certain symbols belong together. The original use in Ancient Greek was to indicate compositions of Greek letters as Greek numerals.^[1] In Latin it indicates Roman numerals multiplied by a thousand and it forms medieval abbreviations (sigla). Marking one or more words with a continuous line above the characters is sometimes called overstriking, though overstriking generally refers to printing one character on top of an already-printed character.

An overline, that is, a single line above a chunk of text, should not be confused with the macron, a diacritical mark placed above (or sometimes below) individual letters. The macron is narrower than the character box.^[2]

Uses

Medicine

An overbar over a letter is a traditional way of specifying certain Latin abbreviations. For example, s̅ (s overbar) stands for Latin, sine (meaning without), c̅ (c overbar) is an abbreviation for Latin, cum (meaning with), ā (a overbar) stands for Latin, "ante" (meaning "before"), and p̄ (p overbar) stands for Latin, "post" (meaning "after").

Math and science

Vinculum

In mathematics, an overline can be used as a vinculum.

The vinculum can indicate a line segment:

• [math]\displaystyle{ \overline{\rm AB} }[/math]

The vinculum can indicate a repeating decimal value:

• ​¹⁄₇ = 0.142857 = 0.1428571428571428571...

When it is not possible to format the number so that the overline is over the digit(s) that repeat, one overline character is placed to the left of the digit(s) that repeat:

• 3.¯3 = 3.3 = 3.333333333333...
• 3.12¯34 = 3.1234 = 3.123434343434...

Historically, the vinculum was used to group together symbols so that they could be treated as a unit. Today, parentheses are more commonly used for this purpose.

Statistics

The overline is used to indicate a sample mean:

• [math]\displaystyle{ \overline x }[/math] is the average value of [math]\displaystyle{ x_i }[/math]

Survival functions or complementary cumulative distribution functions are often denoted by placing an overline over the symbol for the cumulative:[math]\displaystyle{ \overline{F}(x) =1-F(x) }[/math]

Negation

In set theory and some electrical engineering contexts, negation operators can be written as an overline above the term or expression to be negated, for example:

Common set theory notation:

[math]\displaystyle{ \begin{align} \overline{A \cup B} &\equiv \overline{A} \cap \overline{B} \\ \overline{A \cap B} &\equiv \overline{A} \cup \overline{B} \end{align} }[/math]

Electrical engineering notation:

[math]\displaystyle{ \begin{align} \overline{A \cdot B} &\equiv \overline {A} + \overline {B} \\ \overline{A + B} &\equiv \overline {A} \cdot \overline {B} \end{align} }[/math]

in which implied multiplication, the times (cross) and the dot all mean logical AND, and the plus sign means logical OR.

Both illustrate De Morgan's laws and its mnemonic, "break the line, change the sign".

Negative

In common logarithms a bar over the characteristic indicates that it is negative whilst the mantissa remains positive. This notation avoids the need for separate tables to convert positive and negative logarithms back to their original numbers.

[math]\displaystyle{ \log_{10}0.012\approx-2+0.07918=\bar{2}.07918 }[/math]

Reciprocal

Rarely, a bar over a number or expression means its multiplicative inverse^[3] which is more commonly shown as a fraction or negative exponent

[math]\displaystyle{ \overline{2l} = 1/(2l) = (2l)^{-1} }[/math]

Complex numbers

The overline notation can indicate a complex conjugate and analogous operations.

• if [math]\displaystyle{ x = a + ib }[/math], then [math]\displaystyle{ \overline{x} = a - ib. }[/math]

Vector

In physics, an overline sometimes indicates a vector, although boldface and arrows are also commonly used:

• [math]\displaystyle{ \overline x = |x|\hat x }[/math]

Improper rotation

In crystallography, an overline indicates an improper rotation or a negative number:

• [math]\displaystyle{ \overline 3 }[/math] is the Hermann–Mauguin notation for a threefold rotoinversion, used in crystallography.
• [math]\displaystyle{ [\overline 11\overline2] }[/math] is the direction with Miller indices [math]\displaystyle{ h=-1 }[/math], [math]\displaystyle{ k=1 }[/math], [math]\displaystyle{ l=-2 }[/math].

Maximal conductance

In computational neuroscience, an overline is used to indicate the 'maximal' conductances in Hodgkin-Huxley models. This goes back to at least the landmark paper published by Nobel prize winners Alan Lloyd Hodgkin and Andrew Fielding Huxley around 1952.^[4]

[math]\displaystyle{ I_\mathrm{Na}(t)=\bar{g}_\mathrm{Na} m(V_m)^3h(V_m)(V_m-E_\mathrm{Na}) }[/math]

Antiparticles

Overlines are used in subatomic particle physics to denote antiparticles for some particles (with the alternate being distinguishing based on electric charge). For example, the proton is denoted as p, and its corresponding antiparticle is denoted as p.

Engineering

An active low signal is designated by an overline, e.g. RESET, representing logical negation.

Morse (CW)

Some Morse code prosigns can be expressed as two or three characters run together, and an overline is often used to signify this. The most famous is the distress signal, SOS.

Writing

An overline-like symbol is traditionally used in Syriac text to mark abbreviations and numbers. It has dots at each end and the center. In German it is occasionally used to indicate a pair of letters which cannot both be fitted into the available space.^[5][6]

When Morse code is written out as text, overlines are used to distinguish prosigns and other concatenated character groups from strings of individual characters.

Linguistics

X-bar theory makes use of overbar notation to indicate differing levels of syntactic structure. Certain structures are represented by adding an overbar to the unit, as in X. Due to difficulty in typesetting the overbar, the prime symbol is often used instead, as in X′. Contemporary typesetting software, such as LaTeX, has made typesetting overbars considerably simpler; both prime and overbar markers are accepted usages. Some variants of X-bar notation use a double-bar (or double-prime) to represent phrasal-level units.

X-bar theory derives its name from the overbar. One of the core proposals of the theory was the creation of an intermediate syntactic node between phrasal (XP) and unit (X) levels; rather than introduce a different label, the intermediate unit was marked with a bar.

Implementations

HTML with CSS

In HTML using CSS, overline is implemented via the text-decoration property; for example, <span style="text-decoration: overline">text</span> results in: text.

The text decoration property supports also other typographical features with horizontal lines: underline (a line below the text) and strikethrough (a line through the text).

Unicode

As mentioned above, Unicode includes two graphic characters, U+00AF ¯ MACRON and U+203E ‾ OVERLINE. They are compatibility equivalent to the U+0020   SPACE with non-spacing diacritics U+0304 ◌̄ COMBINING MACRON and U+0305 ◌̅ COMBINING OVERLINE respectively; the latter allows an overline to be placed over any character. As with any combining character, it appears in the same character box as the character that logically precedes it: for example, x̅, compared to x‾. A series of overlined characters usually results in an unbroken line, depending on the font (for example, 1̅2̅3̅).

For East Asian (CJK) computing, U+FFE3 ￣ FULLWIDTH MACRON is available. Despite the name, Unicode maps this character to both U+203E and U+00AF.^[7]

Unicode maps the overline-like character from ISO/IEC 8859-1 and code page 850 to the U+00AF ¯ MACRON symbol mentioned above. In a reversal of its official name (and compatibility decomposition), it is much wider than an actual macron diacritic over most letters, and actually wider than U+203E ‾ OVERLINE in most fonts. In Microsoft Windows, U+00AF can be entered with the keystrokes (where numbers are entered from the numeric keypad). In GTK/GTK+, the symbol can be added using the keystrokes to activate Unicode input, then type "00AF" as the code for the character. On a Mac, with the ABC Extended keyboard, use .

The Unicode character U+070F SYRIAC ABBREVIATION MARK is used to mark Syriac abbreviations and numbers. However, several computer environments do not render this line correctly or at all.

The Unicode character U+0B55 ୕ ORIYA SIGN OVERLINE is used as a length mark in Odia script.

Word processors

In Microsoft Word, overstriking of text can be accomplished with the EQ \O() field code. The field code {EQ \O(x,¯)} produces x and the field code {EQ \O(xyz,¯¯¯)} produces xyz. (Doesn't work in Word 2010; it is necessary to insert MS Equation object). Windows: (once before character, one more time after character).

LibreOffice has direct support for several styles of overline in its "Format / Character / Font Effects" dialog.

Overstriking of longer sections of text, such as in 123, can also be produced in many text processors as text markup as a special form of understriking.

TeX

In LaTeX, a text <text> can be overlined with $\overline{\mbox{<text>}}$. The inner \mbox{} is necessary to override the math-mode (here invoked by the dollar signs) which the \overline{} demands.

References

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