An overline, overscore, or overbar, is a
typographical
Typography is the art and technique of arranging type to make written language legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line spacing, letter spac ...
feature of a
horizontal line drawn immediately above the text. In old
mathematical notation
Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...
, an overline was called a ''
vinculum'', a notation for grouping symbols which is expressed in modern notation by parentheses, though it persists for symbols under a radical sign. The original use in
Ancient Greek
Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...
was to indicate compositions of
Greek letters as
Greek numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), ordi ...
. In Latin, it indicates
Roman numeral
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...
s 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
A diacritic (also diacritical mark, diacritical point, diacritical sign, or accent) is a glyph added to a letter or to a basic glyph. The term derives from the Ancient Greek (, "distinguishing"), from (, "to distinguish"). The word ''diacrit ...
placed above (or sometimes below) ''individual'' letters. The macron is narrower than the character box.
Uses
Medicine
In most forms of
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
scribal abbreviation, an overline or
macron indicates omitted letters similar to use of
apostrophe
The apostrophe (, ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes:
* The marking of the omission of one o ...
s in English contractions. Letters with macrons or overlines continue to be used in
medical abbreviations in various European languages, particularly for
prescriptions. Common examples include
* , a̅, or ā for ("before")
* , c̅, or c̄ for ("with")
* , p̅, or p̄ for ("after")
* , q̅, or q̄ for and its inflections ("every", "each")
* , s̅, or s̄ for ("without")
* , x̅, or x̄ for and its inflections ("except")
Note, however, that abbreviations involving the letter h take their macron halfway up the ascending line rather than at the normal height for Unicode overlines and macrons:
ħ. This is separately encoded in
Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
with the symbols using
bar diacritics and appears shorter than other overlines in many fonts.
Math and science
Decimal separator
In the
Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
, from the original
Indian decimal writing, before printing, an overline over the
units digit
A numerical digit (often shortened to just digit) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originate ...
was used to separate the integral part of a number from its
fractional part, as in 995 (meaning 99.95 in decimal point format). A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal separator, as in 99
.
Vinculum
In mathematics, an overline can be used as a
vinculum.
The vinculum can indicate a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
:
The vinculum can indicate a
repeating decimal
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that i ...
value:
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:
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:
*
is the average value of
Survival function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time.
The survival function is also known as the survivor function
or reliability function.
The term ...
s or complementary cumulative distribution functions are often denoted by placing an overline over the symbol for the cumulative:
.
Negation
In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
and some
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
contexts,
negation
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
operators (also known as
complement) can be written as an overline above the term or expression to be negated. For example:
Common set theory notation:
:
Electrical engineering notation:
:
in which the times (cross) means multiplication, the dot means logical AND, and the plus sign means logical OR.
Both illustrate
De Morgan's laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are nam ...
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.
:
Complex numbers
The overline notation can indicate a
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
and analogous operations.
*if
, then
Vector
In physics, an overline sometimes indicates a
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
, although
boldface
In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of prosody stress in speech.
Methods and use
The most common methods in We ...
and
arrows are also commonly used:
*
Congruence classes
Congruence modulo is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
, and the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of the integer , denoted by , is the set . This set, consisting of all the integers congruent to modulo , is called the congruence class,
residue class
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mod ...
, or simply residue of the integer modulo . When the modulus is known from the context, that residue may also be denoted or .
Topological closure
In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, the
closure of a subset ''S'' of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is often denoted or
.
Improper rotation
In
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, an overline indicates an
improper rotation or a
negative number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
:
*
is the
Hermann–Mauguin notation for a threefold rotoinversion, used in
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
.
*