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 (leading), and ...
feature of a
horizontal line drawn immediately above the text. In old
mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
, 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 ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
was to indicate compositions of
Greek letters
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as we ...
as
Greek numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those ...
. 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 a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
scribal abbreviation
Scribal abbreviations or sigla (singular: siglum) are abbreviations used by ancient and medieval scribes writing in various languages, including Latin, Greek, Old English and Old Norse. In modern manuscript editing (substantive and mechanica ...
, an overline or
macron indicates omitted letters similar to use of
apostrophe
The apostrophe ( or ) 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 ...
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, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
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 late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
, from the original
Indian decimal writing, before printing, an overline over the
units digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits ( Lati ...
was used to separate the integral part of a number from its
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part ca ...
, 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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
:
The vinculum can indicate a
repeating decimal
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational 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
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables.
The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
:
*
is the average value of
Survival functions 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 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 mathematics, is mostly concern ...
and some
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
contexts,
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
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 logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
and its mnemonic, "break the line, change the sign".
Negative
In
common logarithms
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
, 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, then) the complex conjugate of a + bi is equal to 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
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, 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 W ...
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.
Each equivalence relatio ...
, 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 a ...
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, 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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the
closure of a subset ''S'' of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is often denoted or
.
Improper rotation
In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...
, an overline indicates an
improper rotation
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
or a
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...
:
*
is the
Hermann–Mauguin notation
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...
for a threefold rotoinversion, used in
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...
.
*