Positive number

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the sign of a
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is its property of being either positive, negative, or
zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in languag ...

. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention. In some contexts, it makes sense to consider a
signed zero Signed zero is zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be re ...
(such as
floating-point representation In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
s of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
(negation, or multiplication by
−1 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathematical objects that resemble positivity and negativity, such as odd and even (
sign of a permutation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
), sense of
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
or rotation (),
one sided limit In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
s, and other concepts described in below.

# Sign of a number

Number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s from various number systems, like
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
, rationals,
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s,
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s,
octonion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, ... may have multiple attributes, that fix certain properties of a number. If a number system bears the structure of an
ordered ring 350px, The real numbers are an ordered ring which is also an ordered field. The integers">ordered_field.html" ;"title="real numbers are an ordered ring which is also an ordered field">real numbers are an ordered ring which is also an ordered field. ...
, for example, the integers, it must contain a number that does not change any number when it is added to it (an additive
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). This number is generally denoted as Because of the
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
in this ring, there are numbers greater than zero, called the ''positive'' numbers. For other properties required within a ring, for each such positive number there exists a number less than which, when added to the positive number, yields the result These numbers less than are called the ''negative'' numbers. The numbers in each such pair are their respective
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
s. This attribute of a number, being exclusively either ''zero'' , ''positive'' , or ''negative'' , is called its sign, and is often encoded to the real numbers and respectively (similar to the way the
sign function 300px, Signum function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ...
is defined). Since rational and real numbers are also ordered rings (even
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
), these number systems share the same ''sign'' attribute. While in
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
, a minus sign is usually thought of as representing the binary operation of subtraction, in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, it is usually thought of as representing the
unary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
yielding the
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
(sometimes called ''negation'') of the operand. While is its own additive inverse the additive inverse of a positive number is negative, and the additive inverse of a negative number is positive. A double application of this operation is written as The plus sign is predominantly used in algebra to denote the binary operation of addition, and only rarely to emphasize the positivity of an expression. In common numeral notation (used in
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
and elsewhere), the sign of a number is often made explicit by placing a plus or a minus sign before the number. For example, denotes "positive three", and denotes "negative three" (algebraically: the additive inverse of ). Without specific context (or when no explicit sign is given), a number is interpreted per default as positive. This notation establishes a strong association of the minus sign "" with negative numbers, and the plus sign "+" with positive numbers.

## Sign of zero

Within the convention of
zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in languag ...

being neither positive nor negative, a specific sign-value may be assigned to the number value . This is exploited in the $\sgn$-function, as defined for real numbers. In arithmetic, and both denote the same number . There is generally no danger of confusing the value with its sign, although the convention of assigning both signs to does not immediately allow for this discrimination. In some contexts, especially in
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d ...

, it is useful to consider signed versions of zero, with
signed zero Signed zero is zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be re ...
s referring to different, discrete number representations (see
signed number representations In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwar ...
for more). The symbols and rarely appear as substitutes for and used in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

and
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
for
one-sided limit In calculus, a one-sided limit is either of the two Limit of a function, limits of a function (mathematics), function ''f''(''x'') of a real number, real variable ''x'' as ''x'' approaches a specified point either from the left or from the right. ...
s (right-sided limit and left-sided limit, respectively). This notation refers to the behaviour of a function as its real input variable approaches along positive (resp., negative) values; the two limits need not exist or agree.

## Terminology for signs

When is said to be neither positive nor negative, the following phrases may refer to the sign of a number: * A number is positive if it is greater than zero. * A number is negative if it is less than zero. * A number is non-negative if it is greater than or equal to zero. * A number is non-positive if it is less than or equal to zero. When is said to be both positive and negative, modified phrases are used to refer to the sign of a number: * A number is strictly positive if it is greater than zero. * A number is strictly negative if it is less than zero. * A number is positive if it is greater than or equal to zero. * A number is negative if it is less than or equal to zero. For example, the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of a real number is always "non-negative", but is not necessarily "positive" in the first interpretation, whereas in the second interpretation, it is called "positive"—though not necessarily "strictly positive". The same terminology is sometimes used for
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that yield real or other signed values. For example, a function would be called a ''positive function'' if its values are positive for all arguments of its domain, or a ''non-negative function'' if all of its values are non-negative.

## Complex numbers

Complex numbers are impossible to order, so they cannot carry the structure of an ordered ring, and, accordingly, cannot be partitioned into positive and negative complex numbers. They do, however, share an attribute with the reals, which is called ''
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

'' or ''magnitude''. Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

. For example, the absolute value of and the absolute value of are both equal to This is written in symbols as and In general, any arbitrary real value can be specified by its magnitude and its sign. Using the standard encoding, any real value is given by the product of the magnitude and the sign in standard encoding. This relation can be generalized to define a ''sign'' for complex numbers. Since the real and complex numbers both form a field and contain the positive reals, they also contain the reciprocals of the magnitudes of all non-zero numbers. This means that any non-zero number may be multiplied with the reciprocal of its magnitude, that is, divided by its magnitude. It is immediate that the quotient of any non-zero real number by its magnitude yields exactly its sign. By analogy, the can be defined as the quotient and its Since the magnitude of the complex number is ''divided out'', the resulting sign of the complex number represents in some sense its complex argument. This is to be compared to the sign of real numbers, except with $e^= -1.$ For the definition of a complex sign-function. see below.

## Sign functions

When dealing with numbers, it is often convenient to have their sign available as a number. This is accomplished by functions that extract the sign of any number, and map it to a predefined value before making it available for further calculations. For example, it might be advantageous to formulate an intricate algorithm for positive values only, and take care of the sign only afterwards.

### Real sign function

The sign function or signum function extracts the sign of a real number, by mapping the set of real numbers to the set of the three reals $\.$ It can be defined as follows: :$\sgn: \mathbb R \to \$ :$x \mapsto \sgn\left(x\right) = \begin -1 & \text x < 0, \\ ~~\, 0 & \text x = 0, \\ ~~\, 1 & \text x > 0. \end$ Thus is 1 when is positive, and is −1 when is negative. For non-zero values of , this function can also be defined by the formula :$\sgn\left(x\right) = \frac = \frac,$ where is the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of .

### Complex sign function

While a real number has a 1-dimensional direction, a complex number has a 2-dimensional direction. The complex sign function requires the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of its argument which can be calculated as :$, z, = \sqrt = \sqrt.$ Analogous to above, the complex sign function extracts the complex sign of a complex number by mapping the set of non-zero complex numbers to the set of unimodular complex numbers, and to $\\cup \.$ It may be defined as follows: Let be also expressed by its magnitude and one of its arguments as then :$\sgn\left(z\right) = \begin 0 &\text z=0\\ \displaystyle\frac = e^ &\text. \end$ This definition may also be recognized as a normalized vector, that is, a vector whose direction is unchanged, and whose length is fixed to
unity Unity may refer to: Buildings * Unity Building The Unity Building, in Oregon, Illinois, is a historic building in that city's Oregon Commercial Historic District. As part of the district the Oregon Unity Building has been listed on the National R ...
. If the original value was R,θ in polar form, then sign(R, θ) is 1 θ. Extension of sign() or signum() to any number of dimensions is obvious, but this has already been defined as normalizing a vector.

# Signs per convention

In situations where there are exactly two possibilities on equal footing for an attribute, these are often labelled by convention as ''plus'' and ''minus'', respectively. In some contexts, the choice of this assignment (i.e., which range of values is considered positive and which negative) is natural, whereas in other contexts, the choice is arbitrary, making an explicit sign convention necessary, the only requirement being consistent use of the convention.

## Sign of an angle

In many contexts, it is common to associate a sign with the measure of an
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

, particularly an oriented angle or an angle of
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
. In such a situation, the sign indicates whether the angle is in the
clockwise Two-dimensional rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line ...

or counterclockwise direction. Though different conventions can be used, it is common in
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
to have counterclockwise angles count as positive, and clockwise angles count as negative. It is also possible to associate a sign to an angle of rotation in three dimensions, assuming that the
axis of rotation Rotation around a fixed axis is a special case of rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotati ...
has been oriented. Specifically, a
right-handed In human biology, handedness is the better, faster, or more precise performance or individual preference for use of a hand, known as the dominant hand. The incapable, less capable or less preferred hand is called the non-dominant hand. Right-ha ...

rotation around an oriented axis typically counts as positive, while a left-handed rotation counts as negative.

## Sign of a change

When a quantity ''x'' changes over time, the
change Change or Changing may refer to: Alteration * Impermanence Impermanence, also known as the philosophical problem This is a list of the major unsolved problems in philosophy Philosophy (from , ) is the study of general and fundam ...
in the value of ''x'' is typically defined by the equation :$\Delta x = x_\text - x_\text.$ Using this convention, an increase in ''x'' counts as positive change, while a decrease of ''x'' counts as negative change. In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

, this same convention is used in the definition of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

. As a result, any
increasing function Figure 3. A function that is not monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
has positive derivative, while any decreasing function has negative derivative.

## Sign of a direction

In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
and
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, it is common to label certain directions as positive or negative. For a basic example, the
number line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as abstraction for real numbers, denoted by \mathbb. Every point of a number line is assumed to correspond to a real number, and ever ...

is usually drawn with positive numbers to the right, and negative numbers to the left: As a result, when discussing
linear motion Linear motion, also called rectilinear motion, is one-dimensional motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position (mathematics), ...
,
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
or
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (p ...

, a motion to the right is usually thought of as being positive, while similar motion to the left is thought of as being negative. On the
Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, the rightward and upward directions are usually thought of as positive, with rightward being the positive ''x''-direction, and upward being the positive ''y''-direction. If a displacement or velocity
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
is separated into its
vector component In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, then the horizontal part will be positive for motion to the right and negative for motion to the left, while the vertical part will be positive for motion upward and negative for motion downward.

## Signedness in computing

In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d ...

, an integer value may be either signed or unsigned, depending on whether the computer is keeping track of a sign for the number. By restricting an integer variable to non-negative values only, one more
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both ...
can be used for storing the value of a number. Because of the way integer arithmetic is done within computers,
signed number representation In computing, signed number representations are required to encode negative numbers in binary number systems. In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in computer hardwa ...
s usually do not store the sign as a single independent bit, instead using e.g.
two's complement Two's complement is a mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The number of operands is the arity of the ...
. In contrast, real numbers are stored and manipulated as
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
values. The floating point values are represented using three separate values, mantissa, exponent, and sign. Given this separate sign bit, it is possible to represent both positive and negative zero. Most programming languages normally treat positive zero and negative zero as equivalent values, albeit, they provide means by which the distinction can be detected.

## Other meanings

In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and other sciences: * Words ''
up toTwo mathematics, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respec ...
sign'' mean that, for a quantity , it is known that either or for certain . It is often expressed as . For real numbers, it means that only the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of the quantity is known. For
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
and
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, a quantity known up to sign is a stronger condition than a quantity with known
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
: aside and , there are many other possible values of such that {{math, 1={{!''q''{{! = {{!''Q''{{! . * The
sign of a permutation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is defined to be positive if the permutation is even, and negative if the permutation is odd. * In
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, a
signed graph In the area of graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex ...
is a graph in which each edge has been marked with a positive or negative sign. * In
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
, a
signed measure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is a generalization of the concept of measure in which the measure of a set may have positive or negative values. * In a
signed-digit representation In mathematical notation Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, ana ...
, each digit of a number may have a positive or negative sign. * The ideas of signed area and signed volume are sometimes used when it is convenient for certain areas or volumes to count as negative. This is particularly true in the theory of
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. In an (abstract) oriented vector space, each ordered basis for the vector space can be classified as either positively or negatively oriented. * In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, any
electric charge Electric charge is the physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
comes with a sign, either positive or negative. By convention, a positive charge is a charge with the same sign as that of a
proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collecti ...

, and a negative charge is a charge with the same sign as that of an
electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...

. *