View source for Sign (mathematics)

{{Short description|Number property of being positive or negative}}
{{for|symbols named "... sign"|List of mathematical symbols}} [[File:PlusMinus.svg|thumb|right|150px|The [[Plus and minus signs|plus and minus symbols]] are used to show the sign of a number.]]

In [[Mathematics|mathematics]], the '''sign''' of a [[Real number|real number]] is its property of being either positive, [[Negative number|negative]], or zero. 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).{{fact|date=July 2020}} Whenever not specifically mentioned, this article adheres to the first convention.

In some contexts, it makes sense to consider a [[Signed zero|signed zero]] (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the [[Additive inverse|additive inverse]] (negation, or multiplication by [[−1|−1]]) 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 ([[Parity of a permutation|sign of a permutation]]), sense of [[Orientation (vector space)|orientation]] or rotation ([[Physics:Clockwise|cw/ccw]]), one sided limits, and other concepts described in {{Section link||Other meanings}} below.

==Sign of a number==
[[Number]]s from various number systems, like integers, [[Rational number|rationals]], [[Complex number|complex number]]s, [[Quaternion|quaternion]]s, [[Octonion|octonion]]s, ... may have multiple attributes, that fix certain properties of a number. A number system that bears the structure of an [[Ordered ring|ordered ring]] contains a unique number that when added with any number leaves the latter unchanged. This unique number is known as the system's additive [[Identity element|identity element]]. For example, the integers has the structure of an ordered ring. This number is generally denoted as {{math|0.}} Because of the [[Total order|total order]] in this ring, there are numbers greater than zero, called the ''positive'' numbers. Another property required for a ring to be ordered is that, for each positive number, there exists a unique corresponding number less than {{math|0}} whose sum with the original positive number is {{math|0.}} These numbers less than {{math|0}} are called the ''negative'' numbers. The numbers in each such pair are their respective [[Additive inverse|additive inverse]]s. This attribute of a number, being exclusively either ''zero'' {{math|(0)}}, ''positive'' {{math|(+)}}, or ''negative'' {{math|(−)}}, is called its '''sign''', and is often encoded to the real numbers {{math|0}}, {{math|1}}, and {{math|−1}}, respectively (similar to the way the [[Sign function|sign function]] is defined).<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.| title=Sign|url=https://mathworld.wolfram.com/Sign.html|access-date=2020-08-26| website=mathworld.wolfram.com|language=en}}</ref> Since rational and real numbers are also ordered rings (in fact ordered [[Field (mathematics)|fields]]), the ''sign'' attribute also applies to these number systems.

When a minus sign is used in between two numbers, it represents the binary operation of subtraction. When a minus sign is written before a single number, it represents the [[Unary operation|unary operation]] of yielding the [[Additive inverse|additive inverse]] (sometimes called ''negation'') of the operand. Abstractly then, the difference of two number is the sum of the minuend with the additive inverse of the subtrahend. While {{math|0}} is its own additive inverse ({{math|1=−0 = 0}}), 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 {{math|1=−(−3) = 3}}. 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 system|numeral notation]] (used in [[Arithmetic|arithmetic]] and elsewhere), the sign of a number is often made explicit by placing [[Plus and minus signs|a plus or a minus sign]] before the number. For example, {{math|+3}} denotes "positive three", and {{math|−3}} denotes "negative three" (algebraically: the additive inverse of {{math|3}}). 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 "{{math|−}}" with negative numbers, and the plus sign "+" with positive numbers.

===Sign of zero===
Within the convention of zero being neither positive nor negative, a specific sign-value {{math|0}} may be assigned to the number value {{math|0}}. This is exploited in the [[Sign function|<math>\sgn</math>-function]], as defined for real numbers.<ref name=":0" /> In arithmetic, {{math|+0}} and {{math|−0}} both denote the same number {{math|0}}. There is generally no danger of confusing the value with its sign, although the convention of assigning both signs to {{math|0}} does not immediately allow for this discrimination.

In some contexts, especially in [[Computing|computing]], it is useful to consider signed versions of zero, with [[Signed zero|signed zero]]s referring to different, discrete number representations (see [[Signed number representations|signed number representations]] for more).

The symbols {{math|+0}} and {{math|−0}} rarely appear as substitutes for {{math|0<sup>+</sup>}} and {{math|0<sup>−</sup>}}, used in [[Calculus|calculus]] and [[Mathematical analysis|mathematical analysis]] for [[One-sided limit|one-sided limit]]s (right-sided limit and left-sided limit, respectively). This notation refers to the behaviour of a function as its real input variable approaches {{math|0}} along positive (resp., negative) values; the two limits need not exist or agree.

==={{anchor|non-negative and non-positive}} Terminology for signs===
<!-- Several articles link to the above anchor -->
When {{math|0}} 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 {{math|0}} is said to be both positive and negative {{fact|date=June 2023}}, 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|absolute value]] 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 [[Function (mathematics)|functions]] 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|absolute value]]'' or ''magnitude''. Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its [[Absolute value|absolute value]]. 

For example, the absolute value of {{math|−3}} and the absolute value of {{math|3}} are both equal to {{math|3}}. This is written in symbols as {{math|1={{abs|−3}} = 3}} and {{math|1={{abs|3}} = 3}}.

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 {{nowrap|'''sign of a complex number''' {{mvar|z}}}} can be defined as the quotient {{nowrap|of {{mvar|z}}}} and its {{nowrap|magnitude {{math|{{abs|''z''}}}}.}} The sign of a complex number is the exponential of the product of its argument with the imaginary unit. represents in some sense its complex argument. This is to be compared to the sign of real numbers, except with <math>e^{i \pi}= -1.</math> For the definition of a complex sign-function. see {{Section link||Complex sign function}} below.

===Sign functions===
[[Image:Signum function.svg|thumb|200px|Real sign function {{math|1=''y'' = sgn(''x'')}}]]
{{main|Sign function}}

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 <math>\{-1,\; 0,\; 1\}.</math> It can be defined as follows:<ref name=":0" />
<math display="block">\begin{align}
\sgn : {} & \Reals \to \{-1, 0, 1\} \\
& x \mapsto \sgn(x) = \begin{cases}
-1 & \text{if } x < 0, \\ 
~~\, 0 & \text{if } x = 0, \\
~~\, 1 & \text{if } x > 0.
\end{cases}
\end{align}</math>
Thus {{math|sgn(''x'')}} is 1 when {{mvar|x}} is positive, and {{math|sgn(''x'')}} is −1 when {{mvar|x}} is negative. For non-zero values of {{mvar|x}}, this function can also be defined by the formula
<math display="block"> \sgn(x) = \frac{x}{|x|} = \frac{|x|}{x},</math>
where {{math|{{abs|''x''}}}} is the [[Absolute value|absolute value]] of {{mvar|x}}.

====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 [[Absolute value#Complex numbers|magnitude]] of its argument {{math|1=''z'' = ''x'' + ''iy''}}, which can be calculated as
<math display="block">|z| = \sqrt{z\bar z} = \sqrt{x^2 + y^2}.</math>

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 {{math|0}} to {{math|0}}: <math>\{z \in \Complex : |z| = 1\} \cup \{0\}.</math> It may be defined as follows:

Let {{mvar|z}} be also expressed by its magnitude and one of its arguments {{mvar|φ}} as {{math|1=''z'' = {{abs|''z''}}⋅''e<sup>iφ</sup>'',}} then<ref>{{Cite web|title=SignumFunction| url=http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/SpecialFunctions/SignumFunction.htm|access-date=2020-08-26| website=www.cs.cas.cz}}</ref>
<math display="block">\sgn(z) = \begin{cases}
0  &\text{for } z=0\\
\dfrac{z}{|z|} = e^{i\varphi}  &\text{otherwise}.
\end{cases}</math>

This definition may also be recognized as a normalized vector, that is, a vector whose direction is unchanged, and whose length is fixed to [[Unit vector|unity]]. 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==
{{main|Sign 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{{anchor|Angles}}===
[[File:Angles on the unit circle.svg|right|thumb|Measuring from the x-axis, angles on the [[Unit circle|unit circle]] count as positive in the counterclockwise direction, and negative in the clockwise direction.]]

In many contexts, it is common to associate a sign with the measure of an [[Angle|angle]], particularly an oriented angle or an angle of [[Rotation (mathematics)|rotation]]. In such a situation, the sign indicates whether the angle is in the [[Physics:Clockwise|clockwise]] or counterclockwise direction. Though different conventions can be used, it is common in [[Mathematics|mathematics]] to have counterclockwise angles count as positive, and clockwise angles count as negative.<ref>{{Cite web|title=Sign of Angles {{!}} What is An Angle? {{!}} Positive Angle {{!}} Negative Angle|url=https://www.math-only-math.com/sign-of-angles.html|access-date=2020-08-26|website=Math Only Math}}</ref>

It is also possible to associate a sign to an angle of rotation in three dimensions, assuming that the [[Physics:Axis of rotation|axis of rotation]] has been oriented. Specifically, a [[Physics:Right-hand rule|right-handed]] 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 [[Finite difference|change]] in the value of ''x'' is typically defined by the equation
<math display="block">\Delta x = x_\text{final} - x_\text{initial}. </math>

Using this convention, an increase in ''x'' counts as positive change, while a decrease of ''x'' counts as negative change. In [[Calculus|calculus]], this same convention is used in the definition of the [[Derivative|derivative]]. As a result, any [[Monotonic function|increasing function]] has positive derivative, while any decreasing function has negative derivative.

===Sign of a direction===
In [[Analytic geometry|analytic geometry]] and [[Physics:Physics|physics]], it is common to label certain directions as positive or negative. For a basic example, the [[Number line|number line]] is usually drawn with positive numbers to the right, and negative numbers to the left:
[[File:Number-line.svg|center|600px]]
As a result, when discussing [[Physics:Linear motion|linear motion]], [[Displacement (vector)|displacement]] or [[Physics:Velocity|velocity]], 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, 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 [[Euclidean vector|vector]] is separated into its vector components, 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===
{| class="floatright" style="width:20em;"
|align="center" style="background-color:#ddeeff;"| [[Most significant bit|most-significant bit]] ||align="center" colspan="9"|
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''127'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''126'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''2'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''1'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''0'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''−1'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''−2'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''−127'''
|-
|align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''1'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-right:1px solid #aaaaaa;"| '''0'''
|align="center" style="width:2em;"| '''=''' ||align="right" style="width:2em;"| '''−128'''
|-
| colspan=9 style="text-align:center;font-size:90%" | Most computers use [[Two's complement|two's complement]] to represent the sign of an integer.
|}
{{main|Signedness}}
In [[Computing|computing]], 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|bit]] can be used for storing the value of a number. Because of the way integer arithmetic is done within computers, signed number representations usually do not store the sign as a single independent bit, instead using e.g. [[Two's complement|two's complement]].

In contrast, real numbers are stored and manipulated as floating point 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===
[[File:VFPt dipole electric.svg|thumb|right|[[Physics:Electric charge|Electric charge]] may be positive or negative.]]
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 to|up to]] sign'' mean that, for a quantity {{mvar|q}}, it is known that either {{math|1=''q'' = ''Q''}} or {{math|1=''q'' = −''Q''}} for certain {{mvar|Q}}. It is often expressed as {{math|1=''q'' = ±''Q''}}. For real numbers, it means that only the [[Absolute value|absolute value]] {{math|{{!}}''q''{{!}}}} of the quantity is known. For complex numbers and [[Vector space|vectors]], a quantity known up to sign is a stronger condition than a quantity with known [[Norm (mathematics)|magnitude]]: aside {{mvar|Q}} and {{math|−''Q''}}, there are many other possible values of {{mvar|q}} such that {{math|1={{!}}''q''{{!}} = {{!}}''Q''{{!}}}}.
* The [[Parity of a permutation|sign of a permutation]] is defined to be positive if the permutation is even, and negative if the permutation is odd.
* In [[Graph theory|graph theory]], a [[Signed graph|signed graph]] is a graph in which each edge has been marked with a positive or negative sign.
* In [[Mathematical analysis|mathematical analysis]], a [[Signed measure|signed measure]] is a generalization of the concept of [[Measure (mathematics)|measure]] in which the measure of a set may have positive or negative values.
* In a [[Signed-digit representation|signed-digit representation]], each digit of a number may have a positive or negative sign.
* The ideas of [[Signed area|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|determinant]]s. In an (abstract) [[Orientation (vector space)|oriented vector space]], each ordered basis for the vector space can be classified as either positively or negatively oriented.
* In [[Physics:Physics|physics]], any [[Physics:Electric charge|electric charge]] comes with a sign, either positive or negative. By convention, a positive charge is a charge with the same sign as that of a [[Physics:Proton|proton]], and a negative charge is a charge with the same sign as that of an [[Physics:Electron|electron]].

== See also ==
* [[Plus–minus sign]]
* [[Positive element]]
* Signed distance
* [[Signedness]]
* [[Symmetry in mathematics]]

== References ==
<references />
[[Category:Elementary arithmetic]]
[[Category:Numbers]]
[[Category:Mathematical terminology]]

{{Sourceattribution|Sign (mathematics)|1}}