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| [[File:PlusMinus.svg|thumb|right|150px|The [[plus and minus signs|plus and minus symbols]] are used to show the sign of a number.]]
| | Claude is her title and she completely digs that title. My job is a production and distribution officer and I'm performing fairly good monetarily. Kansas is our birth location and my parents reside close by. To perform handball is the thing she enjoys most of all.<br><br>Feel free to visit my site - [http://14.63.168.193/index.php?document_srl=249897&mid=freeboard auto warranty] |
| {{distinguish2|the [[sine|sine function]] in trigonometry}}
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| {{for|symbols named "… sign"|table of mathematical symbols}}
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| In [[mathematics]], the concept of '''sign''' originates from the property of every non-zero [[real number]] to be positive or [[negative number|negative]]. [[0 (number)|Zero]] itself is signless, although in some contexts it makes sense to consider a [[signed zero]]. Along its application to real numbers, "change of sign" is used throughout mathematics and [[physics]] to denote the [[additive inverse]] (multiplication to [[−1]]), even for quantities which are not real numbers (so, which are not prescribed to be either positive, negative, or zero). Also, the word "sign" can indicate aspects of mathematical objects that resemble positivity and negativity, such as the [[Parity of a permutation|sign of a permutation]] (see [[#Other meanings|below]]).
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| ==Sign of a number==
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| A [[real number]] is said to be positive if it is [[inequality (mathematics)|greater than]] zero, and [[negative number|negative]] if it is less than zero. The attribute of being positive or negative is called the '''sign''' of the number. Zero itself is not considered to have a sign. Also, signs are not defined for [[complex numbers]], although the [[argument (complex analysis)|argument]] generalizes it in some sense.
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| In common [[numeral system|numeral notation]] (which is used in [[arithmetic]] and elsewhere), the sign of a number is often denoted by placing [[plus and minus signs|a plus sign or a minus sign]] before the number. For example, +3 would denote a positive 3, and −3 would denote a negative 3. When no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of negative numbers through [[subtraction]], the [[minus sign]] is perceived to have a strong association with negative numbers (of the negative sign). Likewise, "+" associates with positivity.
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| In [[algebra]], a minus sign is usually thought of as representing the operation of [[additive inverse]] (sometimes called ''negation''), with the additive inverse of a positive number being negative and the additive inverse of a negative number being positive. In this context, it makes sense to write −(−3) = +3.
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| Any non-zero number can be changed to a positive one using the [[absolute value]] function. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| = 3 and |3| = 3.
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| ===Sign of zero===
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| The number [[0 (number)|zero]] is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the additive inverse of itself.
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| In some contexts, such as [[signed number representations]] in [[computing]], it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers (see [[signed zero]]).
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| One also sees +0 and −0 in [[calculus]] and [[mathematical analysis]] when evaluating [[one-sided limit]]s. This notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively; these behaviours are not necessarily the same.
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| ==={{anchor|non-negative and non-positive}} Terminology for signs===
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| <!-- Several articles link to the above anchor -->
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| Because zero is neither positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number:
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| * A number is '''positive''' if it is greater than zero.
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| * A number is '''negative''' if it is less than zero.
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| * A number is '''non-negative''' if it is greater than or equal to zero.
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| * A number is '''non-positive''' if it is less than or equal to zero.
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| Thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero. For example, the [[absolute value]] of a real number is always non-negative, but is not necessarily positive.
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| The same terminology is sometimes used for [[Function (mathematics)|functions]] that take real or integer values. For example, a function would be called positive if all of its values are positive, or non-negative if all of its values are non-negative.
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| ===Sign convention===
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| {{main|Sign convention}}
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| In many contexts the choice of sign convention (which range of values is considered positive and which negative) is natural, whereas in others the choice is arbitrary subject only to consistency, the latter necessitating an explicit sign convention.
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| ==Sign function==
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| [[Image:Signum function.svg|thumb|200px|Signum function ''y'' = sgn(''x'')]]
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| {{main|Sign function}}
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| The '''sign function''' or '''signum function''' is sometimes used to extract the sign of a number. This function is usually defined as follows:
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| :<math> \sgn(x) = \begin{cases}
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| -1 & \text{if } x < 0, \\
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| ~~\, 0 & \text{if } x = 0, \\
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| ~~\, 1 & \text{if } x > 0. \end{cases}</math>
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| Thus sgn(''x'') is 1 when ''x'' is positive, and sgn(''x'') is −1 when ''x'' is negative. For nonzero values of ''x'', this function can also be defined by the formula
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| :<math> \sgn(x) = \frac{x}{|x|}</math>
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| where |''x''| is the [[absolute value]] of ''x''.
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| ==Meanings of sign==
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| ===Sign of an angle===
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| [[File:Yhvyvczsa7.png|right|thumb|Measuring from the [[x-axis]], angles on the [[unit circle]] count as positive in the [[counterclockwise]] direction, and negative in the [[clockwise]] direction.]]
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| In many contexts, it is common to associate a sign with the measure of an [[angle]], particularly an oriented angle or an angle of [[rotation]]. In such a situation, the sign indicates whether the angle is in the [[clockwise]] or counterclockwise direction. Though different conventions can be used, it is common in [[mathematics]] to have counterclockwise angles count as positive, and clockwise angles count as negative.
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| It is also possible to associate a sign to an angle of rotation in three dimensions, assuming the [[axis of rotation]] has been oriented. Specifically, a [[right-hand rule|right-handed]] rotation around an oriented axis typically counts as positive, while a left-handed rotation counts as negative.
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| ===Sign of a change===
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| When a quantity ''x'' changes over time, the [[Finite difference|change]] in the value of ''x'' is typically defined by the equation
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| :<math>\Delta x = x_\text{final} - x_\text{initial}. \,</math> | |
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| Using this convention, an increase in ''x'' counts as positive change, while a decrease of ''x'' counts as negative change. In [[calculus]], this same convention is used in the definition of the [[derivative]]. As a result, any [[Monotonic function|increasing function]] has positive derivative, while a decreasing function has negative derivative.
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| ===Sign of a direction===
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| In [[analytic geometry]] and [[physics]], it is common to label certain directions as positive or negative. For a basic example, the [[number line]] is usually drawn with positive numbers to the right, and negative numbers to the left:
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| [[File:Number-line.svg|center|600px]]
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| As a result, when discussing [[linear motion]], [[Displacement (vector)|displacement]] or [[velocity]] to the right is usually thought of as being positive, while similar motion to the left is thought of as being negative.
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| 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 component]]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.
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| ===Signedness in computing===
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| <div class="thumb tright"><div style="width:20em;">
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| {|style="width:100%; margin:0;" cellspacing="0"
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| |align="center" style="background-color:#ddeeff;"| [[Most significant bit|most-significant bit]] ||align="center" colspan="9"|
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| |-
| |
| |align="center" style="width:2em; background-color:#ddeeff; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa; border-left:1px solid #aaaaaa;"| '''0'''
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| |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'''
| |
| |-
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| |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'''
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| |align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
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| |align="center" style="width:2em; border-top:1px solid #aaaaaa; border-bottom:1px solid #aaaaaa;"| '''1'''
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| |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'''
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| |-
| |
| |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'''
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| |}
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| {{caption|Most computers use [[two's complement]] to represent the sign of an integer.}}
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| </div></div>
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| {{main|Signedness}}
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| In [[computing]], a numeric value may be either signed or unsigned, depending on whether the computer is keeping track of a sign for the number. By restricting a [[Variable (programming)|variable]] to non-negative values only, one more [[bit]] can be used for storing the value of a number.
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| Because of the way arithmetic is done within computers, the sign of a signed variable is usually not stored as a single independent bit, but is instead stored using [[two's complement]] or some other [[signed number representation]].
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| ===Other meanings===
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| [[File:VFPt_dipole_electric.svg|thumb|right|[[Electric charge]] may be positive or negative.]]
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| In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and the sciences:
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| * Words ''[[up to]] sign'' mean that for a quantity {{mvar|q}} is known that either {{math|1=''q'' = ''Q''}} or {{math|1=''q'' = −''Q''}} for certain {{mvar|Q}}. In is often expressed as {{math|1=''q'' = [[±]]''Q''}}. For real numbers, it means that only the [[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''{{!}}}}.
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| * 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.
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| * In [[graph theory]], a [[signed graph]] is a graph in which each edge has been marked with a positive or negative sign.
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| * In [[mathematical analysis]], a [[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.
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| * In a [[signed-digit representation]], each digit of a number may have a positive or negative sign.
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| * 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]]s.
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| * In [[physics]], any [[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 [[proton]], and a negative charge is a charge with the same sign as that of an [[electron]].
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| == See also ==
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| * [[Positive element]]
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| * [[Symmetry in mathematics]]
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| [[Category:Elementary arithmetic]]
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| [[Category:Numbers]]
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| [[Category:Mathematical terminology]]
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