Sign (mathematics)

Not to be confused withsine functionin trigonometry.

For symbols named "... sign", seeList of mathematical symbols.

Theplus and minus symbolsare used to show the sign of a number.

Inmathematics,thesignof areal numberis its property of being either positive,negative,or0.Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish betweena positive and a negative zero.

In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for itsadditive inverse(multiplication with−1,negation), an operation which 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 binary aspects of mathematical or scientific objects, such as odd and even (sign of a permutation), sense oforientationor rotation (cw/ccw),one sided limits,and other concepts described in§ Other meaningsbelow.

Sign of a number

Numbersfrom various number systems, likeintegers,rationals,complex numbers,quaternions,octonions,... may have multiple attributes, that fix certain properties of a number. A number system that bears the structure of anordered ringcontains a unique number that when added with any number leaves the latter unchanged. This unique number is known as the system's additiveidentity element.For example, the integers has the structure of an ordered ring. This number is generally denoted as0.Because of thetotal orderin this ring, there are numbers greater than zero, called thepositivenumbers. Another property required for a ring to be ordered is that, for each positive number, there exists a unique corresponding number less than0whose sum with the original positive number is0.These numbers less than0are called thenegativenumbers. The numbers in each such pair are their respectiveadditive inverses.This attribute of a number, being exclusively eitherzero(0),positive(+),ornegative(−),is called itssign,and is often encoded to the real numbers0,1,and−1,respectively (similar to the way thesign functionis defined).Since rational and real numbers are also ordered rings (in fact orderedfields), thesignattribute 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 theunary operationof yielding theadditive inverse(sometimes callednegation) of the operand. Abstractly then, the difference of two number is the sum of the minuend with the additive inverse of the subtrahend. While0is its own additive inverse (−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−(−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 commonnumeral notation(used inarithmeticand elsewhere), the sign of a number is often made explicit by placinga plus or a minus signbefore the number. For example,+3denotes "positive three", and−3denotes "negative three" (algebraically: the additive inverse of3). 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 ofzerobeing neither positive nor negative, a specific sign-value0may be assigned to the number value0.This is exploited in the$\operatorname {sgn}$-function,as defined for real numbers.In arithmetic,+0and−0both denote the same number0.There is generally no danger of confusing the value with its sign, although the convention of assigning both signs to0does not immediately allow for this discrimination.

In certain European countries, e.g. in Belgium and France,0is considered to bebothpositive and negative following the convention set forth byNicolas Bourbaki.

In some contexts, such asfloating-point representationsof real numbers within computers, it is useful to consider signed versions of zero, withsigned zerosreferring to different, discrete number representations (seesigned number representationsfor more).

The symbols+0and−0rarely appear as substitutes for0⁺and0⁻,used incalculusandmathematical analysisforone-sided limits(right-sided limit and left-sided limit, respectively). This notation refers to the behaviour of a function as its real input variable approaches0along positive (resp., negative) values; the two limits need not exist or agree.

Terminology for signs

When0is said to be neither positive nor negative, the following phrases may refer to the sign of a number:

• A number ispositiveif it is greater than zero.
• A number isnegativeif it is less than zero.
• A number isnon-negativeif it is greater than or equal to zero.
• A number isnon-positiveif it is less than or equal to zero.

When0is said to be both positive and negative,modified phrases are used to refer to the sign of a number:

• A number isstrictly positiveif it is greater than zero.
• A number isstrictly negativeif it is less than zero.
• A number ispositiveif it is greater than or equal to zero.
• A number isnegativeif it is less than or equal to zero.

For example, theabsolute valueof 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 forfunctionsthat yield real or other signed values. For example, a function would be called apositive functionif its values are positive for all arguments of its domain, or anon-negative functionif 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 calledabsolute valueormagnitude.Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, itsabsolute value.

For example, the absolute value of−3and the absolute value of3are both equal to3.This is written in symbols as|−3| = 3and|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 asignfor 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, thesign of a complex numberzcan be defined as the quotientofzand itsmagnitude|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$e^{i\pi }=-1.$For the definition of a complex sign-function. see§ Complex sign functionbelow.

Sign functions

Real sign functiony= sgn(x)

Main article: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

Thesign functionorsignum functionextracts the sign of a real number, by mapping the set of real numbers to the set of the three reals$\{-1,\;0,\;1\}.$It can be defined as follows:

$${\begin{aligned}\operatorname {sgn}:{}&\mathbb {R} \to \{-1,0,1\}\\&x\mapsto \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\~~\,0&{\text{if }}x=0,\\~~\,1&{\text{if }}x>0.\end{cases}}\end{aligned}}$$

Thussgn(x)is 1 whenxis positive, andsgn(x)is −1 whenxis negative. For non-zero values ofx,this function can also be defined by the formula

$$\operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}},$$

where|x|is theabsolute valueofx.

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 themagnitudeof its argumentz=x+iy,which can be calculated as

$$|z|={\sqrt {z{\bar {z}}}}={\sqrt {x^{2}+y^{2}}}.$$

Analogous to above, thecomplex sign functionextracts the complex sign of a complex number by mapping the set of non-zero complex numbers to the set of unimodular complex numbers, and0to0:$\{z\in \mathbb {C}:|z|=1\}\cup \{0\}.$It may be defined as follows:

Letzbe also expressed by its magnitude and one of its argumentsφasz= |z|⋅e^iφ,then

$$\operatorname {sgn}(z)={\begin{cases}0&{\text{for }}z=0\\{\dfrac {z}{|z|}}=e^{i\varphi }&{\text{otherwise}}.\end{cases}}$$

This definition may also be recognized as a normalized vector, that is, a vector whose direction is unchanged, and whose length is fixed tounity.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 article:Sign convention

In situations where there are exactly two possibilities on equal footing for an attribute, these are often labelled by convention asplusandminus,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

Main article:Angle § Sign

Measuring from thex-axis,angles on theunit circlecount as positive in thecounterclockwisedirection, and negative in theclockwisedirection.

In many contexts, it is common to associate a sign with the measure of anangle,particularly an oriented angle or anangle of rotation.In such a situation, the sign indicates whether the angle is in theclockwiseor counterclockwise direction. Though different conventions can be used, it is common inmathematicsto 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 theaxis of rotationhas been oriented. Specifically, aright-handedrotation around an oriented axis typically counts as positive, while a left-handed rotation counts as negative.

An angle which is the negative of a given angle has an equal arc, but theopposite axis.

Sign of a change

When a quantityxchanges over time, thechangein the value ofxis typically defined by the equation

$$\Delta x=x_{\text{final}}-x_{\text{initial}}.$$

Using this convention, an increase inxcounts as positive change, while a decrease ofxcounts as negative change. Incalculus,this same convention is used in the definition of thederivative.As a result, anyincreasing functionhas positive derivative, while any decreasing function has negative derivative.

Sign of a direction

When studying one-dimensionaldisplacementsandmotionsinanalytic geometryandphysics,it is common to label the two possibledirectionsas positive and negative. Because thenumber lineis usually drawn with positive numbers to the right, and negative numbers to the left, a common convention is for motions to the right to be given a positive sign, and for motions to the left to be given a negative sign.

On theCartesian plane,the rightward and upward directions are usually thought of as positive, with rightward being the positivex-direction, and upward being the positivey-direction. If a displacementvectoris separated into itsvector 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.

Likewise, a negativespeed(rate of change of displacement) implies avelocityin theopposite direction,i.e., receding instead of advancing; a special case is theradial speed.

In3D space,notions related to sign can be found in the twonormal orientationsandorientabilityin general.

Signedness in computing

most-significant bit
0	1	1	1	1	1	1	1	=	127
0	1	1	1	1	1	1	0	=	126
0	0	0	0	0	0	1	0	=	2
0	0	0	0	0	0	0	1	=	1
0	0	0	0	0	0	0	0	=	0
1	1	1	1	1	1	1	1	=	−1
1	1	1	1	1	1	1	0	=	−2
1	0	0	0	0	0	0	1	=	−127
1	0	0	0	0	0	0	0	=	−128
Most computers usetwo's complementto represent the sign of an integer.

Main article:Signedness

Incomputing,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 integervariableto non-negative values only, one morebitcan be used for storing the value of a number. Because of the way integer arithmetic is done within computers,signed number representationsusually do not store the sign as a single independent bit, instead using e.g.two's complement.

In contrast, real numbers are stored and manipulated asfloating pointvalues. 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

Electric chargemay 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:

• Wordsup tosignmean that, for a quantityq,it is known that eitherq=Qorq= −Qfor certainQ.It is often expressed asq=±Q.For real numbers, it means that only theabsolute value|q|of the quantity is known. Forcomplex numbersandvectors,a quantity known up to sign is a stronger condition than a quantity with knownmagnitude:asideQand−Q,there are many other possible values ofqsuch that|q| = |Q|.
• Thesign of a permutationis defined to be positive if the permutation is even, and negative if the permutation is odd.
• Ingraph theory,asigned graphis a graph in which each edge has been marked with a positive or negative sign.
• Inmathematical analysis,asigned measureis a generalization of the concept ofmeasurein which the measure of a set may have positive or negative values.
  • The concept ofsigned distanceis used to conveyside,inside or out.
  • The ideas ofsigned areaandsigned volumeare sometimes used when it is convenient for certain areas or volumes to count as negative. This is particularly true in the theory ofdeterminants.In an (abstract)oriented vector space,each ordered basis for the vector space can be classified as either positively or negatively oriented.
• In asigned-digit representation,each digit of a number may have a positive or negative sign.
• Inphysics,anyelectric chargecomes with a sign, either positive or negative. By convention, a positive charge is a charge with the same sign as that of aproton,and a negative charge is a charge with the same sign as that of anelectron.

See also

• Percent sign
• Plus–minus sign
• Positive element
• Signedness
• Symmetry in mathematics