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Inmathematics, thesign of areal number is 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 oforientation or rotation (cw/ccw),one sided limits, and other concepts described in§ Other meanings below.
Numbers from 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 ring contains 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 order in this ring, there are numbers greater than zero, called thepositive numbers. Another property required for a ring to be ordered is that, for each positive number, there exists a unique corresponding number less than0 whose sum with the original positive number is0. These numbers less than0 are called thenegative numbers. 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 function is defined).[1] Since rational and real numbers are also ordered rings (in fact orderedfields), thesign 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 theunary operation of 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. While0 is 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 inarithmetic and elsewhere), the sign of a number is often made explicit by placinga plus or a minus sign before the number. For example,+3 denotes "positive three", and−3 denotes "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.
Within the convention ofzero being neither positive nor negative, a specific sign-value0 may be assigned to the number value0. This is exploited in the-function, as defined for real numbers.[1] In arithmetic,+0 and−0 both denote the same number0. There is generally no danger of confusing the value with its sign, although the convention of assigning both signs to0 does not immediately allow for this discrimination.
In certain European countries, e.g. in Belgium and France,0 is considered to beboth positive and negative following the convention set forth byNicolas Bourbaki.[2]
In some contexts, such asfloating-point representations of real numbers within computers, it is useful to consider signed versions of zero, withsigned zeros referring to different, discrete number representations (seesigned number representations for more).
The symbols+0 and−0 rarely appear as substitutes for0+ and0−, used incalculus andmathematical analysis forone-sided limits (right-sided limit and left-sided limit, respectively). This notation refers to the behaviour of a function as its real input variable approaches0 along positive (resp., negative) values; the two limits need not exist or agree.
When0 is said to be neither positive nor negative, the following phrases may refer to the sign of a number:
When0 is said to be both positive and negative,[2][verification needed] modified phrases are used to refer to the sign of a number:
For example, theabsolute 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 forfunctions that yield real or other signed values. For example, a function would be called apositive function if its values are positive for all arguments of its domain, or anon-negative function if all of its values are non-negative.
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 value ormagnitude. 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−3 and the absolute value of3 are both equal to3. This is written in symbols as|−3| = 3 and|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 asign 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, thesign of a complex numberz can be defined as the quotientofz and 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 For the definition of a complex sign-function. see§ Complex sign function below.
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.
Thesign function orsignum 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:[1]Thussgn(x) is 1 whenx is positive, andsgn(x) is −1 whenx is negative. For non-zero values ofx, this function can also be defined by the formulawhere|x| is theabsolute value ofx.
While a real number has a 1-dimensional direction, a complex number has a 2-dimensional direction. The complex sign function requires themagnitude of its argumentz =x +iy, which can be calculated as
Analogous to above, thecomplex 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, and0 to0: It may be defined as follows:
Letz be also expressed by its magnitude and one of its argumentsφ asz = |z|⋅eiφ, then[3]
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.
In situations where there are exactly two possibilities on equal footing for an attribute, these are often labelled by convention asplus andminus, 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.
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 theclockwise or counterclockwise direction. Though different conventions can be used, it is common inmathematics to have counterclockwise angles count as positive, and clockwise angles count as negative.[4]
It is also possible to associate a sign to an angle of rotation in three dimensions, assuming that theaxis of rotation has been oriented. Specifically, aright-handed rotation 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.[5]
When a quantityx changes over time, thechange in the value ofx is typically defined by the equation
Using this convention, an increase inx counts as positive change, while a decrease ofx counts as negative change. Incalculus, this same convention is used in the definition of thederivative. As a result, anyincreasing function has positive derivative, while any decreasing function has negative derivative.
When studying one-dimensionaldisplacements andmotions inanalytic geometry andphysics, it is common to label the two possibledirections as positive and negative. Because thenumber line is 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 displacementvector is 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 avelocity in 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 orientations andorientability in general.
| 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 complement to represent the sign of an integer. | |||||||||
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 integervariable to non-negative values only, one morebit 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.
In contrast, real numbers are stored and manipulated asfloating 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.
In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and other sciences:
