Sign (mathematics)

Within the convention of zero being neither positive nor negative, a specific sign-value 0 may be assigned to the number value 0. This is exploited in the ${\displaystyle \operatorname {sgn} }$-function, as defined for real numbers. In arithmetic, +0 and −0 both denote the same number 0. There is generally no danger of confusing the value with its sign.

The convention of assigning both signs to 0 does not immediately allow for this discrimination.

In some contexts, especially in computing, it is useful to consider signed versions of zero, with signed zeros referring to different, discrete number representations (see also signed number representations.)

The symbols +0 and −0 rarely appear as substitutes for 0⁺ and 0⁻, used in calculus and mathematical analysis for one-sided limits. This notation refers to the behaviour of a function as its real input variable approaches 0 along positive or negative values, respectively; the two limits need not exist or agree.

When 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 0 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 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" and is not necessarily "strictly positive".

The same terminology is sometimes used for 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 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 or magnitude. Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its absolute value. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3. This is written in symbols as |−3| = 3 and |3| = 3. So 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. So 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 sign of a complex number z can be defined as the quotient of z and its magnitude |z|. Since the magnitude of the complex number is divided out, the resulting sign of the complex number represents in some sense its complex argument. Comparing this with the sign of real numbers, recalls ${\displaystyle e^{i\pi }=-1.}$ See below for the definition of a complex sign-function.

Real sign function y = sgn(x)

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 maps it to a predefined value, then 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.

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 ${\displaystyle \{-1,\;0,\;1\}.}$ It can be defined as follows:

${\displaystyle \operatorname {sgn}$ :\mathbb {R} \to \{x\in \mathbb {R} :|x|=1\}\cup \{0\}}

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

Thus sgn(x) is 1 when x is positive, and sgn(x) is −1 when x is negative. For non-zero values of x, this function can also be defined by the formula

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

where |x| is the absolute value of x.

While a real number has a 1-dimensional direction, a complex number has a 2-dimensional direction. The complex sign function requires the magnitude of its argument z = x + iy, which can be calculated as

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

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 0 to 0: ${\displaystyle \{z\in \mathbb {C}$ :|z|=1\}\cup \{0\}.} It may be defined as follows:

Let z be also expressed by its magnitude and one of its arguments φ as z = |z|⋅e^iφ, then

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

This definition may also be recognized as a normalized vector, that is the vector's direction remains unchanged, and its length is fixed to 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.

Measuring from the x-axis, angles on the 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, 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.

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

When a quantity x changes over time, the change in the value of x is typically defined by the equation

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

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 increasing function has positive derivative, while a decreasing function has negative derivative.

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:

As a result, when discussing linear motion, 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.

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 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.

In 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 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 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.

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 the sciences:

• Words up to sign mean that, for a quantity q, it is known that either q = Q or q = −Q for certain Q. It is often expressed as q = ±Q. For real numbers, it means that only the absolute value |q| of the quantity is known. For complex numbers and vectors, a quantity known up to sign is a stronger condition than a quantity with known magnitude: aside Q and −Q, there are many other possible values of q such that |q| = |Q|.
• The sign of a permutation is defined to be positive if the permutation is even, and negative if the permutation is odd.
• In graph theory, a signed graph is a graph in which each edge has been marked with a positive or negative sign.
• In mathematical analysis, a signed measure 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, 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 determinants. In an (abstract) oriented vector space, each ordered basis for the vector space can be classified as either positively or negatively oriented.
• 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.