Sign (mathematics)

In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive 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 of orientation or rotation (cw/ccw), one sided limits, and other concepts described in § Other meanings below.

Sign of a number

Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions, ... may have multiple attributes, that fix certain properties of a number. A number system that bears the structure of an 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. For example, the integers has the structure of an ordered ring. This number is generally denoted as 0. Because of the 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 0 whose sum with the original positive number is 0. These numbers less than 0 are called the negative numbers. The numbers in each such pair are their respective additive inverses. This attribute of a number, being exclusively either zero (0), positive (+), or negative (−), is called its sign, and is often encoded to the real numbers 0, 1, and −1, respectively (similar to the way the sign function is defined). Since rational and real numbers are also ordered rings (in fact ordered 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 of yielding the 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 0 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 common numeral notation (used in arithmetic and elsewhere), the sign of a number is often made explicit by placing a plus or a minus sign before the number. For example, +3 denotes "positive three", and −3 denotes "negative three" (algebraically: the additive inverse of 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 "−" 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 0 may be assigned to the number value 0. This is exploited in the ${\displaystyle \mathrm{sgn}}$-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, although the convention of assigning both signs to 0 does not immediately allow for this discrimination.

In certain European countries, e.g. in Belgium and France, 0 is considered to be both positive and negative following the convention set forth by Nicolas Bourbaki.

In some contexts, such as floating-point representations of real numbers within computers, it is useful to consider signed versions of zero, with signed zeros referring to different, discrete number representations (see signed number representations for more).

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

Terminology for signs

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"—though 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

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.

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 sign of a complex number z can be defined as the quotient of z and its magnitude |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 ${\displaystyle e^{i\pi }=-1.}$ For the definition of a complex sign-function. see § Complex sign function below.

Sign functions

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 ${\displaystyle \{-1,0,1\}.}$ It can be defined as follows: $${\displaystyle \begin{array}{cc}\mathrm{sgn}:& \mathbb{R}\to \{-1,0,1\}\\ & x\mapsto \mathrm{sgn}(x)=\{\begin{array}{ll}-1& \text{if }x<0,\\ 0& \text{if }x=0,\\ 1& \text{if }x>0.\end{array}\end{array}}$$ 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 $${\displaystyle \mathrm{sgn}(x)=\frac{x}{|x|}=\frac{|x|}{x},}$$ where |x| is the absolute value of 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 magnitude of its argument z = x + iy, which can be calculated as $${\displaystyle |z|=\sqrt{z\overline{z}}=\sqrt{x^2+y^2}.}$$

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 $${\displaystyle \mathrm{sgn}(z)=\{\begin{array}{ll}0& \text{for }z=0\\ {\displaystyle \frac{z}{|z|}}=e^{i\phi }& \text{otherwise}.\end{array}}$$

This definition may also be recognized as a normalized vector, that is, a vector whose direction is unchanged, and whose 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.