Sign (mathematics)


In mathematics, the concept of sign originates from the property of every real number being either positive or negative or zero. Depending on local conventions, zero is either considered as being neither a positive nor a negative number, or as belonging to both negative and positive numbers. If not specifically mentioned, this article adheres to the first convention. In some contexts it makes sense to consider a signed zero, e.g., in floating point representations of real numbers within computers. The phrase "change of sign" is associated throughout mathematics and physics to generate the additive inverse of any object that allows for this construction, and 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 other binary aspects of mathematical objects that resemble positivity and negativity, such as odd and even, sense of orientation or rotation, one sided limits, and others, below.
Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions,... may have multiple attributes, that fix certain properties of a number. If a number system bears the structure of an ordered ring, for example, the integers, it must contain a number that does not change any number when it is added to it. This number is generally denoted as Because of the total order in this ring, there are numbers greater than zero, called the positive numbers. For other properties, required within a ring, for each such positive number there exists a number less than which, when added to the positive number, yields the result These numbers less than 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, positive, or negative, is called its sign, and is often encoded to the real numbers and respectively. Since rational and real numbers are also ordered rings, these number systems share the sign attribute.
While in arithmetic a minus sign is usually thought of as representing the binary operation of subtraction, in algebra it is usually thought of as representing the unary operation yielding the additive inverse of the operand. While is its own additive inverse 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 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, the sign of a number is often made explicit by placing a plus or a minus sign before the number. For example, denotes "positive three", and denotes "negative three". 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, likewise, the plus sign "+" is associated with positivity.
Within the convention of zero being neither positive nor negative, a specific sign-value may be assigned to the number value. This is exploited in the -function, as defined for real numbers. In arithmetic, and both denote the same number. There is generally no danger of confusing the value with its sign.
The convention of assigning both signs to 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
The symbols and rarely appear as substitutes for and 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 along positive or negative values, respectively; the two limits need not exist or agree.
When is said to be neither positive nor negative, the following phrases may refer to the sign of a number:
When is said to be both positive and negative, modified phrases are used to refer to the sign of a number:
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.