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 (having no sign, or a specific sign of its own), or as belonging to both, negative and positive numbers (having both signs). 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 (negation, or multiplication by −1) 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 (sign of a permutation), sense of orientation or rotation (cw/ccw), one sided limits, and others, below. 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 (having no sign, or a specific sign of its own), or as belonging to both, negative and positive numbers (having both signs). 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 (negation, or multiplication by −1) 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 (sign of a permutation), sense of orientation or rotation (cw/ccw), one sided limits, and others, below. Numbers from various number systems, like integers, rationals, complex numbers, quaternions, ... may have multiple attributes, that fix certain properties of a number. If a number system bears the structure of an ordered ring, like, for example, the integers, it must contain a number that does not change any number when it is added to it (an additive identity element). 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. For other properties, required within a ring, for each such positive number there exists a number less than 0, which, when added to the positive number, yields the result 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), or positive (+), or negative (−), is called its sign, and is often encoded to the real numbers 0, 1, and −1, respectively. Since rational and real numbers are also ordered rings (even fields), 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 (sometimes called negation) of the operand. 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 (which is 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, likewise, the plus sign '+' is associated with positivity. 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 sgn {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 be used to refer to the sign of a number: