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In mathematics, the concept of sign originates from the property that every real number is either positive, negative or zero. Depending on local conventions, zero is either considered as being neither a positive number, 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).[citation needed] Whenever not specifically mentioned, this article adheres to the first convention.

In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of 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,[1] 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 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. 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 (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), 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).[2] Since rational and real numbers are also ordered rings (even fields), these number systems share the same 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 (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 signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of 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,[1] 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 other concepts described in § Other meanings 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 (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), 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).[2] Since rational and real numbers are also ordered rings (even fields), these number systems share the same 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 (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 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 (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.

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 -function, as defined for real numbers.[2] 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 some contexts, especially in computing, it is useful to consider signed versions of zero, with signed zeros referring to different, discrete number representations (see 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 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).[3] 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.

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

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 n

    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| = 3For 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|. Since the magnitude of the complex number is divided out, the resulting sign of the complex number 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.

    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