[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/sign-mathematics-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/sign-mathematics-wikipedia\/","headline":"Sign (mathematics) – Wikipedia","name":"Sign (mathematics) – Wikipedia","description":"Number property of being positive or negative In mathematics, the sign of a real number is its property of being","datePublished":"2015-11-07","dateModified":"2015-11-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c5\/PlusMinus.svg\/150px-PlusMinus.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c5\/PlusMinus.svg\/150px-PlusMinus.svg.png","height":"103","width":"150"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/sign-mathematics-wikipedia\/","wordCount":7163,"articleBody":"Number property of being positive or negative  In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (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 \u22121) 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 other concepts described in \u00a7\u00a0Other meanings below.Table of ContentsSign of a number[edit]Sign of zero[edit] Terminology for signs[edit]Complex numbers[edit]Sign functions[edit]Real sign function[edit]Complex sign function[edit]Signs per convention[edit]Sign of an angle[edit]Sign of a change[edit]Sign of a direction[edit]Signedness in computing[edit]Other meanings[edit]See also[edit]References[edit]Sign of a number[edit]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 (\u2212), is called its sign, and is often encoded to the real numbers 0, 1, and \u22121, respectively (similar to the way the sign function is defined).[1] 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 (\u22120 = 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 \u2212(\u22123) = 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 \u22123 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 “\u2212” with negative numbers, and the plus sign “+” with positive numbers.Sign of zero[edit]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.[1] In arithmetic, +0 and \u22120 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 signed number representations for more).The symbols +0 and \u22120 rarely appear as substitutes for 0+ and 0\u2212, 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[edit]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”\u2014though 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[edit]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 \u22123 and the absolute value of 3 are both equal to 3. This is written in symbols as |\u22123| = 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 ei\u03c0=\u22121.{displaystyle e^{ipi }=-1.} For the definition of a complex sign-function. see \u00a7\u00a0Complex sign function below.Sign functions[edit]  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 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[edit]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 {\u22121,0,1}.{displaystyle {-1,;0,;1}.} It can be defined as follows:[1]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/sign-mathematics-wikipedia\/#breadcrumbitem","name":"Sign (mathematics) – Wikipedia"}}]}]