Misplaced Pages

#346653

116-703: In mathematics , the result of 0/0 is typically not defined as a number and may therefore be represented by NaN in computing systems. The square root of a negative number is not a real number , and is therefore also represented by NaN in compliant computing systems. NaNs may also be used to represent missing values in computations. Two separate kinds of NaNs are provided, termed quiet NaNs and signaling NaNs . Quiet NaNs are used to propagate errors resulting from invalid operations or values. Signaling NaNs can support advanced features such as mixing numerical and symbolic computation or other extensions to basic floating-point arithmetic. In floating-point calculations, NaN

232-411: A n ∧ b n ) {\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})} , for all b 1 , b 2 , … , b n ∈ { 0 , 1 } {\displaystyle b_{1},b_{2},\dots ,b_{n}\in \{0,1\}} . Another way to express this

348-491: A n ) {\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})} for all a 1 , … , a n ∈ { 0 , 1 } {\displaystyle a_{1},\dots ,a_{n}\in \{0,1\}} . Negation is a self dual logical operator. In first-order logic , there are two quantifiers, one is the universal quantifier ∀ {\displaystyle \forall } (means "for all") and

464-441: A Heyting algebra . These algebras provide a semantics for classical and intuitionistic logic. The negation of a proposition p is notated in different ways, in various contexts of discussion and fields of application. The following table documents some of these variants: The notation N p {\displaystyle Np} is Polish notation . In set theory , ∖ {\displaystyle \setminus }

580-740: A payload ), these will occasionally be found in string representations of NaNs, too. Some examples are: Not all languages admit the existence of multiple NaNs. For example, ECMAScript only uses one NaN value throughout. Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

696-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

812-573: A subnormal number , or zero ). IEEE 754 NaNs are encoded with the exponent field filled with ones (like infinity values), and some non-zero number in the significand field (to make them distinct from infinity values); this allows the definition of multiple distinct NaN values, depending on which bits are set in the significand field, but also on the value of the leading sign bit (but applications are not required to provide distinct semantics for those distinct NaN values). For example, an IEEE 754 single precision (32-bit) NaN would be encoded as where s

928-399: A NaN and adds 1 five times in a row, each addition results in a NaN, but there is no need to check each calculation because one can just note that the final result is NaN. However, depending on the language and the function, NaNs can silently be removed from a chain of calculations where one calculation in the chain would give a constant result for all other floating-point values. For example,

1044-524: A NaN is signaling or quiet. Two different implementations, with reversed meanings, resulted: The former choice has been preferred as it allows the implementation to quiet a signaling NaN by just setting the signaling/quiet bit to 1. The reverse is not possible with the latter choice because setting the signaling/quiet bit to 0 could yield an infinity. The 2008 and 2019 revisions of the IEEE ;754 standard make formal requirements and recommendations for

1160-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

1276-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

SECTION 10

#1732786815347

1392-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

1508-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

1624-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

1740-1062: A person x in all humans who is not mortal", or "there exists someone who lives forever". There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of P {\displaystyle P} to both Q {\displaystyle Q} and ¬ Q {\displaystyle \neg Q} , infer ¬ P {\displaystyle \neg P} ; this rule also being called reductio ad absurdum ), negation elimination (from P {\displaystyle P} and ¬ P {\displaystyle \neg P} infer Q {\displaystyle Q} ; this rule also being called ex falso quodlibet ), and double negation elimination (from ¬ ¬ P {\displaystyle \neg \neg P} infer P {\displaystyle P} ). One obtains

1856-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

1972-672: A primitive rule ex falso quodlibet . As in mathematics, negation is used in computer science to construct logical statements. The exclamation mark " ! " signifies logical NOT in B , C , and languages with a C-inspired syntax such as C++ , Java , JavaScript , Perl , and PHP . " NOT " is the operator used in ALGOL 60 , BASIC , and languages with an ALGOL- or BASIC-inspired syntax such as Pascal , Ada , Eiffel and Seed7 . Some languages (C++, Perl, etc.) provide more than one operator for negation. A few languages like PL/I and Ratfor use ¬ for negation. Most modern languages allow

2088-441: A proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in intuitionistic logic , the weaker equivalence ¬ ¬ ¬ P ≡ ¬ P {\displaystyle \neg \neg \neg P\equiv \neg P} does hold. This

2204-400: A qNaN. However, math libraries have typically returned 1 for pow(1, y ) for any real number y , and even when y is an infinity . Similarly, they produce 1 for pow( x , 0) even when x is 0 or an infinity. The rationale for returning the value 1 for the indeterminate forms was that the value of functions at singular points can be taken as a particular value if that value is in

2320-467: A quiet NaN ( qNaN ). Most floating-point operations on a signaling NaN ( sNaN ) signal the invalid-operation exception ; the default exception action is then the same as for qNaN operands and they produce a qNaN if producing a floating-point result. The propagation of quiet NaNs through arithmetic operations allows errors to be detected at the end of a sequence of operations without extensive testing during intermediate stages. For example, if one starts with

2436-414: A result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko's theorem . De Morgan's laws provide a way of distributing negation over disjunction and conjunction : Let ⊕ {\displaystyle \oplus } denote the logical xor operation. In Boolean algebra , a linear function

SECTION 20

#1732786815347

2552-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

2668-528: A similar concept, NaR (Not a Real), where NaR = NaR holds. There are three kinds of operations that can return NaN: NaNs may also be explicitly assigned to variables, typically as a representation for missing values. Prior to the IEEE standard, programmers often used a special value (such as −99999999) to represent undefined or missing values, but there was no guarantee that they would be handled consistently or correctly. NaNs are not necessarily generated in all

2784-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

2900-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

3016-451: A trap handler could decode the sNaN and return an index to the computed result. In practice, this approach is faced with many complications. The treatment of the sign bit of NaNs for some simple operations (such as absolute value ) is different from that for arithmetic operations. Traps are not required by the standard. IEEE 754-2019 recommends the operations getPayload , setPayload , and setPayloadSignaling be implemented, standardizing

3132-565: A way of reducing the number of necessary parentheses, one may introduce precedence rules : ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, P ∨ Q ∧ ¬ R → S {\displaystyle P\vee Q\wedge {\neg R}\rightarrow S} is short for ( P ∨ ( Q ∧ ( ¬ R ) ) ) → S . {\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.} Here

3248-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

3364-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

3480-561: Is flat " and "a field is always a ring ". Negation In logic , negation , also called the logical not or logical complement , is an operation that takes a proposition P {\displaystyle P} to another proposition "not P {\displaystyle P} ", written ¬ P {\displaystyle \neg P} , ∼ P {\displaystyle {\mathord {\sim }}P} or P ¯ {\displaystyle {\overline {P}}} . It

3596-420: Is logical consequence and ⊥ {\displaystyle \bot } is absolute falsehood ). Conversely, one can define ⊥ {\displaystyle \bot } as Q ∧ ¬ Q {\displaystyle Q\land \neg Q} for any proposition Q (where ∧ {\displaystyle \land } is logical conjunction ). The idea here

NaN - Misplaced Pages Continue

3712-431: Is NaN or non-NaN. The comparison predicates are either signaling or non-signaling on quiet NaN operands; the signaling versions signal the invalid-operation exception for such comparisons (i.e., by default, this just sets the corresponding status flag in addition to the behavior of the non-signaling versions). The equality and inequality predicates are non-signaling. The other standard comparison predicates associated with

3828-472: Is a table that shows a commonly used precedence of logical operators. Within a system of classical logic , double negation, that is, the negation of the negation of a proposition P {\displaystyle P} , is logically equivalent to P {\displaystyle P} . Expressed in symbolic terms, ¬ ¬ P ≡ P {\displaystyle \neg \neg P\equiv P} . In intuitionistic logic ,

3944-411: Is also used to indicate 'not in the set of': U ∖ A {\displaystyle U\setminus A} is the set of all members of U that are not members of A . Regardless how it is notated or symbolized , the negation ¬ P {\displaystyle \neg P} can be read as "it is not the case that P ", "not that P ", or usually more simply as "not P ". As

4060-449: Is an operation on one logical value , typically the value of a proposition , that produces a value of true when its operand is false, and a value of false when its operand is true. Thus if statement P {\displaystyle P} is true, then ¬ P {\displaystyle \neg P} (pronounced "not P") would then be false; and conversely, if ¬ P {\displaystyle \neg P}

4176-632: Is because in intuitionistic logic, ¬ P {\displaystyle \neg P} is just a shorthand for P → ⊥ {\displaystyle P\rightarrow \bot } , and we also have P → ¬ ¬ P {\displaystyle P\rightarrow \neg \neg P} . Composing that last implication with triple negation ¬ ¬ P → ⊥ {\displaystyle \neg \neg P\rightarrow \bot } implies that P → ⊥ {\displaystyle P\rightarrow \bot } . As

4292-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

4408-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

4524-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

4640-426: Is formulated using a primitive absurdity sign ⊥ {\displaystyle \bot } . In this case the rule says that from P {\displaystyle P} and ¬ P {\displaystyle \neg P} follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity. Typically

4756-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

NaN - Misplaced Pages Continue

4872-541: Is interpreted intuitively as being true when P {\displaystyle P} is false, and false when P {\displaystyle P} is true. For example, if P {\displaystyle P} is "Spot runs", then "not P {\displaystyle P} " is "Spot does not run". Negation is a unary logical connective . It may furthermore be applied not only to propositions, but also to notions , truth values , or semantic values more generally. In classical logic , negation

4988-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

5104-421: Is negative because " x < 0 " yields true) To demonstrate logical negation: Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e. will have identical results for any input (depending on the compiler used, the actual instructions performed by the computer may differ). In C (and some other languages descended from C), double negation ( !!x )

5220-513: Is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic , according to the Brouwer–Heyting–Kolmogorov interpretation , the negation of a proposition P {\displaystyle P} is the proposition whose proofs are the refutations of P {\displaystyle P} . An operand of a negation is a negand , or negatum . Classical negation

5336-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

5452-412: Is not the same as infinity , although both are typically handled as special cases in floating-point representations of real numbers as well as in floating-point operations. An invalid operation is also not the same as an arithmetic overflow (which would return an infinity or the largest finite number in magnitude) or an arithmetic underflow (which would return the smallest normal number in magnitude,

5568-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

5684-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

5800-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

5916-474: Is one such that: If there exists a 0 , a 1 , … , a n ∈ { 0 , 1 } {\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}} , f ( b 1 , b 2 , … , b n ) = a 0 ⊕ ( a 1 ∧ b 1 ) ⊕ ⋯ ⊕ (

SECTION 50

#1732786815347

6032-408: Is particularly acute for the exponentiation function pow( x , y ) = x . The expressions 0, ∞ and 1 are considered indeterminate forms when they occur as limits (just like ∞ × 0), and the question of whether zero to the zero power should be defined as 1 has divided opinion. If the output is considered as undefined when a parameter is undefined, then pow(1, qNaN) should produce

6148-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

6264-476: Is supported, the operation yields an "invalid" floating-point exception (as required by the IEEE standard) and an unspecified value. Perl 's Math::BigInt package uses "NaN" for the result of strings that do not represent valid integers. Different operating systems and programming languages may have different string representations of NaN. Since, in practice, encoded NaNs have a sign, a quiet/signaling bit and optional 'diagnostic information' (sometimes called

6380-454: Is that any contradiction is false, and while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic , where contradictions are not necessarily false. As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR. Algebraically, classical negation corresponds to complementation in a Boolean algebra , and intuitionistic negation to pseudocomplementation in

6496-405: Is that each variable always makes a difference in the truth-value of the operation, or it never makes a difference. Negation is a linear logical operator. In Boolean algebra , a self dual function is a function such that: f ( a 1 , … , a n ) = ¬ f ( ¬ a 1 , … , ¬

6612-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

6728-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

6844-493: Is the sign (most often ignored in applications) and the x sequence represents a non-zero number (the value zero encodes infinities). In practice, the most significant bit from x is used to determine the type of NaN: " quiet NaN " or " signaling NaN " (see details in Encoding ). The remaining bits encode a payload (most often ignored in applications). Floating-point operations other than ordered comparisons normally propagate

6960-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

7076-489: Is true, then P {\displaystyle P} would be false. The truth table of ¬ P {\displaystyle \neg P} is as follows: Negation can be defined in terms of other logical operations. For example, ¬ P {\displaystyle \neg P} can be defined as P → ⊥ {\displaystyle P\rightarrow \bot } (where → {\displaystyle \rightarrow }

SECTION 60

#1732786815347

7192-527: Is used as an idiom to convert x to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not special in this regard, it is sometimes important to ensure that a canonical value is used, for example for printing or if the number is subsequently used for arithmetic operations. The convention of using ! to signify negation occasionally surfaces in ordinary written speech, as computer-related slang for not . For example,

7308-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

7424-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

7540-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

7656-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

7772-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

7888-501: The <, ≤, =, ≥, > mathematical symbols (or equivalent notation in programming languages) return false on an unordered relation. So, for instance, NOT ( x < y ) is not logically equivalent to x ≥ y : on unordered, i.e. when x or y is NaN, the former returns true while the latter returns false. However, ≠ is defined as the negation of =, thus it returns true on unordered. From these rules, comparing x with itself, x ≠ x or x = x , can be used to test whether x

8004-504: The 'payload' of the NaN. If an operation has a single NaN input and propagates it to the output, the result NaN's payload should be that of the input NaN (this is not always possible for binary formats when the signaling/quiet state is encoded by an is_signaling flag, as explained above). If there are multiple NaN inputs, the result NaN's payload should be from one of the input NaNs; the standard does not specify which. A number of systems have

8120-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

8236-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

8352-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

8468-406: The IEEE 754 floating-point formats), the first number may be either less than, equal to, or greater than the second number. This gives three possible relations. But when at least one operand of a comparison is NaN, this trichotomy does not apply, and a fourth relation is needed: unordered . In particular, two NaN values compare as unordered, not as equal. As specified, the predicates associated with

8584-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

8700-477: The above cases. If an operation can produce an exception condition and traps are not masked then the operation will cause a trap instead. If an operand is a quiet NaN, and there is also no signaling NaN operand, then there is no exception condition and the result is a quiet NaN. Explicit assignments will not cause an exception even for signaling NaNs. In general, quiet NaNs, or qNaNs, do not raise any additional exceptions, as they propagate through most operations. But

8816-421: The above mathematical symbols are all signaling if they receive a NaN operand. The standard also provides non-signaling versions of these other predicates. The predicate isNaN( x ) determines whether a value is a NaN and never signals an exception, even if x is a signaling NaN. The IEEE floating-point standard requires that NaN ≠ NaN hold. In contrast, the 2022 private standard of posit arithmetic has

8932-471: The above statement to be shortened from if (!(r == t)) to if (r != t) , which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs. In computer science there is also bitwise negation . This takes the value given and switches all the binary 1s to 0s and 0s to 1s. See bitwise operation . This is often used to create ones' complement or " ~ " in C or C++ and two's complement (just simplified to " - " or

9048-417: The access to payloads to streamline application use. According to the IEEE 754-2019 background document, this recommendation should be interpreted as "required for new implementations, with reservation for backward compatibility". In IEEE 754 interchange formats, NaNs are identified by specific, pre-defined bit patterns unique to NaNs. The sign bit does not matter. Binary format NaNs are represented with

9164-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

9280-489: The calculation x may produce the result 1, even where x is NaN, so checking only the final result would obscure the fact that a calculation before the x resulted in a NaN. In general, then, a later test for a set invalid flag is needed to detect all cases where NaNs are introduced (see Function definition below for further details). In section 6.2 of the old IEEE 754-2008 standard, there are two anomalous functions (the maxNum and minNum functions, which return

9396-400: The concept of a "canonical NaN", where one specific NaN value is chosen to be the only possible qNaN generated by floating-point operations not having a NaN input. The value is usually chosen to be a quiet NaN with an all-zero payload and an arbitrarily-defined sign bit. Using a limited amount of NaN representations allows the system to use other possible NaN values for non-arithmetic purposes,

9512-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

9628-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

9744-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

9860-643: The encoding of the signaling/quiet state. For IEEE 754-2008 conformance, the meaning of the signaling/quiet bit in recent MIPS processors is now configurable via the NAN2008 field of the FCSR register. This support is optional in MIPS Release ;3 and required in Release ;5. The state/value of the remaining bits of the significand field are not defined by the standard. This value is called

9976-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

10092-650: The exponent must be an integer, and powr( x , y ) , which returns a NaN whenever a parameter is a NaN or the exponentiation would give an indeterminate form . Most fixed-size integer formats cannot explicitly indicate invalid data. In such a case, when converting NaN to an integer type, the IEEE ;754 standard requires that the invalid-operation exception be signaled. For example in Java , such operations throw instances of java.lang.ArithmeticException . In C , they lead to undefined behavior , but if annex F

10208-423: The exponential field filled with ones (like infinity values), and some non-zero number in the significand field (to make them distinct from infinity values). The original IEEE 754 standard from 1985 ( IEEE 754-1985 ) only described binary floating-point formats, and did not specify how the signaling/quiet state was to be tagged. In practice, the most significant bit of the significand field determined whether

10324-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

10440-458: The indication of an error. Another view, and the one taken by the ISO ;C99 and IEEE 754-2008 standards in general, is that if the function has multiple arguments and the output is uniquely determined by all the non-NaN inputs (including infinity), then that value should be the result. Thus for example the value returned by hypot(±∞, qNaN) and hypot(qNaN, ±∞) is +∞. The problem

10556-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

10672-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

10788-430: The intuitionistic negation ¬ P {\displaystyle \neg P} of P {\displaystyle P} is defined as P → ⊥ {\displaystyle P\rightarrow \bot } . Then negation introduction and elimination are just special cases of implication introduction ( conditional proof ) and elimination ( modus ponens ). In this case one must also add as

10904-549: The invalid-operation exception is signaled by some operations that do not return a floating-point value, such as format conversions or certain comparison operations. Signaling NaNs, or sNaNs, are special forms of a NaN that, when consumed by most operations, should raise the invalid operation exception and then, if appropriate, be "quieted" into a qNaN that may then propagate. They were introduced in IEEE 754 . There have been several ideas for how these might be used: When encountered,

11020-400: The limit the value for all but a vanishingly small part of a ball around the limit value of the parameters. The 2008 version of the IEEE 754 standard says that pow(1, qNaN) and pow(qNaN, 0) should both return 1 since they return 1 whatever else is used instead of quiet NaN. Moreover, ISO C99, and later IEEE 754-2008, chose to specify pow(−1, ±∞) = 1 instead of qNaN;

11136-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

11252-503: The maximum and the minimum, respectively, of two operands that are expected to be numbers) that favor numbers — if just one of the operands is a NaN then the value of the other operand is returned. The IEEE 754-2019 revision has replaced these functions as they are not associative (when a signaling NaN appears in an operand). Comparisons are specified by the IEEE 754 standard to take into account possible NaN operands. When comparing two real numbers, or extended real numbers (as in

11368-421: The most important being "NaN-boxing", i.e. using the payload for arbitrary data. (This concept of "canonical NaN" is not the same as the concept of a "canonical encoding" in IEEE 754.) There are differences of opinion about the proper definition for the result of a numeric function that receives a quiet NaN as input. One view is that the NaN should propagate to the output of the function in all cases to propagate

11484-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

11600-412: The negative sign since this is equivalent to taking the arithmetic negative value of the number) as it basically creates the opposite (negative value equivalent) or mathematical complement of the value (where both values are added together they create a whole). To get the absolute (positive equivalent) value of a given integer the following would work as the " - " changes it from negative to positive (it

11716-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

11832-600: The other is the existential quantifier ∃ {\displaystyle \exists } (means "there exists"). The negation of one quantifier is the other quantifier ( ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) {\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)} and ¬ ∃ x P ( x ) ≡ ∀ x ¬ P ( x ) {\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)} ). For example, with

11948-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

12064-494: The predicate P as " x is mortal" and the domain of x as the collection of all humans, ∀ x P ( x ) {\displaystyle \forall xP(x)} means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is ¬ ∀ x P ( x ) ≡ ∃ x ¬ P ( x ) {\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)} , meaning "there exists

12180-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

12296-414: The reason of this choice is given in the C rationale: "Generally, C99 eschews a NaN result where a numerical value is useful. ... The result of pow(−2, ∞) is +∞, because all large positive floating-point values are even integers." To satisfy those wishing a more strict interpretation of how the power function should act, the 2008 standard defines two additional power functions: pown( x , n ) , where

12412-517: The rules for intuitionistic negation the same way but by excluding double negation elimination. Negation introduction states that if an absurdity can be drawn as conclusion from P {\displaystyle P} then P {\displaystyle P} must not be the case (i.e. P {\displaystyle P} is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination

12528-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

12644-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

12760-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

12876-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

12992-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

13108-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

13224-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

13340-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

13456-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

#346653