158-496: An integer is the number zero ( 0 ), a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number ( −1 , −2, −3, . . .). The negations or additive inverses of the positive natural numbers are referred to as negative integers . The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } . The set of natural numbers N {\displaystyle \mathbb {N} }
316-443: A . To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule: precisely when Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using [( a , b )] to denote the equivalence class having ( a , b ) as a member, one has: The negation (or additive inverse) of an integer
474-524: A Sanskrit word Shunye or shunya to refer to the concept of void . In mathematics texts this word often refers to the number zero. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi , an early example of an algebraic grammar for the Sanskrit language (also see Pingala ). There are other uses of zero before Brahmagupta, though the documentation
632-521: A finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the Cantor set during this period. The fifth paper in this series, " Grundlagen einer allgemeinen Mannigfaltigkeitslehre" (" Foundations of
790-404: A , b positive and the other negative. The incorrect use of this identity, and the related identity in the case when both a and b are negative even bedeviled Euler . This difficulty eventually led him to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. The 18th century saw
948-550: A 1-to-1 correspondence between the points of the unit square and the points of a unit line segment . In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n , there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n -dimensional space . About this discovery Cantor wrote to Dedekind: " Je le vois, mais je ne le crois pas! " ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and
1106-461: A 1-to-1 correspondence with the natural numbers , and proved that the rational numbers are denumerable. He also proved that n -dimensional Euclidean space R has the same power as the real numbers R , as does a countably infinite product of copies of R . While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension , stressing that his mapping between
1264-416: A General Theory of Aggregates" ), published in 1883, was the most important of the six and was also published as a separate monograph . It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines
1422-546: A Latin commentary on Book 1 of Spinoza's Ethica . Trendelenburg was also the examiner of Cantor's Habilitationsschrift . In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim , as well as theologians such as Cardinal Johann Baptist Franzelin , who once replied by equating
1580-603: A base 4, base 5 "finger" abacus. By 130 AD, Ptolemy , influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals . Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica ( Almagest ),
1738-664: A center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867. Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle , where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle. In 1874, Cantor married Vally Guttmann. They had six children,
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#17327720619631896-423: A colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of
2054-509: A construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers. Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in
2212-552: A devout Lutheran Christian , believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics ) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet
2370-514: A fatal heart attack on 6 January 1918, in the sanatorium where he had spent the last year of his life. Cantor's work between 1874 and 1884 is the origin of set theory . Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle . No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which
2528-472: A few basic operations (e.g., zero , succ , pred ) and using natural numbers , which are assumed to be already constructed (using the Peano approach ). There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2), and the types of arguments accepted by these operations;
2686-458: A finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1) . In fact, Z {\displaystyle \mathbb {Z} } under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to Z {\displaystyle \mathbb {Z} } . The first four properties listed above for multiplication say that Z {\displaystyle \mathbb {Z} } under multiplication
2844-439: A given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing. The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD , when he considered the volume of an impossible frustum of a pyramid . They became more prominent when in
3002-586: A member of the Saint Petersburg stock exchange ; when he became ill, the family moved to Germany in 1856, first to Wiesbaden , then to Frankfurt , seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt ; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from
3160-451: A notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis 's De algebra tractatus . In the same year, Gauss provided the first generally accepted proof of the fundamental theorem of algebra , showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of
3318-594: A paper presented by Julius König at the Third International Congress of Mathematicians . The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Cantor suffered from chronic depression for
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#17327720619633476-451: A part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol ∞ {\displaystyle {\text{∞}}}
3634-503: A placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems . Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting . Indian texts used
3792-541: A reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. In 1889, Cantor was instrumental in founding the German Mathematical Society , and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as
3950-425: A rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz . A modern geometrical version of infinity is given by projective geometry , which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in
4108-400: A set P − {\displaystyle P^{-}} which is disjoint from P {\displaystyle P} and in one-to-one correspondence with P {\displaystyle P} via a function ψ {\displaystyle \psi } . For example, take P − {\displaystyle P^{-}} to be
4266-517: A set A is strictly larger than the cardinality of A . This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem . Cantor wrote on the Goldbach conjecture in 1894. In 1895 and 1897, Cantor published
4424-460: A set of limit points S ω+1 , and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω , ω + 1, ω + 2, ... Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers . Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in
4582-415: A set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent . From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent:
4740-467: A subset of B and B equivalent to a subset of A , then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem . Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence , though he did not use that phrase. He then began looking for
4898-679: A system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero , which was developed by ancient Indian mathematicians around 500 AD. The first known documented use of zero dates to AD 628, and appeared in
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5056-532: A two-part paper in Mathematische Annalen under Felix Klein 's editorship; these were his last significant papers on set theory. The first paper begins by defining set, subset , etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to
5214-479: A way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as exponents , but referred to them as "absurd numbers". As recently as
5372-461: Is countably infinite . An integer may be regarded as a real number that can be written without a fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2 , 5/4, and √ 2 are not. The integers form the smallest group and the smallest ring containing the natural numbers . In algebraic number theory , the integers are sometimes qualified as rational integers to distinguish them from
5530-433: Is a commutative monoid . However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that Z {\displaystyle \mathbb {Z} } under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication
5688-422: Is a commutative ring with unity . It is the prototype of all objects of such algebraic structure . Only those equalities of expressions are true in Z {\displaystyle \mathbb {Z} } for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings. The lack of zero divisors in the integers (last property in
5846-409: Is a subset of Z {\displaystyle \mathbb {Z} } , which in turn is a subset of the set of all rational numbers Q {\displaystyle \mathbb {Q} } , itself a subset of the real numbers R {\displaystyle \mathbb {R} } . Like the set of natural numbers, the set of integers Z {\displaystyle \mathbb {Z} }
6004-444: Is a subset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as A more complete list of number sets appears in the following diagram. The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0
6162-491: Is a "law of thought". Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph . First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that
6320-469: Is called the quotient and r is called the remainder of the division of a by b . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. The above says that Z {\displaystyle \mathbb {Z} } is a Euclidean domain . This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain , and any positive integer can be written as
6478-579: Is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2 . Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency. The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to Pythagoras , more specifically to
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6636-474: Is equivalent to the statement that any Noetherian valuation ring is either a field —or a discrete valuation ring . In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero , and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms , call this P {\displaystyle P} . Then construct
6794-467: Is greater than zero , and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: Thus it follows that Z {\displaystyle \mathbb {Z} } together with the above ordering is an ordered ring . The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered . This
6952-401: Is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety. The difficulty Cantor had in proving
7110-401: Is identified with the class [( n ,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [( n ,0)] ), and the class [(0, n )] is denoted − n (this covers all remaining classes, and gives the class [(0,0)] a second time since –0 = 0. Thus, [( a , b )] is denoted by If the natural numbers are identified with the corresponding integers (using
7268-538: Is largely due to Ernst Kummer , who also invented ideal numbers , which were expressed as geometrical entities by Felix Klein in 1893. In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points . This eventually led to the concept of the extended complex plane . Prime numbers have been studied throughout recorded history. They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of
7426-417: Is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare ), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by
7584-588: Is not as complete as it is in the Brāhmasphuṭasiddhānta . Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on
7742-437: Is not defined on Z {\displaystyle \mathbb {Z} } , the division "with remainder" is defined on them. It is called Euclidean division , and possesses the following important property: given two integers a and b with b ≠ 0 , there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b | , where | b | denotes the absolute value of b . The integer q
7900-476: Is not free since the integer 0 can be written pair (0,0), or pair (1,1), or pair (2,2), etc.. This technique of construction is used by the proof assistant Isabelle ; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers. Number A number is a mathematical object used to count, measure, and label. The most basic examples are
8058-431: Is obtained by reversing the order of the pair: Hence subtraction can be defined as the addition of the additive inverse: The standard ordering on the integers is given by: It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form ( n ,0) or (0, n ) (or both at once). The natural number n
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#17327720619638216-405: Is often used to represent an infinite quantity. Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity —the general consensus being that only the latter had true value. Galileo Galilei 's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in
8374-485: Is the Hindu–Arabic numeral system , which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits . In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers ), and for codes (as with ISBNs ). In common usage, a numeral is not clearly distinguished from
8532-434: Is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply
8690-464: Is the set of limit points of S . If S k+1 is the set of limit points of S k , then he could construct a trigonometric series whose zeros are S k+1 . Because the sets S k were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S , S 1 , S 2 , S 3 ,... formed a limit set, which we would now call S ω , and then he noticed that S ω would also have to have
8848-508: Is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite , so there is an uncountably infinite number of transcendental numbers. The earliest known conception of mathematical infinity appears in the Yajur Veda , an ancient Indian script, which at one point states, "If you remove
9006-502: Is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p -adic integers . The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers referred to the natural numbers , excluding negative numbers, while integer included
9164-568: The Brāhmasphuṭasiddhānta , the main work of the Indian mathematician Brahmagupta . He treated 0 as a number and discussed operations involving it, including division . By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals , and documentation shows the idea later spreading to China and the Islamic world . Brahmagupta's Brāhmasphuṭasiddhānta
9322-649: The Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell , whose newly published Principia Mathematica repeatedly cited Cantor's work, but
9480-464: The Burali-Forti paradox (which was just mentioned), Cantor's paradox , and Russell's paradox . Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes. In 1908, Zermelo published his axiom system for set theory . He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of
9638-402: The Cantor set , discovered by Henry John Stephen Smith in 1875, is nowhere dense , but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers . Cantor introduced fundamental constructions in set theory, such as
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#17327720619639796-697: The Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic , and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. In 1796, Adrien-Marie Legendre conjectured
9954-783: The Pythagorean Hippasus of Metapontum , who produced a (most likely geometrical) proof of the irrationality of the square root of 2 . The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news. The 16th century brought final European acceptance of negative integral and fractional numbers. By
10112-461: The absolute . The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it
10270-460: The absolute infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory. Debate among mathematicians grew out of opposing views in
10428-726: The complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields , and the application of the term "number" is a matter of convention, without fundamental significance. Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks . These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered
10586-504: The intension of a set of cardinal or real numbers with its extension , thus conflating the concept of rules for generating a set with an actual set. Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view
10744-462: The natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals ; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system , which is an organized way to represent any number. The most common numeral system
10902-479: The number that it represents. In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers , rational numbers such as one half ( 1 2 ) {\displaystyle \left({\tfrac {1}{2}}\right)} , real numbers such as the square root of 2 ( 2 ) {\displaystyle \left({\sqrt {2}}\right)} and π , and complex numbers which extend
11060-426: The ordered pairs ( 1 , n ) {\displaystyle (1,n)} with the mapping ψ = n ↦ ( 1 , n ) {\displaystyle \psi =n\mapsto (1,n)} . Finally let 0 be some object not in P {\displaystyle P} or P − {\displaystyle P^{-}} , for example the ordered pair (0,0). Then
11218-412: The paradise that Cantor has created ." Georg Cantor, born in 1845 in Saint Petersburg , Russian Empire, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm 's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been
11376-474: The philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought ( constructivism and its two offshoots, intuitionism and finitism ) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with
11534-399: The power set of a set A , which is the set of all possible subsets of A . He later proved that the size of the power set of A is strictly larger than the size of A , even when A is an infinite set; this result soon became known as Cantor's theorem . Cantor developed an entire theory and arithmetic of infinite sets , called cardinals and ordinals , which extended the arithmetic of
11692-540: The prime number theorem , describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture , which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis , formulated by Bernhard Riemann in 1859. The prime number theorem
11850-427: The unit interval and the unit square was not a continuous one. This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle. Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power
12008-435: The well-ordering theorem . Zermelo had proved this theorem in 1904 using the axiom of choice , but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets. In 1923, John von Neumann developed an axiom system that eliminates
12166-708: The "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt . In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin , attending lectures by Leopold Kronecker , Karl Weierstrass and Ernst Kummer . He spent the summer of 1866 at the University of Göttingen , then and later
12324-461: The 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolò Fontana Tartaglia and Gerolamo Cardano . It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since they did not even consider negative numbers to be on firm ground at
12482-467: The 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since Euclid . In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine , Georg Cantor , and Richard Dedekind
12640-595: The 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. It is likely that the concept of fractional numbers dates to prehistoric times . The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus . Classical Greek and Indian mathematicians made studies of
12798-611: The 3rd century AD in Greece. Diophantus referred to the equation equivalent to 4 x + 20 = 0 (the solution is negative) in Arithmetica , saying that the equation gave an absurd result. During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta , in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce
12956-603: The Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70). Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus ), but as a word, nulla meaning nothing , not as a symbol. When division produced 0 as a remainder, nihil , also meaning nothing , was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N,
13114-402: The addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem : the cardinality of the power set of
13272-424: The algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to
13430-578: The chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor. In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica . But in 1885, Mittag-Leffler
13588-431: The class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and
13746-440: The class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom . The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in
13904-457: The continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness. This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of Elizabethan literature , thinking there might be evidence that Francis Bacon wrote
14062-417: The continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as " ZFC "). In 1883, Cantor divided the infinite into the transfinite and
14220-416: The development of Greek mathematics , stimulating the investigation of many problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers , which consist of various extensions or modifications of
14378-407: The embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} . Some examples are: In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines . Integers are represented as algebraic terms built using
14536-483: The encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person. Cantor retired in 1913, and lived in poverty and suffered from malnourishment during World War I . The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had
14694-520: The end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder . The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal , the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, "No one shall expel us from
14852-603: The existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer , while Ludwig Wittgenstein raised philosophical objections ; see Controversy over Cantor's theory . Cantor,
15010-655: The first kind of abstract numeral system. The first known system with place value was the Mesopotamian base 60 system ( c. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt . Numbers should be distinguished from numerals , the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals,
15168-524: The first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians , which took place in Zürich, Switzerland, in 1897. After Cantor's 1884 hospitalization there
15326-417: The form a + bi , where a and b are integers (now called Gaussian integers ) or rational numbers. His student, Gotthold Eisenstein , studied the type a + bω , where ω is a complex root of x − 1 = 0 (now called Eisenstein integers ). Other such classes (called cyclotomic fields ) of complex numbers derive from the roots of unity x − 1 = 0 for higher values of k . This generalization
15484-744: The founders of the constructive viewpoint in mathematics , disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle. In 1881, Cantor's Halle colleague Eduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind, Heinrich M. Weber and Franz Mertens , in that order, but each declined
15642-744: The general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots". European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci , 1202) and later as losses (in Flos ). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found
15800-505: The history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works. Cantor's first ten papers were on number theory , his thesis topic. At the suggestion of Eduard Heine , the Professor at Halle, Cantor turned to analysis . Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet , Rudolf Lipschitz , Bernhard Riemann , and Heine himself:
15958-579: The idea of a cut (Schnitt) in the system of real numbers , separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker , and Méray. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem ( Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formulas involving only arithmetical operations and roots). Hence it
16116-583: The idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in
16274-586: The inconsistency of infinitesimals . The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper , "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having
16432-1179: The integers are defined to be the union P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}} . The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows: − x = { ψ ( x ) , if x ∈ P ψ − 1 ( x ) , if x ∈ P − 0 , if x = 0 {\displaystyle -x={\begin{cases}\psi (x),&{\text{if }}x\in P\\\psi ^{-1}(x),&{\text{if }}x\in P^{-}\\0,&{\text{if }}x=0\end{cases}}} The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey
16590-458: The integers are not (since the result can be a fraction when the exponent is negative). The following table lists some of the basic properties of addition and multiplication for any integers a , b , and c : The first five properties listed above for addition say that Z {\displaystyle \mathbb {Z} } , under addition, is an abelian group . It is also a cyclic group , since every non-zero integer can be written as
16748-448: The integers as a subring is the field of rational numbers . The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z {\displaystyle \mathbb {Z} } as its subring . Although ordinary division
16906-447: The integers into this ring. This universal property , namely to be an initial object in the category of rings , characterizes the ring Z {\displaystyle \mathbb {Z} } . Z {\displaystyle \mathbb {Z} } is not closed under division , since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation ,
17064-439: The intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as L. E. J. Brouwer and especially Henri Poincaré adopted an intuitionist stance against Cantor's work. Finally, Wittgenstein 's attacks were finitist: he believed that Cantor's diagonal argument conflated
17222-560: The last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains , Cantor spent much time in mathematical discussions with Richard Dedekind , whom he had met at Interlaken in Switzerland two years earlier while on holiday. Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain
17380-411: The latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as
17538-509: The more general algebraic integers . In fact, (rational) integers are algebraic integers that are also rational numbers . The word integer comes from the Latin integer meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). " Entire " derives from the same origin via the French word entier , which means both entire and integer . Historically the term
17696-455: The natural numbers. His notation for the cardinal numbers was the Hebrew letter ℵ {\displaystyle \aleph } ( ℵ , aleph ) with a natural number subscript; for the ordinals he employed the Greek letter ω {\displaystyle \omega } ( ω , omega ). This notation is still in use today. The Continuum hypothesis , introduced by Cantor,
17854-434: The negative numbers. The whole numbers remain ambiguous to the present day. Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra Like the natural numbers , Z {\displaystyle \mathbb {Z} } is closed under the operations of addition and multiplication , that is,
18012-445: The notion of dimension . In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of " power " (a term he took from Jakob Steiner ) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into
18170-469: The ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof. Cantor avoided paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form
18328-456: The paper where he first set out his celebrated definition of real numbers by Dedekind cuts . While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond , describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of
18486-440: The paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and
18644-472: The plays attributed to William Shakespeare (see Shakespearean authorship question ); this ultimately resulted in two pamphlets, published in 1896 and 1897. Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem . However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891. He eventually sought, and achieved,
18802-702: The positive integers, Z 0 + {\displaystyle \mathbb {Z} ^{0+}} or Z ≥ {\displaystyle \mathbb {Z} ^{\geq }} for non-negative integers, and Z ≠ {\displaystyle \mathbb {Z} ^{\neq }} for non-zero integers. Some authors use Z ∗ {\displaystyle \mathbb {Z} ^{*}} for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the group of units of Z {\displaystyle \mathbb {Z} } ). Additionally, Z p {\displaystyle \mathbb {Z} _{p}}
18960-727: The presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms. The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair ( x , y ) {\displaystyle (x,y)} that takes as arguments two natural numbers x {\displaystyle x} and y {\displaystyle y} , and returns an integer (equal to x − y {\displaystyle x-y} ). This operation
19118-411: The products of primes in an essentially unique way. This is the fundamental theorem of arithmetic . Z {\displaystyle \mathbb {Z} } is a totally ordered set without upper or lower bound . The ordering of Z {\displaystyle \mathbb {Z} } is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... . An integer is positive if it
19276-509: The properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is often regarded as unlucky , and " a million " may signify "a lot" rather than an exact quantity. Though it is now regarded as pseudoscience , belief in a mystical significance of numbers, known as numerology , permeated ancient and medieval thought. Numerology heavily influenced
19434-411: The real algebraic numbers as a sequence a 1 , a 2 , a 3 , .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in
19592-403: The real numbers with a square root of −1 (and its combinations with real numbers by adding or subtracting its multiples). Calculations with numbers are done with arithmetical operations, the most familiar being addition , subtraction , multiplication , division , and exponentiation . Their study or usage is called arithmetic , a term which may also refer to number theory , the study of
19750-400: The rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ( Burali-Forti paradox , Cantor's paradox , and Russell's paradox ) to a meeting of
19908-445: The same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre , where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified
20066-534: The same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable . His proof differs from the diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers . Transcendental numbers were first constructed by Joseph Liouville in 1844. Cantor established these results using two constructions. His first construction shows how to write
20224-440: The sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains
20382-633: The set of all natural numbers is N , also written N {\displaystyle \mathbb {N} } , and sometimes N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} when it is necessary to indicate whether the set should start with 0 or 1, respectively. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( / ˈ k æ n t ɔːr / KAN -tor ; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯] ; 3 March [ O.S. 19 February] 1845 – 6 January 1918 )
20540-557: The set of the integers was not used before the end of the 19th century, when Georg Cantor introduced the concept of infinite sets and set theory . The use of the letter Z to denote the set of integers comes from the German word Zahlen ("numbers") and has been attributed to David Hilbert . The earliest known use of the notation in a textbook occurs in Algèbre written by the collective Nicolas Bourbaki , dating to 1947. The notation
20698-403: The sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0 ), Z {\displaystyle \mathbb {Z} } , unlike the natural numbers, is also closed under subtraction . The integers form a ring which is the most basic one, in the following sense: for any ring, there is a unique ring homomorphism from
20856-422: The table) means that the commutative ring Z {\displaystyle \mathbb {Z} } is an integral domain . The lack of multiplicative inverses, which is equivalent to the fact that Z {\displaystyle \mathbb {Z} } is not closed under division, means that Z {\displaystyle \mathbb {Z} } is not a field . The smallest field containing
21014-567: The theory of rational numbers, as part of the general study of number theory . The best known of these is Euclid's Elements , dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra , which also covers number theory as part of a general study of mathematics. The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it
21172-403: The theory of transfinite numbers with pantheism . Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him. Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from
21330-408: The theory was made by Georg Cantor ; in 1895 he published a book about his new set theory , introducing, among other things, transfinite numbers and formulating the continuum hypothesis . In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents
21488-430: The time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation seemed capriciously inconsistent with the algebraic identity which is valid for positive real numbers a and b , and was also used in complex number calculations with one of
21646-462: The traditional areas of mathematics (such as algebra , analysis , and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics. In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers ; this showed, for the first time, that there exist infinite sets of different sizes . He
21804-578: The uncertain interpretation of 0. (The ancient Greeks even questioned whether 1 was a number.) The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph , in the New World, possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar . Maya arithmetic used base 4 and base 5 written as base 20. George I. Sánchez in 1961 reported
21962-504: The uniqueness of the representation of a function by trigonometric series . Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the n th derived set S n of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S 1 as its set of zeros, where S 1
22120-411: The various laws of arithmetic. In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers ( a , b ) . The intuition is that ( a , b ) stands for the result of subtracting b from
22278-420: The work of Abraham de Moivre and Leonhard Euler . De Moivre's formula (1730) states: while Euler's formula of complex analysis (1748) gave us: The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received
22436-472: The writings of Joseph Louis Lagrange . Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus first connected the subject with determinants , resulting, with the subsequent contributions of Heine, Möbius , and Günther, in the theory of Kettenbruchdeterminanten . The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e
22594-520: The year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica ." Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence: ... I don't know when I shall return to
22752-409: Was a mathematician who played a pivotal role in the creation of set theory , which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets , and proved that the real numbers are more numerous than the natural numbers . Cantor's method of proof of this theorem implies
22910-585: Was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.". Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God. Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and
23068-445: Was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets , subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets). Cantor developed important concepts in topology and their relation to cardinality . For example, he showed that
23226-456: Was brought about. In 1869, Charles Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on
23384-426: Was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta . He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until
23542-446: Was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all
23700-539: Was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted. Numbers can be classified into sets , called number sets or number systems , such as the natural numbers and the real numbers . The main number systems are as follows: N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} are sometimes used. Each of these number systems
23858-402: Was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). The objections to Cantor's work were occasionally fierce: Leopold Kronecker 's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that
24016-434: Was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory . Simple continued fractions , closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler , and at the opening of the 19th century were brought into prominence through
24174-654: Was not adopted immediately. For example, another textbook used the letter J, and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers. The symbol Z {\displaystyle \mathbb {Z} } is often annotated to denote various sets, with varying usage amongst different authors: Z + {\displaystyle \mathbb {Z} ^{+}} , Z + {\displaystyle \mathbb {Z} _{+}} , or Z > {\displaystyle \mathbb {Z} ^{>}} for
24332-424: Was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 ( cardinality of the empty set , i.e. 0 elements, where 0 is thus the smallest cardinal number ) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for
24490-751: Was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on
24648-538: Was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs. It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 Grundlagen , he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. In making these claims, Cantor may have been influenced by F. A. Trendelenburg , whose lecture courses he attended at Berlin, and in turn Cantor produced
24806-439: Was used for a number that was a multiple of 1, or to the whole part of a mixed number . Only positive integers were considered, making the term synonymous with the natural numbers . The definition of integer expanded over time to include negative numbers as their usefulness was recognized. For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers. The phrase
24964-519: Was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol. The abstract concept of negative numbers was recognized as early as 100–50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients , black for negative. The first reference in a Western work was in
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