Contents 1 History 1.1 Origins 1.1.1 Dawn of arithmetic
1.1.2 Classical Greece and the early Hellenistic period
1.1.3 Diophantus
1.1.4 Āryabhaṭa, Brahmagupta, Bhāskara
1.1.5
1.2 Early modern number theory 1.2.1 Fermat 1.2.2 Euler 1.2.3 Lagrange, Legendre, and Gauss 1.3 Maturity and division into subfields 2 Main subdivisions 2.1 Elementary tools
2.2 Analytic number theory
2.3
3 Recent approaches and subfields 3.1 Probabilistic number theory
3.2
4 Applications 5 Prizes 6 See also 7 Notes 8 References 9 Sources 10 Further reading 11 External links History[edit]
Origins[edit]
Dawn of arithmetic[edit]
The first historical find of an arithmetical nature is a fragment of a
table: the broken clay tablet
( a , b , c ) displaystyle (a,b,c) such that a 2 + b 2 = c 2 displaystyle a^ 2 +b^ 2 =c^ 2 . The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal which has been subtracted such that the width..."[2] The
The table's layout suggests[3] that it was constructed by means of what amounts, in modern language, to the identity ( 1 2 ( x − 1 x ) ) 2 + 1 = ( 1 2 ( x + 1 x ) ) 2 , displaystyle left( frac 1 2 left(x- frac 1 x right)right)^ 2 +1=left( frac 1 2 left(x+ frac 1 x right)right)^ 2 , which is implicit in routine Old Babylonian exercises.[4] If some other method was used,[5] the triples were first constructed and then reordered by c / a displaystyle c/a , presumably for actual use as a "table", i.e., with a view to
applications.
It is not known what these applications may have been, or whether
there could have been any; Babylonian astronomy, for example, truly
came into its own only later. It has been suggested instead that the
table was a source of numerical examples for school problems.[6][note
3]
While Babylonian number theory—or what survives of Babylonian
mathematics that can be called thus—consists of this single,
striking fragment, Babylonian algebra (in the secondary-school sense
of "algebra") was exceptionally well developed.[7] Late Neoplatonic
sources[8] state that
2 displaystyle sqrt 2 is irrational.[11] Pythagorean mystics gave great importance to the odd and the even.[12] The discovery that 2 displaystyle sqrt 2 is irrational is credited to the early Pythagoreans
(pre-Theodorus).[13] By revealing (in modern terms) that numbers could
be irrational, this discovery seems to have provoked the first
foundational crisis in mathematical history; its proof or its
divulgation are sometimes credited to Hippasus, who was expelled or
split from the Pythagorean sect.[14] This forced a distinction between
numbers (integers and the rationals—the subjects of arithmetic), on
the one hand, and lengths and proportions (which we would identify
with real numbers, whether rational or not), on the other hand.
The Pythagorean tradition spoke also of so-called polygonal or
figurate numbers.[15] While square numbers, cubic numbers, etc., are
seen now as more natural than triangular numbers, pentagonal numbers,
etc., the study of the sums of triangular and pentagonal numbers would
prove fruitful in the early modern period (17th to early 19th
century).
We know of no clearly arithmetical material in ancient Egyptian or
"In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad."[19] Aristotle claimed that the philosophy of
3 , 5 , … , 17 displaystyle sqrt 3 , sqrt 5 ,dots , sqrt 17 are irrational. Theaetetus was, like Plato, a disciple of
Theodorus's; he worked on distinguishing different kinds of
incommensurables, and was thus arguably a pioneer in the study of
number systems. (
Title page of the 1621 edition of Diophantus's Arithmetica, translated
into
Very little is known about
f ( x , y ) = z 2 displaystyle f(x,y)=z^ 2 or f ( x , y , z ) = w 2 displaystyle f(x,y,z)=w^ 2 . Thus, nowadays, we speak of
f ( x 1 , x 2 , x 3 ) = 0 displaystyle f(x_ 1 ,x_ 2 ,x_ 3 )=0 , his aim was to find (in essence) three rational functions g 1 , g 2 , g 3 displaystyle g_ 1 ,g_ 2 ,g_ 3 such that, for all values of r displaystyle r and s displaystyle s , setting x i = g i ( r , s ) displaystyle x_ i =g_ i (r,s) for i = 1 , 2 , 3 displaystyle i=1,2,3 gives a solution to f ( x 1 , x 2 , x 3 ) = 0. displaystyle f(x_ 1 ,x_ 2 ,x_ 3 )=0.
n ≡ a 1 mod m 1 displaystyle nequiv a_ 1 bmod m _ 1 , n ≡ a 2 mod m 2 displaystyle nequiv a_ 2 bmod m _ 2 could be solved by a method he called kuṭṭaka, or pulveriser;[27]
this is a procedure close to (a generalisation of) the Euclidean
algorithm, which was probably discovered independently in India.[28]
In the early ninth century, the caliph
Pierre de Fermat
a p − 1 ≡ 1 mod p . displaystyle a^ p-1 equiv 1 bmod p . [note 7] If a and b are coprime, then a 2 + b 2 displaystyle a^ 2 +b^ 2 is not divisible by any prime congruent to −1 modulo 4;[41] and every prime congruent to 1 modulo 4 can be written in the form a 2 + b 2 displaystyle a^ 2 +b^ 2 .[42] These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent.[43] Fermat and Frenicle also did some work (some of it erroneous)[44] on other quadratic forms. Fermat posed the problem of solving x 2 − N y 2 = 1 displaystyle x^ 2 -Ny^ 2 =1 as a challenge to English mathematicians (1657). The problem was
solved in a few months by Wallis and Brouncker.[45] Fermat considered
their solution valid, but pointed out they had provided an algorithm
without a proof (as had Jayadeva and Bhaskara, though Fermat would
never know this). He states that a proof can be found by descent.
Fermat developed methods for (doing what in our terms amounts to)
finding points on curves of genus 0 and 1. As in Diophantus, there are
many special procedures and what amounts to a tangent construction,
but no use of a secant construction.[46]
Fermat states and proves (by descent) in the appendix to Observations
on
x 4 + y 4 = z 4 displaystyle x^ 4 +y^ 4 =z^ 4 has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that x 3 + y 3 = z 3 displaystyle x^ 3 +y^ 3 =z^ 3 has no non-trivial solutions, and that this could be proven by descent.[48] The first known proof is due to Euler (1753; indeed by descent).[49] Fermat's claim ("Fermat's last theorem") to have shown there are no solutions to x n + y n = z n displaystyle x^ n +y^ n =z^ n for all n ≥ 3 displaystyle ngeq 3 appears only in his annotations on the margin of his copy of Diophantus. Euler[edit] Leonhard Euler The interest of
Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that p = x 2 + y 2 displaystyle p=x^ 2 +y^ 2 if and only if p ≡ 1 mod 4 displaystyle pequiv 1 bmod 4 ; initial work towards a proof that every integer is the sum of four
squares (the first complete proof is by
x 4 + y 4 = z 2 displaystyle x^ 4 +y^ 4 =z^ 2 (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method). Pell's equation, first misnamed by Euler.[55] He wrote on the link between continued fractions and Pell's equation.[56] First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.[57] Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form x 2 + N y 2 displaystyle x^ 2 +Ny^ 2 , some of it prefiguring quadratic reciprocity.[58] [59][60]
Diophantine equations. Euler worked on some
Lagrange, Legendre, and Gauss[edit] Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition
m X 2 + n Y 2 displaystyle mX^ 2 +nY^ 2 ) — defining their equivalence relation, showing how to put them in
reduced form, etc.
a x 2 + b y 2 + c z 2 = 0 displaystyle ax^ 2 +by^ 2 +cz^ 2 =0 [64] and worked on quadratic forms along the lines later developed fully by Gauss.[65] In his old age, he was the first to prove "Fermat's last theorem" for n = 5 displaystyle n=5 (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).[66] Carl Friedrich Gauss In his
The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.[68] In this way,
Ernst Kummer Peter Gustav Lejeune Dirichlet Starting early in the nineteenth century, the following developments gradually took place: The rise to self-consciousness of number theory (or higher arithmetic) as a field of study.[69] The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra. The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.
The action of the modular group on the upper half plane. The region in grey is the standard fundamental domain.
in terms of its tools, as the study of the integers by means of tools from real and complex analysis;[70] or in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.[79] Some subjects generally considered to be part of analytic number
theory, e.g., sieve theory,[note 9] are better covered by the second
rather than the first definition: some of sieve theory, for instance,
uses little analysis,[note 10] yet it does belong to analytic number
theory.
The following are examples of problems in analytic number theory: the
prime number theorem, the
f ( x ) = 0 displaystyle f(x)=0 with rational coefficients; for example, every solution x displaystyle x of x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 displaystyle x^ 5 +(11/2)x^ 3 -7x^ 2 +9=0 (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields.[83] Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form a + b d displaystyle a+b sqrt d , where a displaystyle a and b displaystyle b are rational numbers and d displaystyle d is a fixed rational number whose square root is not rational.) For
that matter, the 11th-century chakravala method amounts—in modern
terms—to an algorithm for finding the units of a real quadratic
number field. However, neither Bhāskara nor
− 5 displaystyle sqrt -5 , the number 6 displaystyle 6 can be factorised both as 6 = 2 ⋅ 3 displaystyle 6=2cdot 3 and 6 = ( 1 + − 5 ) ( 1 − − 5 ) displaystyle 6=(1+ sqrt -5 )(1- sqrt -5 ) ; all of 2 displaystyle 2 , 3 displaystyle 3 , 1 + − 5 displaystyle 1+ sqrt -5 and 1 − − 5 displaystyle 1- sqrt -5 are irreducible, and thus, in a naïve sense, analogous to primes
among the integers.) The initial impetus for the development of ideal
numbers (by Kummer) seems to have come from the study of higher
reciprocity laws,[84]i.e., generalisations of quadratic reciprocity.
x 2 + y 2 = 1 displaystyle x^ 2 +y^ 2 =1 ; we would like to study its rational solutions, i.e., its solutions ( x , y ) displaystyle (x,y) such that x and y are both rational. This is the same as asking for all integer solutions to a 2 + b 2 = c 2 displaystyle a^ 2 +b^ 2 =c^ 2 ; any solution to the latter equation gives us a solution x = a / c displaystyle x=a/c , y = b / c displaystyle y=b/c to the former. It is also the same as asking for all points with rational coordinates on the curve described by x 2 + y 2 = 1 displaystyle x^ 2 +y^ 2 =1 . (This curve happens to be a circle of radius 1 around the origin.) Two examples of an elliptic curve, i.e., a curve of genus 1 having at least one rational point. (Either graph can be seen as a slice of a torus in four-dimensional space.) The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation f ( x , y ) = 0 displaystyle f(x,y)=0 , where f displaystyle f is a polynomial in two variables—turns out to depend crucially on the genus of the curve. The genus can be defined as follows:[note 12] allow the variables in f ( x , y ) = 0 displaystyle f(x,y)=0 to be complex numbers; then f ( x , y ) = 0 displaystyle f(x,y)=0 defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, i.e., four dimensions). Count the number of (doughnut) holes in the surface; call this number the genus of f ( x , y ) = 0 displaystyle f(x,y)=0 . Other geometrical notions turn out to be just as crucial. There is also the closely linked area of Diophantine approximations: given a number x displaystyle x , how well can it be approximated by rationals? (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call a / q displaystyle a/q (with gcd ( a , q ) = 1 displaystyle gcd(a,q)=1 ) a good approximation to x displaystyle x if
x − a / q
< 1 q c displaystyle x-a/q< frac 1 q^ c , where c displaystyle c is large.) This question is of special interest if x displaystyle x is an algebraic number. If x displaystyle x cannot be well approximated, then some equations do not have integer
or rational solutions. Moreover, several concepts (especially that of
height) turn out to be crucial both in
0 displaystyle 0 must happen sometimes; one may say with equal justice that many
applications of probabilistic number theory hinge on the fact that
whatever is unusual must be rare. If certain algebraic objects (say,
rational or integer solutions to certain equations) can be shown to be
in the tail of certain sensibly defined distributions, it follows that
there must be few of them; this is a very concrete non-probabilistic
statement following from a probabilistic one.
At times, a non-rigorous, probabilistic approach leads to a number of
heuristic algorithms and open problems, notably Cramér's conjecture.
A displaystyle A , does it contain many elements in arithmetic progression: a displaystyle a , a + b , a + 2 b , a + 3 b , … , a + 10 b displaystyle a+b,a+2b,a+3b,ldots ,a+10b , say? Should it be possible to write large integers as sums of elements of A displaystyle A ? These questions are characteristic of arithmetic combinatorics. This is a presently coalescing field; it subsumes additive number theory (which concerns itself with certain very specific sets A displaystyle A of arithmetic significance, such as the primes or the squares) and, arguably, some of the geometry of numbers, together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with ergodic theory, finite group theory, model theory, and other fields. The term additive combinatorics is also used; however, the sets A displaystyle A being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of A + A displaystyle A+A and A displaystyle A · A displaystyle A may be compared.
Computations in number theory[edit]
Main article: Computational number theory
While the word algorithm goes back only to certain readers of
al-Khwārizmī, careful descriptions of methods of solution are older
than proofs: such methods (that is, algorithms) are as old as any
recognisable mathematics—ancient Egyptian, Babylonian, Vedic,
Chinese—whereas proofs appeared only with the Greeks of the
classical period. An interesting early case is that of what we now
call the Euclidean algorithm. In its basic form (namely, as an
algorithm for computing the greatest common divisor) it appears as
Proposition 2 of
a x + b y = c displaystyle ax+by=c , or, what is the same, for finding the quantities whose existence is
assured by the Chinese remainder theorem) it first appears in the
works of
This section needs expansion with: Modern applications of
The number-theorist
Algebraic function field Finite field p-adic number Notes[edit] ^ Already in 1921,
[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (Robson 2001, p. 202). Robson takes issue with the notion that the scribe who produced
^ Sunzi Suanjing, Ch. 3, Problem 26, in Lam & Ang 2004, pp. 219–220: [26] Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. Answer: 23. Method: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When [a number] exceeds 106, the result is obtained by subtracting 105. ^ See, e.g., Sunzi Suanjing, Ch. 3, Problem 36, in Lam & Ang 2004, pp. 223–224: [36] Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. Answer: Male. Method: Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female. This is the last problem in Sunzi's otherwise matter-of-fact treatise. ^ Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in medieval times – whether in the West or the Arab-speaking world – due in part to the importance given to them by the Neopythagorean (and hence mystical) Nicomachus (ca. 100 CE), who wrote a primitive but influential "Introduction to Arithmetic". See van der Waerden 1961, Ch. IV. ^ Here, as usual, given two integers a and b and a non-zero integer m, we write a ≡ b mod m displaystyle aequiv b bmod m (read "a is congruent to b modulo m") to mean that m divides
a − b, or, what is the same, a and b leave the same
residue when divided by m. This notation is actually much later than
Fermat's; it first appears in section 1 of Gauss's Disquisitiones
Arithmeticae.
x a ≡ 1 mod p displaystyle xaequiv 1 bmod p ); this fact (which, in modern language, makes the residues mod p into
a group, and which was already known to Āryabhaṭa; see above) was
familiar to Fermat thanks to its rediscovery by Bachet (Weil 1984,
p. 7). Weil goes on to say that Fermat would have recognised that
Bachet's argument is essentially Euclid's algorithm.
^ Up to the second half of the seventeenth century, academic positions
were very rare, and most mathematicians and scientists earned their
living in some other way (Weil 1984, pp. 159, 161). (There were
already some recognisable features of professional practice, viz.,
seeking correspondents, visiting foreign colleagues, building private
libraries (Weil 1984, pp. 160–161). Matters started to shift in
the late 17th century (Weil 1984, p. 161); scientific academies
were founded in England (the Royal Society, 1662) and France (the
Académie des sciences, 1666) and Russia (1724). Euler was offered a
position at this last one in 1726; he accepted, arriving in St.
Petersburg in 1727 (Weil 1984, p. 163 and Varadarajan 2006,
p. 7). In this context, the term amateur usually applied to
Goldbach is well-defined and makes some sense: he has been described
as a man of letters who earned a living as a spy (Truesdell 1984,
p. xv); cited in Varadarajan 2006, p. 9). Notice, however,
that Goldbach published some works on mathematics and sometimes held
academic positions.
^
y 2 = x 3 + 7 displaystyle y^ 2 =x^ 3 +7 . We allow x and y to be complex numbers: ( a + b i ) 2 = ( c + d i ) 3 + 7 displaystyle (a+bi)^ 2 =(c+di)^ 3 +7 . This is, in effect, a set of two equations on four variables, since both the real and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four-dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate a), we can plot the resulting projection, which is a surface in ordinary three-dimensional space. It then becomes clear that the result is a torus, i.e., the surface of a doughnut (somewhat stretched). A doughnut has one hole; hence the genus is 1. References[edit] ^ Long 1972, p. 1. ^ Neugebauer & Sachs 1945, p. 40. The term takiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Robson 2001, p. 192 ^ Robson 2001, p. 189. Other sources give the modern formula ( p 2 − q 2 , 2 p q , p 2 + q 2 ) displaystyle (p^ 2 -q^ 2 ,2pq,p^ 2 +q^ 2 ) . Van der Waerden gives both the modern formula and what amounts to
the form preferred by Robson.(van der Waerden 1961, p. 79)
^ van der Waerden 1961, p. 184.
^ Neugebauer (Neugebauer 1969, pp. 36–40) discusses the table
in detail and mentions in passing Euclid's method in modern notation
(Neugebauer 1969, p. 39).
^ Friberg 1981, p. 302.
^ van der Waerden 1961, p. 43.
^ Iamblichus, Life of Pythagoras,(trans. e.g. Guthrie 1987) cited in
van der Waerden 1961, p. 108. See also Porphyry, Life of
Pythagoras, paragraph 6, in Guthrie 1987 Van der Waerden (van der
Waerden 1961, pp. 87–90) sustains the view that
Sources[edit] Apostol, Tom M. (1976). Introduction to analytic number theory.
Undergraduate Texts in Mathematics. Springer.
ISBN 978-0-387-90163-3. Retrieved 2016-02-28.
Apostol, Tom M. (n.d.). "An Introduction to the Theory of Numbers".
(Review of Hardy & Wright.)
This article incorporates material from the
Further reading[edit] Two of the most popular introductions to the subject are: G. H. Hardy; E. M. Wright (2008) [1938]. An introduction to the theory of numbers (rev. by D. R. Heath-Brown and J. H. Silverman, 6th ed.). Oxford University Press. ISBN 978-0-19-921986-5. Retrieved 2016-03-02. Vinogradov, I. M. (2003) [1954]. Elements of
Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods (Apostol n.d.). Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are: Ivan M. Niven; Herbert S. Zuckerman;
Popular choices for a second textbook include: Borevich, A. I.; Shafarevich, Igor R. (1966).
Serre, Jean-Pierre (1996) [1973]. A course in arithmetic. Graduate texts in mathematics. 7. Springer. ISBN 978-0-387-90040-7. External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "
v t e
Fields
Key concepts Numbers
Natural numbers
Prime numbers
Rational numbers
Irrational numbers
Algebraic numbers
Transcendental numbers
p-adic numbers
Arithmetic
Modular arithmetic
Advanced concepts Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Continued fractions Category Portal List of topics List of recreational topics Wikibook Wikversity v t e Areas of mathematics outline topic lists Branches Arithmetic History of mathematics
Philosophy of mathematics
Algebra
Calculus Analysis Differential equations / Dynamical systems Numerical analysis Optimization Functional analysis Geometry Discrete Algebraic Analytic Differential Finite Topology Trigonometry Applied Probability
Mathematical physics
Mathematical statistics
Statistics
Computer science
Game theory
Recreational mathematics
Divisions Pure Applied Discrete Computational Category Portal Commons WikiProject v t e Major fields of computer science Note: This template roughly follows the 2012 ACM Computing Classification System. Hardware Printed circuit board Peripheral Integrated circuit Very-large-scale integration Energy consumption Electronic design automation Computer systems organization Computer architecture Embedded system Real-time computing Dependability Networks Network architecture
Network protocol
Network components
Network scheduler
Software organization Interpreter Middleware Virtual machine Operating system Software quality Software notations and tools Programming paradigm Programming language Compiler Domain-specific language Modeling language Software framework Integrated development environment Software configuration management Software library Software repository Software development
Theory of computation Model of computation Formal language Automata theory Computational complexity theory Logic Semantics Algorithms
Mathematics of computing Discrete mathematics Probability Statistics Mathematical software Information theory Mathematical analysis Numerical analysis Information systems Database management system
Information storage systems
Enterprise information system
Social information systems
Geographic information system
Decision support system
Security Cryptography Formal methods Security services Intrusion detection system Hardware security Network security Information security Application security Human–computer interaction Interaction design Social computing Ubiquitous computing Visualization Accessibility Concurrency Concurrent computing Parallel computing Distributed computing Multithreading Multiprocessing Artificial intelligence Natural language processing Knowledge representation and reasoning Computer vision Automated planning and scheduling Search methodology Control method Philosophy of artificial intelligence Distributed artificial intelligence Machine learning Supervised learning Unsupervised learning Reinforcement learning Multi-task learning Cross-validation Graphics Animation Rendering Image manipulation Graphics processing unit Mixed reality Virtual reality Image compression Solid modeling Applied computing E-commerce Enterprise software Computational mathematics Computational physics Computational chemistry Computational biology Computational social science Computational engineering Computational healthcare Digital art Electronic publishing Cyberwarfare Electronic voting Video game Word processing Operations research Educational technology Document management Book Category Portal WikiProject Commons Authority control LCCN: sh85093222 GND: 4067277-3 SUDOC: 027270475 BNF: cb131627085 (data) NDL: 00570 |