Английская Википедия:4

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description Шаблон:Hatnote Шаблон:Use dmy dates Шаблон:Infobox number

4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures.

Evolution of the Hindu-Arabic digit

Файл:Evolution4glyph.png
Файл:Vier.jpg
Two modern handwritten fours

Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross.[1]

While the shape of the character for the digit 4 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in Файл:TextFigs148.svg.

On the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4 is seen with an open top: Файл:Seven-segment 4.svg.[2]

Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".[3]

Mathematics

Four is the smallest composite number, its proper divisors being Шаблон:Num and Шаблон:Num.[4] Four is the sum and product of two with itself: <math>2+2=4=2\times 2</math>, the only number <math>b</math> such that <math>a+a=b=a\times a</math>, which also makes four the smallest and only even squared prime number <math>2^{2}</math> and hence the first squared prime of the form <math>p^2</math>, where <math>p</math> is a prime. Four, as the first composite number, has a prime aliquot sum of 3; and as such it is part of the first aliquot sequence with a single composite member, expressly (4, 3, 1, 0).

  • In Knuth's up-arrow notation, <math>2+2=2\times2=2^{2}=2\uparrow\uparrow 2=2\uparrow\uparrow\uparrow2=\;...\; = 4</math>, and so forth, for any number of up arrows.[5] By consequence, four is the only square one more than a prime number, specifically three.
  • The sum of the first four prime numbers two + three + five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other hand, the square of four (<math>4^2</math>), equivalently the fourth power of two (<math>2^4</math>), is sixteen; the only number that has <math>a^{b}=b^{a}</math> as a form of factorization.

Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (−), multiplication (×), and division (÷); and four basic number systems, the real numbers <math>\mathbb{R}</math>, rational numbers <math>\mathbb{Q}</math>, integers <math>\mathbb{Z}</math>, and natural numbers <math>\mathbb{N}</math>.

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. <math>4x=y^{2}-z^{2}</math>. A number is a multiple of 4 if its last two digits are a multiple of 4 (for example, 1092 is a multiple of 4 because Шаблон:Nowrap).[6]

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers.[7] Three are not always sufficient; Шаблон:Num for instance cannot be written as the sum of three squares.[8]

There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal.

A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square.

Four is the highest degree general polynomial equation for which there is a solution in radicals.[9]

The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.[10] Three colors are not, in general, sufficient to guarantee this.[11] The largest planar complete graph has four vertices.[12]

A solid figure with four faces as well as four vertices is a tetrahedron, which is the smallest possible number of faces and vertices a polyhedron can have.[13][14] The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid.[15] It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron.[16] Tetrahedra can be inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb.

The third dimension holds a total of four Coxeter groups that generate convex uniform polyhedra: the tetrahedral group, the octahedral group, the icosahedral group, and a dihedral group (of orders 24, 48, 120, and 4<math>n</math>, respectively). There are also four general Coxeter groups of generalized uniform prisms, where two are hosoderal and dihedral groups that form spherical tilings, with another two general prismatic and antiprismatic groups that represent truncated hosohedra (or simply, prisms) and snub antiprisms, respectively.

Four-dimensional space is the highest-dimensional space featuring more than three regular convex figures:

The fourth dimension is also the highest dimension where regular self-intersecting figures exist:

Altogether, sixteen (or 16 = 42) regular convex and star polychora are generated from symmetries of four (4) Coxeter Weyl groups and point groups in the fourth dimension: the <math>\mathrm A_{4}</math> simplex, <math>\mathrm B_{4}</math> hypercube, <math>\mathrm F_{4}</math> icositetrachoric, and <math>\mathrm H_{4}</math> hexacosichoric groups; with the <math>\mathrm D_{4}</math> demihypercube group generating two alternative constructions. There are also sixty-four (or 64 = 43) four-dimensional Bravais lattices, alongside sixty-four uniform polychora in the fourth dimension based on the same <math>\mathrm A_{4}</math>, <math>\mathrm B_{4}</math>, <math>\mathrm F_{4}</math> and <math>\mathrm H_{4}</math> Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. Two infinite families of duoprisms and antiprismatic prisms exist in the fourth dimension.

There are only four polytopes with radial equilateral symmetry: the hexagon, the cuboctahedron, the tesseract, and the 24-cell.

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on <math>\mathbb{R}^n</math> except when <math>n</math> = <math>4</math>, in which case there are uncountably many.

The smallest non-cyclic group has four elements; it is the Klein four-group.[17] AШаблон:Sub alternating groups are not simple for values <math>n</math> ≤ <math>4</math>.

There are four Hopf fibrations of hyperspheres:

<math display=block> \begin{align} S^0 & \hookrightarrow S^1 \to S^1, \\ S^1 & \hookrightarrow S^3 \to S^2, \\ S^3 & \hookrightarrow S^7 \to S^4, \\ S^7 & \hookrightarrow S^{15}\to S^8. \\ \end{align}</math>

They are defined as locally trivial fibrations that map <math>f : S^{2n-1} \rightarrow S^{n}</math> for values of <math>n=2,4,8</math> (aside from the trivial fibration mapping between two points and a circle).[18]

Further extensions of the real numbers under Hurwitz's theorem states that there are four normed division algebras: the real numbers <math>\mathbb{R}</math>, the complex numbers <math>\mathbb C</math>, the quaternions <math>\mathbb H</math>, and the octonions <math>\mathbb O</math>. Under Cayley–Dickson constructions, the sedenions <math>\mathbb S</math> constitute a further fourth extension over <math>\mathbb{R}</math>. The real numbers are ordered, commutative and associative algebras, as well as alternative algebras with power-associativity. The complex numbers <math>\mathbb C</math> share all four multiplicative algebraic properties of the reals <math>\mathbb{R}</math>, without being ordered. The quaternions loose a further commutative algebraic property, while holding associative, alternative, and power-associative properties. The octonions are alternative and power-associative, while the sedenions are only power-associative. The sedenions and all further extensions of these four normed division algebras are solely power-associative with non-trivial zero divisors, which makes them non-division algebras. <math>\mathbb{R}</math> has a vector space of dimension 1, while <math>\mathbb C</math>, <math>\mathbb H</math>, <math>\mathbb O</math> and <math>\mathbb S</math> work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively.

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
4 × x 4 Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num Шаблон:Num
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 ÷ x 4 2 1.Шаблон:Overline 1 0.8 0.Шаблон:Overline 0.Шаблон:Overline 0.5 0.Шаблон:Overline 0.4 0.Шаблон:Overline 0.Шаблон:Overline 0.Шаблон:Overline 0.Шаблон:Overline 0.2Шаблон:Overline 0.25
x ÷ 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4Шаблон:Sup 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456 1073741824 4294967296
xШаблон:Sup 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625 65536

In religion

Buddhism

Judeo-Christian symbolism

Hinduism

Islam

Taoism

Other

In politics

  • Four Freedoms: four fundamental freedoms that Franklin D. Roosevelt declared ought to be enjoyed by everyone in the world: Freedom of Speech, Freedom of Religion, Freedom from Want, Freedom from Fear.[60]
  • Gang of Four: Popular name for four Chinese Communist Party leaders who rose to prominence during China's Cultural Revolution, but were ousted in 1976 following the death of Chairman Mao Zedong. Among the four was Mao's widow, Jiang Qing. Since then, many other political factions headed by four people have been called "Gangs of Four".[61]

In computing

In science

In astronomy

In biology

In chemistry

  • Valency of carbon (that is basis of life on the Earth) is four. Also because of its tetrahedral crystal bond structure, diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth's crust.[76]
  • The atomic number of beryllium[77]
  • There are four basic states of matter: solid, liquid, gas, and plasma.[78]

In physics

In logic and philosophy

Файл:Mugs of tea viewed from above.jpg
Four mugs
  • The symbolic meanings of the number four are linked to those of the cross and the square. "Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself". Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called "lord of the four suns" or "lord of the four quarters of the earth",[82] which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects' doings.
  • The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R").
  • In regard to whether two given propositions can have the same truth value, there are four separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (both may be true, but not that both are false); contraries (both may be false, but not that both are true); or contradictories (it is not possible that both are true, and it is not possible that both are false).
  • Aristotle held that there are basically four causes in nature: the material, the formal, the efficient, and the final.[83]
  • The Stoics held with four basic categories, all viewed as bodies (substantial and insubstantial): (1) substance in the sense of substrate, primary formless matter; (2) quality, matter's organization to differentiate and individualize something, and coming down to a physical ingredient such as pneuma, breath; (3) somehow holding (or disposed), as in a posture, state, shape, size, action, and (4) somehow holding (or disposed) toward something, as in relative location, familial relation, and so forth.
  • Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) Relation, (4) Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories.
  • Arthur Schopenhauer's doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason.
  • Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly progressing with scientific interest and results; then declining through the remaining phases, (2) practical, (3) increasingly skeptical, and (4) literary, mystical, and scientifically worthless—until philosophy is renewed through a new period's first phase. (See Brentano's essay "The Four Phases of Philosophy and Its Current State" 1895, tr. by Mezei and Smith 1998.)
  • C. S. Peirce, usually a trichotomist, discussed four methods for overcoming troublesome uncertainties and achieving secure beliefs: (1) the method of tenacity (policy of sticking to initial belief), (2) the method of authority, (3) the method of congruity (following a fashionable paradigm), and (4) the fallibilistic, self-correcting method of science (see "The Fixation of Belief", 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is unknowable; (3) maintaining that something is inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena (see "F.R.L." [First Rule of Logic], 1899).
  • Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatiotemporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss's Modes of Being, 1958).
  • Karl Popper outlined a tetradic schema to describe the growth of theories and, via generalization, also the emergence of new behaviors and living organisms: (1) problem, (2) tentative theory, (3) (attempted) error-elimination (especially by way of critical discussion), and (4) new problem(s). (See Popper's Objective Knowledge, 1972, revised 1979.)
  • John Boyd (military strategist) made his key concept the decision cycle or OODA loop, consisting of four stages: (1) observation (data intake through the senses), (2) orientation (analysis and synthesis of data), (3) decision, and (4) action.[84] Boyd held that his decision cycle has philosophical generality, though for strategists the point remains that, through swift decisions, one can disrupt an opponent's decision cycle.
  • Richard McKeon outlined four classes (each with four subclasses) of modes of philosophical inquiry: (1) Modes of Being (Being); (2) Modes of Thought (That which is); (3) Modes of Fact (Existence); (4) Modes of Simplicity (Experience)—and, corresponding to them, four classes (each with four subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See McKeon's "Philosophic Semantics and Philosophic Inquiry" in Freedom and History and Other Essays, 1989.)
  • Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: kinds (substantial universals), attributes (relational universals and property-universals), objects (substantial particulars), and modes (relational particulars and property-particulars, also known as "tropes"). (See Lowe's "Recent Advances in Metaphysics," 2001, Eprint)
  • Four opposed camps of the morality and nature of evil: moral absolutism, amoralism, moral relativism, and moral universalism.

In technology

Файл:U+2676 DejaVu Sans.svg
4 as a resin identification code, used in recycling

In transport

Файл:BKV m 4 jms.svg

In sports

  • In the Australian Football League, the top level of Australian rules football, each team is allowed 4 "interchanges" (substitute players), who can be freely substituted at any time, subject to a limit on the total number of substitutions.
  • In baseball:
    • There are four bases in the game: first base, second base, third base, and home plate; to score a run, an offensive player must complete, in the sequence shown, a circuit of those four bases.
    • When a batter receives four pitches that the umpire declares to be "balls" in a single at-bat, a base on balls, informally known as a "walk", is awarded, with the batter sent to first base.
    • For scoring, number 4 is assigned to the second baseman.
    • Four is the most runs that can be scored on any single at bat, whereby all three baserunners and the batter score (the most common being via a grand slam).
    • The fourth batter in the batting lineup is called the cleanup hitter.
  • In basketball, the number four is used to designate the power forward position, often referred to as "the four spot" or "the four".[91]
  • In cricket, a four is a specific type of scoring event, whereby the ball crosses the boundary after touching the ground at least one time, scoring four runs. Taking four wickets in four consecutive balls is typically referred to as a double hat trick (two consecutive, overlapping hat tricks).
  • In American Football teams get four downs to reach the line of gain.
  • In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature, 4− represents a coxless four and 4+ represents a coxed four.
  • In rugby league:
    • A try is worth 4 points.
    • One of the two starting centres wears the jersey number 4. (An exception to this rule is the Super League, which uses static squad numbering.)
  • In rugby union:
    • One of the two starting locks wears the jersey number 4.
    • In the standard bonus points system, a point is awarded in the league standings to a team that scores at least 4 tries in a match, regardless of the match result.
  • In women’s artistic gymnastics, there are four events/apparatus uneven bars, balance beam, floor exercise, and vault.

In other fields

Шаблон:See also

Файл:ICS Four.svg
International maritime signal flag for 4
Файл:I-p-c-s org journal.TIF
Playing cards for 4

In music

Groups of four

See also

References

Шаблон:Reflist

External links

Шаблон:Wiktionary Шаблон:Commons category

Шаблон:Integers Шаблон:Authority control

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64
  2. ↑ Шаблон:Cite web
  3. ↑ Шаблон:Cite news
  4. ↑ Шаблон:Cite book
  5. ↑ Шаблон:Cite book
  6. ↑ Шаблон:Cite book
  7. ↑ Шаблон:Citation
  8. ↑ Шаблон:Cite book
  9. ↑ Шаблон:Cite book
  10. ↑ Шаблон:Cite book
  11. ↑ Шаблон:Cite book
  12. ↑ Шаблон:Cite book
  13. ↑ Шаблон:Cite web
  14. ↑ Шаблон:Cite book
  15. ↑ Шаблон:Cite book
  16. ↑ Шаблон:Cite book
  17. ↑ Шаблон:Cite book
  18. ↑ Шаблон:Cite book
  19. ↑ 19,0 19,1 19,2 Шаблон:Cite book
  20. ↑ Шаблон:Cite book
  21. ↑ Шаблон:Cite book
  22. ↑ Шаблон:Cite book
  23. ↑ Шаблон:Cite book
  24. ↑ Шаблон:Cite book
  25. ↑ Шаблон:Cite book
  26. ↑ Шаблон:Cite book
  27. ↑ Шаблон:Cite book
  28. ↑ Шаблон:Cite book
  29. ↑ Шаблон:Cite book
  30. ↑ Шаблон:Cite book
  31. ↑ Шаблон:Cite book
  32. ↑ Шаблон:Cite book
  33. ↑ 33,0 33,1 33,2 Шаблон:Cite book
  34. ↑ Шаблон:Cite web
  35. ↑ Шаблон:Cite book
  36. ↑ Шаблон:Cite book
  37. ↑ Шаблон:Cite book
  38. ↑ Шаблон:Cite book
  39. ↑ Шаблон:Cite book
  40. ↑ Шаблон:Cite book
  41. ↑ Шаблон:Cite book
  42. ↑ Шаблон:Cite book
  43. ↑ Шаблон:Cite book
  44. ↑ Шаблон:Cite web
  45. ↑ Шаблон:Cite book
  46. ↑ Шаблон:Cite book
  47. ↑ Шаблон:Cite book
  48. ↑ Шаблон:Cite book
  49. ↑ Шаблон:Cite book
  50. ↑ Шаблон:Cite book
  51. ↑ Шаблон:Cite book
  52. ↑ Шаблон:Cite book
  53. ↑ Шаблон:Cite book
  54. ↑ Шаблон:Cite book
  55. ↑ Шаблон:Cite book
  56. ↑ Шаблон:Cite book
  57. ↑ Шаблон:Cite book
  58. ↑ Шаблон:Cite book
  59. ↑ Шаблон:Cite book
  60. ↑ Шаблон:Cite web
  61. ↑ Шаблон:Cite news
  62. ↑ Шаблон:Cite book
  63. ↑ Шаблон:Cite book
  64. ↑ Шаблон:Cite book
  65. ↑ Шаблон:Cite book
  66. ↑ Шаблон:Cite book
  67. ↑ Шаблон:Cite book
  68. ↑ Шаблон:Cite book
  69. ↑ Шаблон:Cite book
  70. ↑ Шаблон:Cite book
  71. ↑ Шаблон:Cite book
  72. ↑ Шаблон:Cite book
  73. ↑ Шаблон:Cite book
  74. ↑ Шаблон:Cite book
  75. ↑ Шаблон:Cite book
  76. ↑ Шаблон:Cite book
  77. ↑ Шаблон:Cite book
  78. ↑ Шаблон:Cite book
  79. ↑ Шаблон:Cite book
  80. ↑ Шаблон:Cite book
  81. ↑ Шаблон:Cite book
  82. ↑ Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.
  83. ↑ Шаблон:Cite book
  84. ↑ Шаблон:Cite book
  85. ↑ Шаблон:Cite web
  86. ↑ Шаблон:Cite book
  87. ↑ Шаблон:Cite book
  88. ↑ Шаблон:Cite book
  89. ↑ Шаблон:Cite journal
  90. ↑ Шаблон:Cite magazine
  91. ↑ Шаблон:Cite book
  92. ↑ Шаблон:Cite web
  93. ↑ Шаблон:Cite book
  94. ↑ Шаблон:Cite book
  95. ↑ Шаблон:Cite book
  96. ↑ Шаблон:Cite book
  97. ↑ Шаблон:Cite book
  98. ↑ Шаблон:Cite web
  99. ↑ Шаблон:Cite web
  100. ↑ Шаблон:Cite book
  101. ↑ Шаблон:Cite book
  102. ↑ Шаблон:Cite book
  103. ↑ Шаблон:Cite book
  104. ↑ Шаблон:Cite book
  105. ↑ Шаблон:Cite book
  106. ↑ Шаблон:Cite book
  107. ↑ Шаблон:Cite book
  108. ↑ Шаблон:Cite book
  109. ↑ Шаблон:Cite book
  110. ↑ Шаблон:Cite book
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:4&oldid=12851746