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# 4  ( Integers )

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4 ( four) is a , , and . It is the following 3 and preceding 5.

In mathematics
Four is the smallest , its proper being and .

4 is the smallest squared prime ( p2) and the only even number in this form. 4 is also the only square one more than a prime number.

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 .

In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, , and so on, for any number of up arrows. (That is, for every positive integer n, where is the .)

A four-sided plane figure is a (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a tetragon. A divided by 4 makes and four quadrants. Because of it, four (4) is the base number of the plane (mathematics). Four cardinal directions, , the system, and the system are based on four.

A solid figure with four faces as well as four vertices is a , and 4 is the smallest possible number of faces (as well as vertices) of a . The regular tetrahedron is the simplest . A , which can also be called a 3-, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.

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

• Two-dimensional: infinitely many convex .
• Three-dimensional: five convex regular polyhedra (the five ).
• Four-dimensional: six convex regular polychora.
• Five-dimensional and every higher-dimensional: three regular convex (, , ).

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on ℝ except when , in which case there are uncountably many.

The smallest non- group has four elements; it is the . Four is also the order of the smallest non-trivial groups that are not .

Four is the only integer n for which the (non trivial) alternating group A is not simple.

Four is the maximum number of dimensions of a real associative (the ), by a theorem of Ferdinand Georg Frobenius.

The four-color theorem states that a (or, equivalently, a flat of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 48 Three colors are not, in general, sufficient to guarantee this. The largest planar has four vertices.

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

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. .

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

The game, there are known solutions for all integers from 0 to 880 (but not 881).

List of basic calculations
4 × x4 ! ! !

4 ÷ x421.10.80.0.0.50.0.4 !0.0.0.0.0.20.25
x ÷ 40.250.50.7511.251.51.7522.252.5 !2.7533.253.53.754

4416642561024409616384655362621441048576 !41943041677721667108864
x11681256625129624014096656110000 !146412073628561

Evolution of the glyph
Representing 1, 2 and 3 in as many lines as the number represented worked well. The Indians simplified 4 by joining its four lines into a cross that looks like the modern plus sign. The would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where the speed of writing was a secondary concern. The ' 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 numeral less cursive, ending up with a glyph very close to the original Brahmin cross.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

While the shape of the 4 character has an ascender in most modern , in typefaces with the character usually has a , as, for example, in .

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.

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 or the Ϥ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".

In religion

Buddhism
• Four Noble Truths – , Samudaya, , Magga
• – observations which affected Prince Siddhartha deeply and made him realize the sufferings of all beings, and compelled him to begin his spiritual journey—an man, a man, a man, and an
• Four Great Elements – earth, water, fire, and wind
• Four Heavenly Kings
• – contemplation of the body, contemplation of feelings, contemplation of mind, contemplation of mental objects
• Four Right Exertions
• Four jhānas
• Four arūpajhānas
• – loving-kindness, compassion, , and
• Four stages of enlightenment – stream-enterer, , , and arahant
• Four main pilgrimage sites – , , , and

Judeo-Christian symbolism

Hinduism
• There are four Vedas: Rigveda, Samaveda, Yajurveda and Atharvaveda.
• In Puruṣārtha, there are four aims of human life: , , Kāma, .
• The four stages of life (student life), (household life), (retired life) and (renunciation).
• The four primary castes or strata of society: (priest/teacher), (warrior/politician), (landowner/entrepreneur) and (servant/manual laborer).

Islam
• lasts for four days, from the 10th to the 14th of .
• There are four or Rightly Guided : , Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib.
• The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)
• There are four months in Islam war isn't permitted in: , , Dhu al-Qi'dah and .
• There are four schools of : , Shafi`i, and .
• There are four major Sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafi`i, Malik ibn Anas and Ahmad ibn Hanbal.
• There are four books in Islam: Torah, Zaboor, Injeel, Quran.
• Waiting for four months is ordained for those who take an oath for abstention from their wives.
• The waiting period of the woman whose husband dies, is four months and ten days.
• When Abraham said: “My Lord, show me how You give life to the dead,” Allah said: “Why! Do you have no faith?” Abraham replied: “Yes, but in order that my heart be at rest.” He said: “Then take four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will come to you flying. Al-Baqara
• The respite of four months was granted to give time to the mushriks in Surah so that they should consider their position carefully and decide whether to make preparation for war or to emigrate from the country or to accept Islam.
• Those who accuse honorable women (of unchastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after. They are indeed transgressors. An-Noor

Taoism

Other
• In a more general sense, numerous mythological and cosmogonical systems consider Four corners of the world as essentially corresponding to the four points of the compass.
• Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of .
• The Chinese, Vietnamese, the Korean and the Japanese are because it is a homonym for "death" in their languages.

In politics
• : 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.
• 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 . Among the four was Mao's widow, . Since then, many other political factions headed by four people have been called "Gangs of Four".

In computing
• 0x04 is the code of the character End of Transmission, which is abbreviated to EOT.

In science
• A is a thing formed out of four sub-units.

In astronomy

In biology
• Four is the number of types in and , , , ( in ).
• Many have four feet, legs or leglike appendages ().
• The consists of four chambers.
• Many (, ) use four fingers for movement.
• All with wings except have four .
• Insects of the superorder , also known as Holometabola, such as butterflies, ants, bees, beetles, fleas, flies, moths, and wasps, undergo —complete metamorphism in four stages—from (1) embryo (ovum, egg), to (2) larva (such as grub, caterpillar), then (3) pupa (such as the chrysalis), and finally (4) the imago.
• In the common ABO blood group system, there are 4 blood types (A, B, O, AB).
• have four and four .
• The cow's stomach is divided in four digestive compartments: reticulum, rumen, omasum and abomasum.

In chemistry
• Valency of (that is basis of life on the Earth) is four. Also because of its tetrahedral crystal bond structure, (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of , whose compounds form the majority of the mass of the Earth's crust.
• The of
• There are four basic states of : , , , and plasma.

In physics
• Special relativity and general relativity treat nature as four-: 3D regular and one-dimensional are treated together and called . Also, any event E has a composed of four zones of possible communication and cause and effect (outside the light cone is strictly incommunicado).
• There are four fundamental forces (, , the weak nuclear force, and the strong nuclear force).
• In statistical mechanics, the four functions inequality is an inequality for four functions on a finite distributive lattice.

In logic and philosophy
• 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",Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402. 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 , there are four separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (it is possible that both are true, but not that both are false); contraries (it is possible that both are 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).
• held that there are basically four causes in nature: the efficient cause, the matter, the end, and the form.
• The 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.
• 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.
• 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 ), and (4) the , self-correcting method of science (see "", 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is absolutely unknowable; (3) maintaining that something is absolutely 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, 1899).
• Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatio-temporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss's Modes of Being, 1958).
• 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. 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.
• 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.)
• (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: , , , and moral universalism.

In technology
• The resin identification code used in recycling to identify low-density polyethylene.
• Most has four legs – tables, chairs, etc.
• The four color process () is used for .
• Wide use of (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames).
• In the Rich Text Format specification, language code 4 is for the . Codes for regional variants of Chinese are congruent to .
• machines have four-twelve function keys.
• On most phones, the 4 key is associated with the letters G, H, and I, but on the cellular phone, it is the key for D and F.
• On many computer keyboards, the "4" key may also be used to type the (\$) if the is held down.
• It is the number of in a , equivalent to half a
• In , "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
• In , "4" may be used to replace the letter "A".
• The TCP/IP stack consists of four layers.

In transport
• Many internal combustion engines are called engines because they complete one thermodynamic cycle in four distinct steps: Intake, compression, power, and exhaust.
• Most , including , and particularly and light commercial vehicles have four road .
• "Quattro", meaning four in the Italian language, is used by as a trademark to indicate that (4WD) technologies are used on Audi-branded cars. The word "Quattro" was initially used by Audi in 1980 in its original 4WD coupé, the . Audi also has a privately held subsidiary company called .
• List of highways numbered 4

In sports
• In :
• There are four bases in the game: , , , and ; 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 "" 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 .
• Four is the most number of 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).
• In , the number four is used to designate the power forward position, often referred to as "the four spot" or "the four".
• In , 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 rowing, a four refers to a for four rowers, with or without . In rowing nomenclature, 4− represents a coxless four and 4+ represents a coxed four.
• In :
• A is worth 4 points.
• One of the two starting centres wears the jersey number 4. (An exception to this rule is the European , which uses static squad numbering.)
• In :
• 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 other fields
• The phrase "" is used to describe many words in the .
• Four is the only number whose name in English has the same number of letters as its value.
• Four (四, formal writing: 肆, sì) is considered an in , , Vietnamese and Japanese cultures mostly in Eastern Asia because it sounds like the word "death" (死, sǐ). To avoid complaints from people with tetraphobia, many numbered product lines skip the "four": e.g. (there is no series beginning with a 4), Palm PDAs, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See and Numbers in Chinese culture.
• The number of characters in a canonical four-character idiom.
• In the NATO phonetic alphabet, the digit 4 is called "fower".
• In , Cancer is the 4th astrological sign of the .
• In , The Emperor is the fourth trump or card.
• In 's series, four is the number of courses that each cup holds.
• In , a game named for the Greek word for 4, every shape in the game is formed of 4 blocks each.
• 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ of (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision).
• Number Four is a character in the book series

In music
• In , is constructed of four beats per measure and a quarter note receives one beat.
• In popular or , the most common is also founded on four beats, i.e., 4/4 having four beats.
• The common is built on two sets of four notes (e.g., CDEF, GABC), where the first and last notes create an interval (a pair-of-four relationship).
• The interval of a is a foundational element of many genres of music, represented in as the tonic and relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the interval of four in more active harmonic contexts.
• The typical number of movements in a .
• The number of completed, numbered symphonies by .
• The number of strings on a , a , a , , a cuatro and a , and the number of string pairs on a .
• "Four calling birds" is the gift on the fourth day of Christmas in the carol "The Twelve Days of Christmas".

Albums
• Four, British rock band 's fourth full-length album.
• Four, British-Irish boy band 's fourth full-length album.
• 4, American artist Beyoncé's fourth album.
• 4, British–American band Foreigner's fourth studio album.

Groups of four

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55–58

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