# 4

Short description: Natural number
 ← 3 4 5 →
Cardinalfour
Ordinal4th
(fourth)
Numeral systemquaternary
Factorization22
Divisors1, 2, 4
Greek numeralΔ´
Roman numeralIV, iv
Greek prefixtetra-
Binary1002
Ternary113
Quaternary104
Quinary45
Senary46
Octal48
Duodecimal412
Vigesimal420
Base 36436
Arabic, Kurdish٤
Persian, Sindhi۴
Shahmukhi, Urdu۴
Ge'ez
Bengali, Assamese
Chinese numeral四，亖，肆
Devanagari
Telugu
Malayalam
Tamil
Hebrewד
Khmer
Thai
Burmese

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

## In mathematics

Four is the smallest composite number, its proper divisors being 1 and 2.[1]

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.[2] For example, 1092 is a multiple of 4 because 92 = 4 × 23.

In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so on, for any number of up arrows.[3] (That is, 2 [n] 2 = 4 for every positive integer n, where a [n] b is the hyperoperation.)

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

A solid figure with four faces as well as four vertices is a tetrahedron,[4] and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron.[5] The regular tetrahedron is the simplest Platonic solid.[6] A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.[7]

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

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

The smallest non-cyclic group has four elements; it is the Klein four-group.[8] Four is also the order of the smallest non-trivial groups that are not simple.

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

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

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.[9] Three colors are not, in general, sufficient to guarantee this.[10] The largest planar complete graph has four vertices.[11]

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers.[12] Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.[13]

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2z2.

Four is the highest degree general polynomial equation for which there is a solution in radicals.[14]

## 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 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 ÷ x 4 2 1.3 1 0.8 0.6 0.571428 0.5 0.4 0.4 0.36 0.3 0.307692 0.285714 0.26 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
4x 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
x4 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

## Evolution of the Hindu-Arabic digit

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.[15]

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 .

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.[16]

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".[17]

## In religion

### Hinduism

• There are four Vedas: Rigveda, Samaveda, Yajurveda and Atharvaveda.[37]
• In Puruṣārtha, there are four aims of human life: Dharma, Artha, Kāma, Moksha.[38]
• The four stages of life Brahmacharya (student life), Grihastha (household life), Vanaprastha (retired life) and Sannyasa (renunciation).[39]
• The four primary castes or strata of society: Brahmana (priest/teacher), Kshatriya (warrior/politician), Vaishya (landowner/entrepreneur) and Shudra (servant/manual laborer).[40]
• The swastika symbol is traditionally used in Hindu religions as a sign of good luck and signifies good from all four directions.[41]
• The god Brahma has four faces.[42]
• There are four yugas: Satya, Dvapara, Treta and Kali[43]

### Islam

• Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja.[44]
• The four holy cities of Islam: Mecca, Medina, Jerusalem and Damascus.
• The four tombs in the Green Dome: Muhammad, Abu Bakr, Umar ibn Khattab and Isa ibn Maryam (Jesus).
• There are four Rashidun or Rightly Guided Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib.[45]
• The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)[46]
• There are four months in which war is not permitted: Muharram, Rajab, Dhu al-Qi'dah and Dhu al-Hijjah.[47]
• There are four Sunni schools of fiqh: Hanafi, Shafii, Maliki and Hanbali.
• There are four major Sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafii, Malik ibn Anas and Ahmad ibn Hanbal.
• There are four books in Islam: Taurait, Zaboor, Injeel, Quran.[48]
• Waiting for four months is ordained for those who take an oath for abstention from their wives.[49]
• The waiting period of the woman whose husband dies, is four months and ten days.[50]
• 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 2:260][51]
• The respite of four months was granted to give time to the mushriks in Surah At-Tawba 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.[52]
• 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 24:4][53]

### 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.[55]
• Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of New Mexico.[56]
• The Chinese, the Koreans, and the Japanese are superstitious about the number four because it is a homonym for "death" in their languages.[57]
• In Slavic mythology, the god Svetovid has four heads.[58]

## 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.[59]
• 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".[60]

## In science

### 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.[75]
• The atomic number of beryllium[76]
• There are four basic states of matter: solid, liquid, gas, and plasma.[77]

## In logic and philosophy

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",[81] 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.[82]
• 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.[83] 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

4 as a resin identification code, used in recycling
• The resin identification code used in recycling to identify low-density polyethylene.[84]
• Most furniture has four legs – tables, chairs, etc.
• The four color process (CMYK) is used for printing.[85]
• Wide use of rectangles (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 Chinese language. Codes for regional variants of Chinese are congruent to 4 mod 256.
• Credit card machines have four-twelve function keys.
• On most phones, the 4 key is associated with the letters G, H, and I,[86] but on the BlackBerry Pearl, it is the key for D and F.
• On many computer keyboards, the "4" key may also be used to type the dollar sign (\$) if the shift key is held down.
• It is the number of bits in a nibble, equivalent to half a byte[87]
• In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
• In Leetspeak, "4" may be used to replace the letter "A".
• The TCP/IP stack consists of four layers.[88]

## In transport

• Many internal combustion engines are called four-stroke engines because they complete one thermodynamic cycle in four distinct steps: Intake, compression, power, and exhaust.
• Most vehicles, including motor vehicles, and particularly cars/automobiles and light commercial vehicles have four road wheels.
• "Quattro", meaning four in the Italian language, is used by Audi as a trademark to indicate that all-wheel drive (AWD) technologies are used on Audi-branded cars.[89] The word "Quattro" was initially used by Audi in 1980 in its original 4WD coupé, the Audi Quattro. Audi also has a privately held subsidiary company called quattro GmbH.
• List of highways numbered 4

## 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".[90]
• 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 other fields

International maritime signal flag for 4
• The phrase "four-letter word" is used to describe many swear words in the English language.[91]
• Four is the only number whose name in English has the same number of letters as its value.
• Four (, formal writing: , pinyin sì) is considered an unlucky number in Chinese, Korean, Vietnamese and Japanese cultures mostly in Eastern Asia because it sounds like the word "death" (, pinyin sǐ). To avoid complaints from people with tetraphobia, many numbered product lines skip the "four": e.g. Nokia cell phones (there was no series beginning with a 4 until the Nokia 4.2), Palm PDAs, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture.
• In Pythagorean numerology (a pseudocience) the number 4 represents security and stability.
• The number of characters in a canonical four-character idiom.
• In the NATO phonetic alphabet, the digit 4 is called "fower".[92]
• In Astrology, Cancer is the 4th astrological sign of the Zodiac.[93]
• In Tarot, The Emperor is the fourth trump or Major Arcana card.[94]
• In Tetris, a game named for the Greek word for 4, every shape in the game is formed of 4 blocks each.[95][96]
• 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ of certiorari (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision) at the court's current size.[97]
• Number Four is a character in the book series Lorien Legacies.[98]
• In the performing arts, the fourth wall is an imaginary barrier which separates the audience from the performers, and is "broken" when performers communicate directly to the audience.[99]

## In music

• In written music, common time is constructed of four beats per measure and a quarter note receives one beat.[100]
• In popular or modern music, the most common time signature is also founded on four beats, i.e., 4/4 having four quarter note beats.
• The common major scale is built on two sets of four notes (e.g., CDEF, GABC), where the first and last notes create an octave interval (a pair-of-four relationship).
• The interval of a perfect fourth is a foundational element of many genres of music, represented in music theory as the tonic and subdominant 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 symphony.[101]
• The number of completed, numbered symphonies by Johannes Brahms.[102]
• The number of strings on a violin, a viola, a cello, double bass, a cuatro and a ukulele, and the number of string pairs on a mandolin.
• "Four calling birds" is the gift on the fourth day of Christmas in the carol "The Twelve Days of Christmas".[103]

## Groups of four

• Big Four (disambiguation)
• Four basic operations of arithmetic: addition, subtraction, multiplication, division.[104]
• Greek classical elements (fire, air, water, earth).[105]
• Four seasons: spring, summer, autumn, winter.
• The Four Seasons (disambiguation)
• A leap year generally occurs every four years.[106]
• Approximately four weeks (4 times 7 days) to a lunar month (synodic month = 29.53 days). Thus the number four is universally an integral part of primitive sacred calendars.
• Four cardinal directions: north, south, east, west.[107]
• Four Temperaments: sanguine, choleric, melancholic, phlegmatic.
• Four Humors: blood, yellow bile, black bile, phlegm.[108]
• Four Great Ancient Capitals of China.
• Four-corner method.
• Four Asian Tigers, referring to the economies of Hong Kong, Taiwan, South Korea , and Singapore
• Cardinal principles.
• Four cardinal virtues: justice, prudence, temperance, fortitude.
• Four suits of playing cards: hearts, diamonds, clubs, spades.[109]
• Four nations of the United Kingdom : England, Wales, Scotland, Northern Ireland.
• Four provinces of Ireland: Munster, Ulster, Leinster, Connacht.
• Four estates: politics, administration, judiciary, journalism. Especially in the expression "Fourth Estate", which means journalism.
• Four Corners is the only location in the United States where four states come together at a single point: Colorado, Utah, New Mexico, and Arizona.
• Four Evangelists – Matthew, Mark, Luke, and John
• Four Doctors of Western Church – Saint Gregory the Great, Saint Ambrose, Saint Augustine, and Saint Jerome
• Four Doctors of Eastern Church – Saint John Chrysostom, Saint Basil the Great, and Gregory of Nazianzus and Saint Athanasius
• Four Galilean moons of Jupiter – Io, Europa, Ganymede, and Callisto
• The Gang of Four was a Chinese communist political faction.
• The Fantastic Four: Mr. Fantastic, The Invisible Woman, The Human Torch, The Thing.
• The Teenage Mutant Ninja Turtles: Leonardo, Michelangelo, Donatello, Raphael
• The Beatles were also known as the "Fab Four": John Lennon, Ringo Starr, George Harrison, Paul McCartney.
• Gang of Four is a British post-punk rock band formed in the late 1970s.
• Four rivers in the Garden of Eden (Genesis 2:10–14): Pishon (perhaps the Jaxartes or Syr Darya), Gihon (perhaps the Oxus or Amu Darya), Hiddekel (Tigris), and P'rat (Euphrates).
• There are also four years in a single Olympiad (duration between the Olympic Games). Many major international sports competitions follow this cycle, among them the FIFA World Cup and its women's version, the FIBA World Championships for men and women, and the Rugby World Cup.
• There are four limbs on the human body.
• Four Houses of Hogwarts in the Harry Potter series: Gryffindor, Hufflepuff, Ravenclaw, Slytherin.[110]
• Four known continents of the world in the A Song of Ice and Fire series: Westeros, Essos, Sothoryos, Ulthos.
• Each Grand Prix in Nintendo's Mario Kart series is divided into four cups and each cup is divided into four courses. The Mushroom Cup, Flower Cup, Star Cup, and Special Cup make up the Nitro Grand Prix, while the Shell Cup, Banana Cup, Leaf Cup, and the Lightning Cup make up the Retro Grand Prix.

• List of highways numbered 4

## References

1. Fiore, Gregory (1993-08-01) (in en). Basic mathematics for college students: concepts and applications. HarperCollins College. pp. 162. ISBN 978-0-06-042046-8. "The smallest composite number is 4."
2. Prep, Kaplan Test (2017-01-03) (in en). SAT Subject Test Mathematics Level 1. Simon and Schuster. pp. 289. ISBN 978-1-5062-0922-7. "An integer is divisible by 4 if the last two digits form a multiple of 4."
3. Hodges, Andrew (2008-05-17) (in en). One to Nine: The Inner Life of Numbers. W. W. Norton & Company. pp. 249. ISBN 978-0-393-06863-4. "2 ↑↑ ... ↑↑ 2 is always 4"
4. Weisstein, Eric W.. "Tetrahedron" (in en).
5. Grossnickle, Foster Earl; Reckzeh, John (1968) (in en). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. pp. 337. ISBN 9780030676451. "...the smallest possible number of faces that a polyhedron may have is four"
6. Grossnickle, Foster Earl; Reckzeh, John (1968) (in en). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. pp. 337. ISBN 9780030676451. "...face of the platonic solid. The simplest of these shapes is the tetrahedron..."
7. Hilbert, David; Cohn-Vossen, Stephan (1999) (in en). Geometry and the Imagination. American Mathematical Soc.. pp. 143. ISBN 978-0-8218-1998-2. "...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs..."
8. Jeremy, Horne (2017-05-19) (in en). Philosophical Perceptions on Logic and Order. IGI Global. pp. 299. ISBN 978-1-5225-2444-1. "The Klein four-group is the smallest noncyclic group,..."
9. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 48
10. Ben-Menahem, Ari (2009-03-06) (in en). Historical Encyclopedia of Natural and Mathematical Sciences. Springer Science & Business Media. pp. 2147. ISBN 978-3-540-68831-0. "(i.e. That there are maps for which three colors are not sufficient)"
11. Molitierno, Jason J. (2016-04-19) (in en). Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs. CRC Press. pp. 197. ISBN 978-1-4398-6339-8. "... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4."
12. Spencer, Joel (1996), Chudnovsky, David V.; Chudnovsky, Gregory V.; Nathanson, Melvyn B., eds., "Four Squares with Few Squares" (in en), Number Theory: New York Seminar 1991–1995 (New York, NY: Springer US): pp. 295–297, doi:10.1007/978-1-4612-2418-1_22, ISBN 978-1-4612-2418-1
13. Peterson, Ivars (2002) (in en). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA. pp. 95. ISBN 978-0-88385-537-9. "7 is an example of an integer that can't be written as the sum of three squares."
14. Bajnok, Béla (2013-05-13) (in en). An Invitation to Abstract Mathematics. Springer Science & Business Media. ISBN 978-1-4614-6636-9. "There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher."
15. 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
16. "Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check" (in en-US). Digital Check. 2017-02-02.
17. Chwalkowski, Farrin (2016-12-14) (in en). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. pp. 22. ISBN 978-1-4438-5728-4. "The four main pilgrimages sites are: Lumbini, Bodh Gaya, Sarnath and Kusinara....four Noble Truths of Buddhism"
18. Voorst, Robert Van (2012-01-01) (in en). RELG: World. Cengage Learning. pp. 108. ISBN 978-1-111-72620-1. "He first observed the suffering of the world in the Four Passing Sites"
19. Yun, Hsing; Xingyun (2010) (in en). The Great Realizations: A Commentary on the Eight Realizations of a Bodhisattva Sutra. Buddha's Light Publishing. pp. 27. ISBN 978-1-932293-44-9. "The four great elements, earth, water, fire and wind..."
20. Chaudhuri, Saroj Kumar (2003) (in en). Hindu Gods and Goddesses in Japan. Vedams eBooks (P) Ltd. pp. 20. ISBN 978-81-7936-009-5. "The Buddhists adopted him as one of the four Devarajas or Heavenly Kings"
21. Bronkhorst, Johannes (2009-12-22) (in en). Buddhist Teaching in India. Simon and Schuster. pp. 66. ISBN 978-0-86171-566-4. "The four right exertions are..."
22. Mistry, Freny (2011-05-02) (in en). Nietzsche and Buddhism: Prolegomenon to a Comparative Study. Walter de Gruyter. pp. 69. ISBN 978-3-11-083724-7. "these four bases of psychic power"
23. Arbel, Keren (2017-03-16) (in en). Early Buddhist Meditation: The Four Jhanas as the Actualization of Insight. Taylor & Francis. pp. 1. ISBN 978-1-317-38399-4. "This book is about the four jhanas"
24. Jayatilleke, K. N. (2013-10-16) (in en). Early Buddhist Theory of Knowledge. Routledge. ISBN 978-1-134-54294-9. "...the states of the four arupajhanas."
25. Gorkom, Nina van (in en). The Perfections Leading to Enlightenment. Рипол Классик. pp. 171. ISBN 978-5-88139-786-9. "There are four of them: loving-kindness, metta, compassion, karuna, sympathetic joy, mudita and equanimity, upekkha."
26. Rinpoche, Khenchen Konchog Gyaltshen; Milarepa; Sumgon, Jigten (2013-10-08) (in en). Opening the Treasure of the Profound: Teachings on the Songs of Jigten Sumgon and Milarepa. Shambhala Publications. ISBN 978-0-8348-2896-4. "...four types of shravaka (stream enterer, oncereturner, nonreturner, and arhat)"
27. Fahlbusch, Erwin; Bromiley, Geoffrey William; Lochman, Jan Milic; Mbiti, John; Pelikan, Jaroslav (2008-02-14) (in en). The Encyclodedia of Christianity, Vol. 5. Wm. B. Eerdmans Publishing. pp. 823. ISBN 978-0-8028-2417-2.
28. Stevenson, Kenneth; Glerup, Michael (2014-03-19) (in en). Ezekiel, Daniel. InterVarsity Press. pp. xlv. ISBN 978-0-8308-9738-4. "We have already mentioned the four living creatures—the man, the lion, the ox and the eagle"
29. Butnick, Stephanie; Leibovitz, Liel; Oppenheimer, Mark (2019-10-01) (in en). The Newish Jewish Encyclopedia: From Abraham to Zabar's and Everything in Between. Artisan Books. ISBN 978-1-57965-893-9. "...be like Sarah, Rachel, Rebecca, and Leah, the foremothers of Judaism"
30. Kaplan, Aryeh (1990) (in en). Innerspace: Introduction to Kabbalah, Meditation and Prophecy. Moznaim. pp. 109. ISBN 9780940118560. "...as well as to the palm ( lulav ), myrtle ( hadas ), willow ( aravah ) and citron ( etrog ), the four species of plants"
31. Dennis, Geoffrey W. (2007) (in en). The Encyclopedia of Jewish Myth, Magic and Mysticism. Llewellyn Worldwide. pp. 188. ISBN 978-0-7387-0905-5. "The Passover Seder is particularly structured around fours: the Four Questions, the Four Sons, and four cups of wine."
32. "Four Expressions of Redemption to be said on the Jewish holiday of Passover - Google Search". pp. 46. "There are four expressions of redemption in the Torah"
33. Charles, Templeton (1973) (in en). Jesus: the four Gospels, Matthew, Mark, Luke, and John, combined in one narrative and rendered in modern English. Simon and Schuster. ISBN 9780671217150.
34. Wagner, Richard; Helyer, Larry R. (2011-01-31) (in en). The Book of Revelation For Dummies. John Wiley & Sons. pp. 308. ISBN 978-1-118-05086-6. "The four horsemen of the Apocalypse are one of the most familiar images of Revelation"
35. Turfe, Tallal Alie (2013-07-19) (in en). Children of Abraham: United We Prevail, Divided We Fail. iUniverse. pp. 91. ISBN 978-1-4759-9047-8. "The four holy cities of Judaism are Jerusalem, Hebron, Safed, and Tiberius."
36. Frawley, David (2014-10-07) (in en). Vedic Yoga: The Path of the Rishi. Lotus Press. ISBN 978-0-940676-25-1. "There are four Vedas"
37. Fritz, Stephen Martin (2019-05-14) (in en). Our Human Herds: The Theory of Dual Morality (Second Edition, Unabridged). Dog Ear Publishing. pp. 491. ISBN 978-1-4575-6755-1. "that these four proper aims and objects"
38. (in en) Maanas - Individual and Society. Rapid Publications. ISBN 978-1-937192-06-8. "The Four Stages of Life"
39. Chwalkowski, Farrin (2016-12-14) (in en). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. pp. 23. ISBN 978-1-4438-5728-4. "The four primary castes or strata of society:..."
40. Kulendiren, Pon (2012-10-11) (in en). Hinduism a Scientific Religion: & Some Temples in Sri Lanka. iUniverse. pp. 32. ISBN 978-1-4759-3675-9.
41. Jansen, Eva Rudy (1993) (in en). The Book of Hindu Imagery: Gods, Manifestations and Their Meaning. Binkey Kok Publications. pp. 87. ISBN 978-90-74597-07-4. "Brahma has four faces,..."
42. Çakmak, Cenap (2017-05-18) (in en). Islam: A Worldwide Encyclopedia [4 volumes]. ABC-CLIO. pp. 397. ISBN 978-1-61069-217-5. "...Eid al-Adha (Feast of Sacrifice) lasts four days ..."
43. Leonard, Timothy; Willis, Peter (2008-06-11) (in en). Pedagogies of the Imagination: Mythopoetic Curriculum in Educational Practice. Springer Science & Business Media. pp. 144. ISBN 978-1-4020-8350-1. "... four Rightly Guided Caliphs, Abu-Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib,..."
44. Chwalkowski, Farrin (2016-12-14) (in en). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. pp. 23. ISBN 978-1-4438-5728-4. "According to Islam, the Four Arch Angels are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)."
45. Ph.D, Assad Nimer Busool (2010-12-28) (in en). The Wise Qur'an: These are the Verses of the Wise Book: These are the verses of the Wise Book. Xlibris Corporation. pp. 50. ISBN 978-1-4535-2526-5. "The sacred months are four, Rajab, Dhu al-Qi'dah, Dhu al-Hijjah, and al-Muharram. During those four sacred months there were no war..."
46. Shabazz, Hassan (2020-01-06) (in en). Al Islaam, and the Transformation of Society. Lulu.com. pp. 15. ISBN 978-1-7948-3337-1. "There are four books in Islam: Torah, Zaboor, Injeel and Holy Qur'an..."
47. BUKHARI, MUOHAMMAD BEN ISMAIL AL (2007-01-01) (in en). THE CORRECT TRADITIONS OF AL'BUKHARI 1-4 VOL 3: صحيح البخاري 1/4 [عربي/انكليزي ج3]. Dar Al Kotob Al Ilmiyah دار الكتب العلمية. pp. 840. "For those who take an oath for abstention from their wives, awaiting for four months is ordained;"
48. Ahmad, Yusuf Al-Hajj (in en). The Book Of Nikkah: Encyclopaedia of Islamic Law. Darussalam Publishers. "...for four months and ten days."
49. Mawdudi, Sayyid Abul A'la (2016-12-15) (in en). Towards Understanding the Qur'an: English Only Edition. Kube Publishing Ltd. pp. 59. ISBN 978-0-86037-613-2. "Then take four birds, ..."
50. Maudoodi, Syed Abul ʻAla (2000) (in en). Sūrah al-Aʻarāf to Sūrah bani Isrāel. Islamic Publications. pp. 177. "The respite of four months..."
51. Barazangi, Nimat Hafez (2016-03-09) (in en). Woman's Identity and Rethinking the Hadith. Routledge. pp. 138. ISBN 978-1-134-77065-6. "And those who launch a charge against chaste women and do not produce four witnesses..."
52. SK, Lim (in en). Origins of Chinese Auspicious Symbols. Asiapac Books Pte Ltd. pp. 16. ISBN 978-981-317-026-1. "Taoism later incorporated the four symbols into its immortality system..."
53. Terry, Milton Spenser (1883) (in en). Biblical Hermenutics: A Treatise on the Interpretation of the Old and New Testaments. Phillips & Hunt. pp. 382. "the four corners or extremities of the earth (Isa. xi, 12; Ezek. vii, 2.; Rev. vii, 1 ; xx, 8), corresponding, doubtless, with the four points of the compass"
54. (in en) Bulletin - State Department of Education. Department of Education. 1955. pp. 151. "Four was a sacred number of Zia"
55. Lachenmeyer, Nathaniel (2005) (in en). 13: The Story of the World's Most Notorious Superstition. Penguin Group (USA) Incorporated. pp. 187. ISBN 978-0-452-28496-8. "In Chinese , Japanese , and Korean , the word for four is , unfortunately , an exact homonym for death"
56. Maberry, Jonathan; Kramer, David F. (2007) (in en). The Cryptopedia: A Dictionary of the Weird, Strange & Downright Bizarre. Citadel Press. pp. 211. ISBN 978-0-8065-2819-9. "Svetovid is portrayed as having four heads ..."
57.
58. Raphael, Howard A., ed (November 1974). "The Functions Of A Computer: Instruction Register And Decoder". MCS-40 User's Manual For Logic Designers. Santa Clara, California, USA: Intel Corporation. p. viii. Retrieved 2020-03-03. "[...] The characteristic eight bit field is sometimes referred to as a byte, a four bit field can be referred to as a nibble. [...]"
59. Petsko, Gregory A.; Ringe, Dagmar (2004) (in en). Protein Structure and Function. New Science Press. pp. 40. ISBN 978-0-87893-663-2. "Oligomers containing two, three, four, five, six or even more subunits are known as dimers, trimers, tetramers, pentamers, hexamers, and so on."
60. Yaqoob, Tahir (2011) (in en). Exoplanets and Alien Solar Systems. New Earth Labs. pp. 12. ISBN 978-0-9741689-2-0. "The four inner planets (known as terrestrial, or rocky planets"
61. Encrenaz, Therese; Bibring, Jean-Pierre; Blanc, M.; Barucci, Maria-Antonietta; Roques, Francoise; Zarka, Philippe (2004-01-26) (in en). The Solar System. Springer Science & Business Media. pp. 283. ISBN 978-3-540-00241-3. "...the gas giants (Jupiter and Saturn), and the icy giants (Uranus and Neptune)"
62. Pidwirny, Michael (2020-05-07) (in en). Chapter 3: Matter, Energy and the Universe: Single chapter from the eBook Understanding Physical Geography. Our Planet Earth Publishing. pp. 10. "including the four large Galilean moons that are easily visible from a hobby telescope"
63. Pugh, Philip (2011-11-02) (in en). Observing the Messier Objects with a Small Telescope: In the Footsteps of a Great Observer. Springer Science & Business Media. pp. 41. ISBN 978-0-387-85357-4. "M4 is a globular star cluster near Antares in Scorpius."
64. Bok, Bart Jan; Bok, Priscilla Fairfield (1981) (in en). The Milky Way. Harvard University Press. pp. 66. ISBN 978-0-674-57503-5. "IV , subgiants"
65. (in en) Encyclopedia of Cell Biology. Academic Press. 2015-08-07. pp. 25. ISBN 978-0-12-394796-3.
66. Chien, Shu; Chen, Peter C. Y.; Fung, Yuan-cheng (2008) (in en). An Introductory Text to Bioengineering. World Scientific. pp. 54. ISBN 978-981-270-793-2. "The mammalian heart consists of four chambers,..."
67. Committee, Creation Research Society Textbook (1970) (in en). Biology: a search for order in complexity. Zondervan Pub. House. pp. 209. ISBN 978-0-310-29490-0. "Except for the flies, mosquitoes, and some others, insects with wings have four wings."
68. Pittenger, Dennis (2014-12-15) (in en). California Master Gardener Handbook, 2nd Edition. UCANR Publications. pp. 180. ISBN 978-1-60107-857-5. "metamorphosis is marked by four distinct stages"
69. Darpan, Pratiyogita (2008) (in en). Pratiyogita Darpan. Pratiyogita Darpan. pp. 85. "In the 'ABO' system, all blood belongs one of four major groups — A, B, AB or O"
70. Daniels, Patricia; Stein, Lisa (2009) (in en). Body: The Complete Human : how it Grows, how it Works, and how to Keep it Healthy and Strong. National Geographic Books. pp. 94. ISBN 978-1-4262-0449-4. "Four canines for tearing + Eight premolars for crushing +Twelve molars (including four wisdom teeth)"
71. Woodward, Thompson Elwyn; Nystrom, Amer Benjamin (1930) (in en). Feeding Dairy Cows. U.S. Department of Agriculture. pp. 4. "The cow's stomach is divided into four compartments."
72. Lucas, Jerry (1993) (in en). Great unsolved mysteries of science. F & W Pubns Inc. pp. 168. ISBN 978-1-55870-291-2. "Of course, carbon is not the only chemical element with a valence of +4 or -4"
73. Walsh, Kenneth A. (2009-01-01) (in en). Beryllium Chemistry and Processing. ASM International. pp. 93. ISBN 978-0-87170-721-5. "Beryllium has an atomic number of four"
74. Ebeling, Werner; Fortov, Vladimir E.; Filinov, Vladimir (2017-11-27) (in en). Quantum Statistics of Dense Gases and Nonideal Plasmas. Springer. pp. 39. ISBN 978-3-319-66637-2. "Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas."
75. Petkov, Vesselin (2009-06-23) (in en). Relativity and the Nature of Spacetime. Springer Science & Business Media. pp. 124. ISBN 978-3-642-01962-3. "should be regarded as a four-dimensional world"
76. Giordano, Nicholas (2009-02-13) (in en). College Physics: Reasoning and Relationships. Cengage Learning. pp. 1073. ISBN 978-0-534-42471-8. "We have referred to the four fundamental forces in nature,..."
77. Alon, Noga; Spencer, Joel H. (2011-09-20) (in en). The Probabilistic Method. John Wiley & Sons. pp. 6.1. ISBN 978-1-118-21044-4. "The Four Functions Theorem of Ahlswede Daykin"
78. Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.
79. Hennig, Boris (2018-12-05) (in en). Aristotle's Four Causes. Peter Lang. ISBN 978-1-4331-5929-9. "This book examines Aristotle's four causes (material, formal, efficient, and final)"
80. Wilkinson, Amy (2015-02-17) (in en). The Creator's Code: The Six Essential Skills of Extraordinary Entrepreneurs. Simon and Schuster. pp. 79. ISBN 978-1-4516-6609-0. "The OODA loop consists of four steps."
81. Howard, Brian Clark; Abdelrahman, Amina Lake; Institute, Good Housekeeping (2020-02-26). "You Might Be Recycling Wrong — Here's Everything You Need to Know About Recycling Symbols" (in en-US). "Plastic Recycling Symbol #4: LDPE"
82. Conover, Charles (2011-11-08) (in en). Designing for Print. John Wiley & Sons. pp. 62. ISBN 978-1-118-13088-9. "CMYK is the standard four-color model used for all full-color print jobs that will be output on an offset printing press"
83. Vermaat, Misty E.; Sebok, Susan L.; Freund, Steven M.; Campbell, Jennifer T.; Frydenberg, Mark (2015-01-01) (in en). Discovering Computers, Essentials. Cengage Learning. pp. 123. ISBN 978-1-305-53402-5. "...the 4 key (labeled with the letters g,h and i)..."
84. Bunting, Steve; Wei, William (2006-03-06) (in en). EnCase Computer Forensics: The Official EnCE: EnCase?Certified Examiner Study Guide. John Wiley & Sons. pp. 246. ISBN 978-0-7821-4435-2. "A byte also contains two 4-bit nibbles..."
85. Braden, R.. "Requirements for Internet Hosts - Communication Layers" (in en). pp. 9–10.
86. Magazines, Hearst (1982) (in en). Popular Mechanics. Hearst Magazines.
87. Schaller, Bob; Harnish, Dave (2009-09-18) (in en). The Everything Kids' Basketball Book: The all-time greats, legendary teams, today's superstars - and tips on playing like a pro. Simon and Schuster. ISBN 978-1-4405-0177-7. "Power forward Referred to as the number 4 spot"
88. Wells, J. C. (2014-09-25) (in en). Sounds Interesting: Observations on English and General Phonetics. Cambridge University Press. pp. 33. ISBN 978-1-316-12385-0. "But one confused re-spelling is fower for 'four'."
89. Guttman, Ariel; Guttman, Gail; Johnson, Kenneth (1993) (in en). Mythic Astrology: Archetypal Powers in the Horoscope. Llewellyn Worldwide. pp. 263. ISBN 978-0-87542-248-0. "Sign: Cancer, the fourth Zodiacal Sign"
90. Curtiss, Harriette A. (1996) (in en). The Key to the Universe. Health Research Books. pp. 161. ISBN 978-0-7873-1233-6. "The 4th Tarot Card is called "The Emperor.""
91. Weller, David; Lobao, Alexandre Santos; Hatton, Ellen (2004-09-20) (in en). Beginning .NET Game Programming in VB .NET. Apress. pp. 383. ISBN 978-1-4302-0724-5. "...tetraminos (the shapes used in Tetris) are all just a collection of four blocks"
92. Jozefowicz, Chris (2009-08-07) (in en). Video Games. Gareth Stevens Publishing LLLP. pp. 17. ISBN 978-1-4339-4406-2. "Tetra means "four" in Greek. Blocks in the game are made of four squares"
93. Bardes, Barbara; Shelley, Mack; Schmidt, Steffen (2008-12-16) (in en). American Government and Politics Today: The Essentials 2009 - 2010 Edition. Cengage Learning. pp. 453. ISBN 978-0-495-57170-4. "The court will not issue a writ unless at least four justices approve of it. This is called the rule of four."
94. Roberts, Gareth E. (2016-02-15) (in en). From Music to Mathematics: Exploring the Connections. JHU Press. pp. 3. ISBN 978-1-4214-1918-3. "... called common time and denoted by C, which has four beats per measure"
95. Bonds, Mark Evan (2009-01-10) (in en). Music as Thought: Listening to the Symphony in the Age of Beethoven. Princeton University Press. pp. 1. ISBN 978-1-4008-2739-8. "The number, character and sequence of movements in the symphony, moreover, did not stabilize until the 1770s when the familiar format of four movements..."
96. Frisch, Walter (2003) (in en). Brahms: The Four Symphonies. Yale University Press. ISBN 978-0-300-09965-2.
97. Brech, Lewis (2010) (in en). Storybook Advent Carols Collection Songbook. Couples Company, Inc. pp. 26. ISBN 978-1-4524-7763-3.
98. Wright, Robert J.; Ellemor-Collins, David; Tabor, Pamela D. (2011-11-04) (in en). Developing Number Knowledge: Assessment,Teaching and Intervention with 7-11 year olds. SAGE. ISBN 978-1-4462-5368-7.
99. Macauley, David (2010-09-29) (in en). Elemental Philosophy: Earth, Air, Fire, and Water as Environmental Ideas. SUNY Press. ISBN 978-1-4384-3246-5.
100. Brooks, Edward (1876) (in en). Normal Higher Arithmetic Designed for Advanced Classes in Common Schools, Normal Schools, and High Schools, Academics, Etc. Sower. pp. 227. "Every year that is divisible by four, except the Centennial years, and every Centennial year divisible by 400, is a leap year..."
101. Touche, Fred; Price, Anne (2005) (in en). Wilderness Navigation Handbook. Touche Publishing. pp. 48. ISBN 978-0-9732527-0-5. "Each of the familiar cardinal directions is equivalent to a particular true bearing:north (0°), east (90°), south (180°),and west (270°)"
102. Roeckelein, J. E. (2006-01-19) (in en). Elsevier's Dictionary of Psychological Theories. Elsevier. pp. 235. ISBN 978-0-08-046064-2. "...four substances or humors: blood, yellow bile, black bile and phlegm"
103. Medley, H. Anthony (1997) (in en). Bridge. Penguin. pp. 6. ISBN 978-0-02-861735-0. "The four playing card suits, as you probably already know, are spades, hearts, diamonds, and clubs"
104. Baker, Felicity (2017) (in en). Houses of Hogwarts: Cinematic Guide. Scholastic Incorporated. ISBN 978-1-338-12861-1. "...the four houses of Hogwarts School of Witchcraft and Wizardry: Gryffindor, Ravenclaw, Hufflepuff, and Slytherin"
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55–58