Four (number)
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Four Number

4 (four) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. It is the smallest 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 .

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.[3] (That is, for every positive integer n, where is the hyperoperation.)

A four-sided plane figure is a quadrilateral (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a tetragon. A circle divided by 4 makes right angles and four quadrants. Because of it, four (4) is the base number of the plane (mathematics).

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 Rn except when , 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. .

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 glyph

Two modern handwritten fours

Representing 1, 2 and 3 in as many lines as the number represented worked well.[] The Brahmin Indians simplified 4 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 numeral, and the Kshatrapa and Pallava evolved the numeral 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 numeral less cursive, ending up with a glyph very close to the original Brahmin cross.[15]

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

Seven-segment 4.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.[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 ?. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".[17]

In religion


Judeo-Christian symbolism





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

In physics

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",[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 (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).
  • 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 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 Rule of Logic], 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).
  • 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.

In transport

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).
  • 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 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

In music


Groups of four

See also


  1. ^ Fiore, Gregory (1993-08-01). Basic mathematics for college students: concepts and applications. HarperCollins College. p. 162. ISBN 978-0-06-042046-8. The smallest composite number is 4.
  2. ^ Prep, Kaplan Test (2017-01-03). SAT Subject Test Mathematics Level 1. Simon and Schuster. p. 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). One to Nine: The Inner Life of Numbers. W. W. Norton & Company. p. 249. ISBN 978-0-393-06863-4. 2 ... 2 is always 4
  4. ^ Weisstein, Eric W. "Tetrahedron". Retrieved .
  5. ^ Grossnickle, Foster Earl; Reckzeh, John (1968). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. p. 337. ...the smallest possible number of faces that a polyhedron may have is four
  6. ^ Grossnickle, Foster Earl; Reckzeh, John (1968). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. p. 337. ...face of the platonic solid. The simplest of these shapes is the tetrahedron...
  7. ^ Hilbert, David; Cohn-Vossen, Stephan (1999). Geometry and the Imagination. American Mathematical Soc. p. 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). Philosophical Perceptions on Logic and Order. IGI Global. p. 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). Historical Encyclopedia of Natural and Mathematical Sciences. Springer Science & Business Media. p. 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). Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs. CRC Press. p. 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", 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). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA. p. 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). 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. ^ "Seven Segment Displays (7-Segment) | Pinout, Types and Applications". Electronics Hub. 2019-04-22. Retrieved .
  17. ^ "Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check". Digital Check. 2017-02-02. Retrieved .
  18. ^ a b c Chwalkowski, Farrin (2016-12-14). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 22. ISBN 978-1-4438-5728-4. The four main pilgrimages sites are: Lumbini, Bodh Gaya, Sarnath and Kusinara....four Noble Truths of Buddhism
  19. ^ Voorst, Robert Van (2012-01-01). RELG: World. Cengage Learning. p. 108. ISBN 978-1-111-72620-1. He first observed the suffering of the world in the Four Passing Sites
  20. ^ Yun, Hsing; Xingyun (2010). The Great Realizations: A Commentary on the Eight Realizations of a Bodhisattva Sutra. Buddha's Light Publishing. p. 27. ISBN 978-1-932293-44-9. The four great elements, earth, water, fire and wind...
  21. ^ Chaudhuri, Saroj Kumar (2003). Hindu Gods and Goddesses in Japan. Vedams eBooks (P) Ltd. p. 20. ISBN 978-81-7936-009-5. The Buddhists adopted him as one of the four Devarajas or Heavenly Kings
  22. ^ Bronkhorst, Johannes (2009-12-22). Buddhist Teaching in India. Simon and Schuster. p. 66. ISBN 978-0-86171-566-4. The four right exertions are...
  23. ^ Mistry, Freny (2011-05-02). Nietzsche and Buddhism: Prolegomenon to a Comparative Study. Walter de Gruyter. p. 69. ISBN 978-3-11-083724-7. these four bases of psychic power
  24. ^ Arbel, Keren (2017-03-16). Early Buddhist Meditation: The Four Jhanas as the Actualization of Insight. Taylor & Francis. p. 1. ISBN 978-1-317-38399-4. This book is about the four jhanas
  25. ^ Jayatilleke, K. N. (2013-10-16). Early Buddhist Theory of Knowledge. Routledge. ISBN 978-1-134-54294-9. ...the states of the four arupajhanas.
  26. ^ Gorkom, Nina van. The Perfections Leading to Enlightenment. ?. p. 171. ISBN 978-5-88139-786-9. There are four of them: loving-kindness, metta, compassion, karuna, sympathetic joy, mudita and equanimity, upekkha.
  27. ^ Rinpoche, Khenchen Konchog Gyaltshen; Milarepa; Sumgon, Jigten (2013-10-08). 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)
  28. ^ Fahlbusch, Erwin; Bromiley, Geoffrey William; Lochman, Jan Milic; Mbiti, John; Pelikan, Jaroslav (2008-02-14). The Encyclodedia of Christianity, Vol. 5. Wm. B. Eerdmans Publishing. p. 823. ISBN 978-0-8028-2417-2.
  29. ^ Stevenson, Kenneth; Glerup, Michael (2014-03-19). 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
  30. ^ Butnick, Stephanie; Leibovitz, Liel; Oppenheimer, Mark (2019-10-01). The Newish Jewish Encyclopedia: From Abraham to Zabar's and Everything in Between. Artisan Books. ISBN 978-1-57965-893-9. like Sarah, Rachel, Rebecca, and Leah, the foremothers of Judaism
  31. ^ Kaplan, Aryeh (1990). Innerspace: Introduction to Kabbalah, Meditation and Prophecy. Moznaim. p. 109. well as to the palm ( lulav ) , myrtle ( hadas ) , willow ( aravah ) and citron ( etrog ) , the four species of plants
  32. ^ a b c Dennis, Geoffrey W. (2007). The Encyclopedia of Jewish Myth, Magic and Mysticism. Llewellyn Worldwide. p. 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.
  33. ^ "Four Expressions of Redemption to be said on the Jewish holiday of Passover - Google Search". p. 46. Retrieved . There are four expressions of redemption in the Torah
  34. ^ Charles, Templeton (1973). Jesus: the four Gospels, Matthew, Mark, Luke, and John, combined in one narrative and rendered in modern English. Simon and Schuster. ISBN 9780671217150.
  35. ^ Wagner, Richard; Helyer, Larry R. (2011-01-31). The Book of Revelation For Dummies. John Wiley & Sons. p. 308. ISBN 978-1-118-05086-6. The four horsemen of the Apocalypse are one of the most familiar images of Revelation
  36. ^ Turfe, Tallal Alie (2013-07-19). Children of Abraham: United We Prevail, Divided We Fail. iUniverse. p. 91. ISBN 978-1-4759-9047-8. The four holy cities of Judaism are Jerusalem, Hebron, Safed, and Tiberius.
  37. ^ Frawley, David (2014-10-07). Vedic Yoga: The Path of the Rishi. Lotus Press. ISBN 978-0-940676-25-1. There are four Vedas
  38. ^ Fritz, Stephen Martin (2019-05-14). Our Human Herds: The Theory of Dual Morality (Second Edition, Unabridged). Dog Ear Publishing. p. 491. ISBN 978-1-4575-6755-1. that these four proper aims and objects
  39. ^ Maanas - Individual and Society. Rapid Publications. ISBN 978-1-937192-06-8. The Four Stages of Life
  40. ^ Chwalkowski, Farrin (2016-12-14). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 23. ISBN 978-1-4438-5728-4. The four primary castes or strata of society:...
  41. ^ Kulendiren, Pon (2012-10-11). Hinduism a Scientific Religion: & Some Temples in Sri Lanka. iUniverse. p. 32. ISBN 978-1-4759-3675-9.
  42. ^ Jansen, Eva Rudy (1993). The Book of Hindu Imagery: Gods, Manifestations and Their Meaning. Binkey Kok Publications. p. 87. ISBN 978-90-74597-07-4. Brahma has four faces,...
  43. ^ "Definition of yuga |". Retrieved .
  44. ^ Çakmak, Cenap (2017-05-18). Islam: A Worldwide Encyclopedia [4 volumes]. ABC-CLIO. p. 397. ISBN 978-1-61069-217-5. ...Eid al-Adha (Feast of Sacrifice) lasts four days ...
  45. ^ Leonard, Timothy; Willis, Peter (2008-06-11). Pedagogies of the Imagination: Mythopoetic Curriculum in Educational Practice. Springer Science & Business Media. p. 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,...
  46. ^ Chwalkowski, Farrin (2016-12-14). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 23. ISBN 978-1-4438-5728-4. According to Islam, the Four Arch Angels are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael).
  47. ^ Ph.D, Assad Nimer Busool (2010-12-28). The Wise Qur'an: These are the Verses of the Wise Book: These are the verses of the Wise Book. Xlibris Corporation. p. 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...
  48. ^ Shabazz, Hassan (2020-01-06). Al Islaam, and the Transformation of Society. p. 15. ISBN 978-1-7948-3337-1. There are four books in Islam: Torah, Zaboor, Injeel and Holy Qur'an...
  49. ^ BUKHARI, MUOHAMMAD BEN ISMAIL AL (2007-01-01). THE CORRECT TRADITIONS OF AL'BUKHARI 1-4 VOL 3 ? 1/4 [?/?] ?3. Dar Al Kotob Al Ilmiyah ?. p. 840. For those who take an oath for abstention from their wives, awaiting for four months is ordained;
  50. ^ Ahmad, Yusuf Al-Hajj. The Book Of Nikkah: Encyclopaedia of Islamic Law. Darussalam Publishers. ...for four months and ten days.
  51. ^ Mawdudi, Sayyid Abul A'la (2016-12-15). Towards Understanding the Qur'an: English Only Edition. Kube Publishing Ltd. p. 59. ISBN 978-0-86037-613-2. Then take four birds, ...
  52. ^ Maudoodi, Syed Abul ?Ala (2000). S?rah al-A?ar?f to S?rah bani Isr?el. Islamic Publications. p. 177. The respite of four months...
  53. ^ Barazangi, Nimat Hafez (2016-03-09). Woman's Identity and Rethinking the Hadith. Routledge. p. 138. ISBN 978-1-134-77065-6. And those who launch a charge against chaste women and do not produce four witnesses...
  54. ^ SK, Lim. Origins of Chinese Auspicious Symbols. Asiapac Books Pte Ltd. p. 16. ISBN 978-981-317-026-1. Taoism later incorporated the four symbols into its immortality system...
  55. ^ Terry, Milton Spenser (1883). Biblical Hermenutics: A Treatise on the Interpretation of the Old and New Testaments. Phillips & Hunt. p. 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
  56. ^ Bulletin - State Department of Education. Department of Education. 1955. p. 151. Four was a sacred number of Zia
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  58. ^ Maberry, Jonathan; Kramer, David F. (2007). The Cryptopedia: A Dictionary of the Weird, Strange & Downright Bizarre. Citadel Press. p. 211. ISBN 978-0-8065-2819-9. Svetovid is portrayed as having four heads ...
  59. ^ "FDR, "The Four Freedoms," Speech Text". Voices of Democracy. Retrieved .
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  62. ^ Petsko, Gregory A.; Ringe, Dagmar (2004). Protein Structure and Function. New Science Press. p. 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.
  63. ^ Yaqoob, Tahir (2011). Exoplanets and Alien Solar Systems. New Earth Labs. p. 12. ISBN 978-0-9741689-2-0. The four inner planets (known as terrestrial, or rocky planets
  64. ^ Encrenaz, Therese; Bibring, Jean-Pierre; Blanc, M.; Barucci, Maria-Antonietta; Roques, Francoise; Zarka, Philippe (2004-01-26). The Solar System. Springer Science & Business Media. p. 283. ISBN 978-3-540-00241-3. ...the gas giants (Jupiter and Saturn), and the icy giants (Uranus and Neptune)
  65. ^ Pidwirny, Michael (2020-05-07). Chapter 3: Matter, Energy and the Universe: Single chapter from the eBook Understanding Physical Geography. Our Planet Earth Publishing. p. 10. including the four large Galilean moons that are easily visible from a hobby telescope
  66. ^ Pugh, Philip (2011-11-02). Observing the Messier Objects with a Small Telescope: In the Footsteps of a Great Observer. Springer Science & Business Media. p. 41. ISBN 978-0-387-85357-4. M4 is a globular star cluster near Antares in Scorpius.
  67. ^ Bok, Bart Jan; Bok, Priscilla Fairfield (1981). The Milky Way. Harvard University Press. p. 66. ISBN 978-0-674-57503-5. IV , subgiants
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