4 Explained

4 should not be confused with .

Number:4
Numeral:quaternary
Divisor:1, 2, 4
Greek Prefix:tetra-
Latin Prefix:quadri-/quadr-
Lang1:Armenian
Lang1 Symbol:Դ
Lang2:Arabic, Kurdish
Lang3:Persian, Sindhi
Lang4:Shahmukhi, Urdu
Lang5:Ge'ez
Lang6:Bengali, Assamese
Lang7:Chinese numeral
Lang7 Symbol:四,亖,肆
Lang9:Devanagari
Lang10:Telugu
Lang11:Malayalam
Lang12:Tamil
Lang13:Hebrew
Lang14:Khmer
Lang15:Thai
Lang16:Kannada
Lang17:Burmese
Lang18:Babylonian numeral
Lang19:Egyptian hieroglyph, Chinese counting rod
Lang20:Maya numerals
Lang20 Symbol:••••
Lang21:Morse code

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

Evolution of the Hindu-Arabic digit

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

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

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: .[2]

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

Mathematics

Four is the smallest composite number, its proper divisors being and .[4] Four is the sum and product of two with itself:

2+2=4=2 x 2

, the only non-zero number

b

such that

a+a=b=a x a

, which also makes four the smallest and only even squared prime number

22

and hence the first squared prime of the form

p2

, where

p

is a prime. Four, as the first composite number, has a prime aliquot sum of 3; and as such it is part of the first aliquot sequence with a single composite member, expressly (4, 3, 1, 0). Four is therefore also the smallest semiprime, where the fourth such distinct semiprime is the product between the smallest pair of twin primes (3, 5); meanwhile the fourth prime and composite are (7, 9), where twice four lies between these. Also,

2+2=2 x 2=22=2\uparrow\uparrow2=2\uparrow\uparrow\uparrow2= ...=4

, and so forth, for any number of up arrows.[5] By consequence, four is the only square one more than a prime number, specifically three.

42

), equivalently the fourth power of two (

24

), is sixteen; the only number that has

ab=ba

as a form of factorization.

Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (), multiplication (×), and division (÷); and four basic number systems, the real numbers

R

, rational numbers

Q

, integers

Z

, and natural numbers

N

.

Four is the smallest non-unitary tetrahedral number.[6]

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

4x=y2-z2

. 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).[7]

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

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

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

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

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

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

The third dimension holds a total of four Coxeter groups that generate convex uniform polyhedra: the tetrahedral group, the octahedral group, the icosahedral group, and a dihedral group (of orders 24, 48, 120, and 4

n

, respectively). There are also four general Coxeter groups of generalized uniform prisms, where two are hosoderal and dihedral groups that form spherical tilings, with another two general prismatic and antiprismatic groups that represent truncated hosohedra (or simply, prisms) and snub antiprisms, respectively.

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

n

-polytopes, all within the infinite family of regular

n

-simplexes,

n

-hypercubes, and

n

-orthoplexes.

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

n

>

4

are the most symmetric, which mainly originate from stellations of regular

n

-polytopes.

Altogether, sixteen (or 16 = 42) regular convex and star polychora are generated from symmetries of four (4) Coxeter Weyl groups and point groups in the fourth dimension: the

A4

simplex,

B4

hypercube,

F4

icositetrachoric, and

H4

hexacosichoric groups; with the

D4

demihypercube group generating two alternative constructions. There are also sixty-four (or 64 = 43) four-dimensional Bravais lattices, alongside sixty-four uniform polychora in the fourth dimension based on the same

A4

,

B4

,

F4

and

H4

Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. Two infinite families of duoprisms and antiprismatic prisms exist in the fourth dimension.

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

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on

Rn

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.[17] A alternating groups are not simple for values

n

4

.

There are four Hopf fibrations of hyperspheres:

\beginS^0 & \hookrightarrow S^1 \to S^1, \\S^1 & \hookrightarrow S^3 \to S^2, \\S^3 & \hookrightarrow S^7 \to S^4, \\S^7 & \hookrightarrow S^\to S^8. \\\end

They are defined as locally trivial fibrations that map

f:S2n-1Sn

for values of

n=2,4,8

(aside from the trivial fibration mapping between two points and a circle).[18]

The unit (1) is the fourth distinct entry in the continued fraction for pi (

\pi

), by order of appearances of entries.[19]

Further extensions of the real numbers under Hurwitz's theorem states that there are four normed division algebras: the real numbers

R

, the complex numbers

C

, the quaternions

H

, and the octonions

O

. Under Cayley–Dickson constructions, the sedenions

S

constitute a further fourth extension over

R

. The real numbers are ordered, commutative and associative algebras, as well as alternative algebras with power-associativity. The complex numbers

C

share all four multiplicative algebraic properties of the reals

R

, without being ordered. The quaternions loose a further commutative algebraic property, while holding associative, alternative, and power-associative properties. The octonions are alternative and power-associative, while the sedenions are only power-associative. The sedenions and all further extensions of these four normed division algebras are solely power-associative with non-trivial zero divisors, which makes them non-division algebras.

R

has a vector space of dimension 1, while

C

,

H

,

O

and

S

work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively.

List of basic calculations

Division12345678910111213141516
4 ÷ x421.10.80.0.0.50.0.40.0.0.0.0.20.25
x ÷ 40.250.50.7511.251.51.7522.252.52.7533.253.53.754
Exponentiation12345678910111213141516
44166425610244096163846553626214410485764194304167772166710886426843545610737418244294967296
x11681256625129624014096656110000146412073628561384165062565536

In religion

Buddhism

Biblical

Judaism

Hinduism

Islam

Taoism

Other

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.[61]

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".[62]

In computing

In science

In astronomy

In biology

In chemistry

In physics

In logic and philosophy

In technology

In transport

In sports

In other fields

See also: 4 (disambiguation).

In music

In fiction

Other groups of four

spring, summer, autumn, winter.

sanguine, choleric, melancholic, phlegmatic.

blood, yellow bile, black bile, phlegm.[110]

justice, prudence, temperance, fortitude.

See also

References

External links

Notes and References

  1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64
  2. Web site: 2019-04-22. Seven Segment Displays (7-Segment) Pinout, Types and Applications. 2020-07-28. Electronics Hub. en-US. 28 July 2020. https://web.archive.org/web/20200728014108/https://www.electronicshub.org/seven-segment-displays/. live.
  3. News: 2017-02-02. Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check. en-US. Digital Check. 2020-07-28. 3 August 2020. https://web.archive.org/web/20200803161254/https://www.digitalcheck.com/battle-micr-fonts-better-e13b-cmc7/. live.
  4. Book: Fiore, Gregory . Basic mathematics for college students: concepts and applications. 1993-08-01. HarperCollins College . 978-0-06-042046-8 . 162. en. The smallest composite number is 4..
  5. Book: Hodges, Andrew. One to Nine: The Inner Life of Numbers. 2008-05-17. W. W. Norton & Company. 978-0-393-06863-4. 249. en. 2 ↑↑ ... ↑↑ 2 is always 4.
  6. 2024-02-10 .
  7. Book: Kaplan Test Prep. SAT Subject Test Mathematics Level 1. 2017-01-03. Simon and Schuster. 978-1-5062-0922-7. 289. en. An integer is divisible by 4 if the last two digits form a multiple of 4..
  8. Book: Peterson, Ivars. Mathematical Treks: From Surreal Numbers to Magic Circles. 2002. MAA. 978-0-88385-537-9. 95. en. 7 is an example of an integer that can't be written as the sum of three squares..
  9. Book: Bajnok, Béla. An Invitation to Abstract Mathematics. 2013-05-13. Springer Science & Business Media. 978-1-4614-6636-9. en. There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher..
  10. Book: Bunch, Bryan . The Kingdom of Infinite Number . New York . W. H. Freeman & Company . 2000 . 48.
  11. Book: Ben-Menahem, Ari. Historical Encyclopedia of Natural and Mathematical Sciences. 2009-03-06. Springer Science & Business Media. 978-3-540-68831-0. 2147. en. (i.e. That there are maps for which three colors are not sufficient).
  12. Book: Molitierno, Jason J.. Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs. 2016-04-19. CRC Press. 978-1-4398-6339-8. 197. en. ... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4..
  13. Web site: Weisstein. Eric W.. Tetrahedron. 2020-07-28. mathworld.wolfram.com. en. 20 August 2020. https://web.archive.org/web/20200820085424/https://mathworld.wolfram.com/Tetrahedron.html. live.
  14. Book: Grossnickle. Foster Earl. Discovering Meanings in Elementary School Mathematics. Reckzeh. John. 1968. Holt, Rinehart and Winston. 337. 9780030676451. en. ...the smallest possible number of faces that a polyhedron may have is four.
  15. Book: Grossnickle. Foster Earl. Discovering Meanings in Elementary School Mathematics. Reckzeh. John. 1968. Holt, Rinehart and Winston. 337. 9780030676451. en. ...face of the platonic solid. The simplest of these shapes is the tetrahedron....
  16. Book: Hilbert. David. Geometry and the Imagination. Cohn-Vossen . Stephan . 1999. American Mathematical Soc.. 978-0-8218-1998-2. 143. en. ...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs....
  17. Book: Horne, Jeremy. Philosophical Perceptions on Logic and Order. 2017-05-19. IGI Global. 978-1-5225-2444-1. 299. en. The Klein four-group is the smallest noncyclic group,.... 31 October 2022. 31 October 2022. https://web.archive.org/web/20221031005437/https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299. live.
  18. Book: Shokurov, A.V. . Michiel Hazewinkel . Michiel Hazewinkel . Hopf fibration . Encyclopedia of Mathematics . . Helsinki . https://encyclopediaofmath.org/wiki/Hopf_fibration . 2002 . 1402006098 . 1013220521 . 2023-04-30 . 1 May 2023 . https://web.archive.org/web/20230501005558/https://encyclopediaofmath.org/wiki/Hopf_fibration . live .
  19. 2024-06-16 .
  20. Book: Chwalkowski, Farrin. Symbols in Arts, Religion and Culture: The Soul of Nature. 2016-12-14. Cambridge Scholars Publishing. 978-1-4438-5728-4. 22. en. The four main pilgrimages sites are: Lumbini, Bodh Gaya, Sarnath and Kusinara....four Noble Truths of Buddhism.
  21. Book: Van Voorst, Robert. RELG: World. 2012-01-01. Cengage Learning. 978-1-111-72620-1. 108. en. He first observed the suffering of the world in the Four Passing Sites.
  22. Book: Yun. Hsing. The Great Realizations: A Commentary on the Eight Realizations of a Bodhisattva Sutra. Xingyun. 2010. Buddha's Light Publishing. 978-1-932293-44-9. 27. en. The four great elements, earth, water, fire and wind....
  23. Book: Chaudhuri, Saroj Kumar. Hindu Gods and Goddesses in Japan. 2003. Vedams eBooks (P) Ltd. 978-81-7936-009-5. 20. en. The Buddhists adopted him as one of the four Devarajas or Heavenly Kings.
  24. Book: Bronkhorst, Johannes. Buddhist Teaching in India. 2009-12-22. Simon and Schuster. 978-0-86171-566-4. 66. en. The four right exertions are....
  25. Book: Mistry, Freny. Nietzsche and Buddhism: Prolegomenon to a Comparative Study. 2011-05-02. Walter de Gruyter. 978-3-11-083724-7. 69. en. these four bases of psychic power.
  26. Book: Arbel, Keren. Early Buddhist Meditation: The Four Jhanas as the Actualization of Insight. 2017-03-16. Taylor & Francis. 978-1-317-38399-4. 1. en. This book is about the four jhanas.
  27. Book: Jayatilleke, K. N.. Early Buddhist Theory of Knowledge. 2013-10-16. Routledge. 978-1-134-54294-9. en. ...the states of the four arupajhanas..
  28. Book: van Gorkom, Nina. The Perfections Leading to Enlightenment. Рипол Классик. 978-5-88139-786-9. 171. en. There are four of them: loving-kindness, metta, compassion, karuna, sympathetic joy, mudita and equanimity, upekkha..
  29. Book: Rinpoche. Khenchen Konchog Gyaltshen. Opening the Treasure of the Profound: Teachings on the Songs of Jigten Sumgon and Milarepa. Milarepa. Sumgon. Jigten. 2013-10-08. Shambhala Publications. 978-0-8348-2896-4. en. ...four types of shravaka (stream enterer, oncereturner, nonreturner, and arhat).
  30. Book: Fahlbusch. Erwin. The Encyclodedia of Christianity, Vol. 5. Bromiley. Geoffrey William. Lochman. Jan Milic. Mbiti. John. Pelikan. Jaroslav. 2008-02-14. Wm. B. Eerdmans Publishing. 978-0-8028-2417-2. 823. en.
  31. Book: Stevenson. Kenneth. Ezekiel, Daniel. Glerup. Michael. 2014-03-19. InterVarsity Press. 978-0-8308-9738-4. xlv. en. We have already mentioned the four living creatures—the man, the lion, the ox and the eagle.
  32. Book: Templeton, Charles. Jesus: the four Gospels, Matthew, Mark, Luke, and John, combined in one narrative and rendered in modern English. 1973. Simon and Schuster. 9780671217150. en.
  33. Book: Wagner. Richard. The Book of Revelation For Dummies. Helyer. Larry R.. 2011-01-31. John Wiley & Sons. 978-1-118-05086-6. 308. en. The four horsemen of the Apocalypse are one of the most familiar images of Revelation.
  34. Book: Jordan. James B.. The Sociology of the Church: Essays in Reconstruction. Wipf and Stock Publishers. 9781579102487. reprint. Eugene, Oregon. 1999. 86. Sociology: A Biblico-Historical Approach. 18 May 1999 . In a generally ignored but all-important paragraph of Genesis 2, we are told how the world was organized when it was created [...]. In short, the world was organized in terms of a primordial duality between the central sanctuary of Eden, and the outlying world watered by four rivers extending to the four corners of the world.. James B. Jordan. 2024-08-09.
  35. Book: Kaplan, Aryeh. Innerspace: Introduction to Kabbalah, Meditation and Prophecy. 1990. Moznaim. 109. 9780940118560. en. ...as well as to the palm (lulav), myrtle (hadas), willow (aravah) and citron (etrog), the four species of plants.
  36. Book: Butnick. Stephanie. The Newish Jewish Encyclopedia: From Abraham to Zabar's and Everything in Between. Leibovitz. Liel. Oppenheimer. Mark. 2019-10-01. Artisan Books. 978-1-57965-893-9. en. ...be like Sarah, Rachel, Rebecca, and Leah, the foremothers of Judaism.
  37. Book: Dennis, Geoffrey W.. The Encyclopedia of Jewish Myth, Magic and Mysticism. 2007. Llewellyn Worldwide. 978-0-7387-0905-5. 188. en. The Passover Seder is particularly structured around fours: the Four Questions, the Four Sons, and four cups of wine..
  38. Book: Turfe, Tallal Alie. Children of Abraham: United We Prevail, Divided We Fail. 2013-07-19. iUniverse. 978-1-4759-9047-8. 91. en. The four holy cities of Judaism are Jerusalem, Hebron, Safed, and Tiberius..
  39. Book: Frawley, David. Vedic Yoga: The Path of the Rishi. 2014-10-07. Lotus Press. 978-0-940676-25-1. en. There are four Vedas.
  40. Book: Fritz, Stephen Martin. Our Human Herds: The Theory of Dual Morality (Second Edition, Unabridged). 2019-05-14. Dog Ear Publishing. 978-1-4575-6755-1. 491. en. that these four proper aims and objects.
  41. Book: Maanas - Individual and Society. Rapid Publications. 978-1-937192-06-8. en. The Four Stages of Life.
  42. Book: Chwalkowski, Farrin. Symbols in Arts, Religion and Culture: The Soul of Nature. 2016-12-14. Cambridge Scholars Publishing. 978-1-4438-5728-4. 23. en. The four primary castes or strata of society:....
  43. Book: Kulendiren, Pon. Hinduism a Scientific Religion: & Some Temples in Sri Lanka. 2012-10-11. iUniverse. 978-1-4759-3675-9. 32. en.
  44. Book: Jansen, Eva Rudy. The Book of Hindu Imagery: Gods, Manifestations and Their Meaning. 1993. Binkey Kok Publications. 978-90-74597-07-4. 87. en. Brahma has four faces,....
  45. Web site: Definition of yuga. Dictionary.com. 2020-07-28. en. 28 July 2020. https://web.archive.org/web/20200728214059/https://www.dictionary.com/browse/yuga. live.
  46. Book: Çakmak, Cenap. Islam: A Worldwide Encyclopedia [4 volumes]]. 2017-05-18. ABC-CLIO. 978-1-61069-217-5. 397. en. ...Eid al-Adha (Feast of Sacrifice) lasts four days ....
  47. Book: Leonard. Timothy. Pedagogies of the Imagination: Mythopoetic Curriculum in Educational Practice. Willis. Peter. 2008-06-11. Springer Science & Business Media. 978-1-4020-8350-1. 144. en. ... four Rightly Guided Caliphs, Abu-Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib,....
  48. Book: Chwalkowski, Farrin. Symbols in Arts, Religion and Culture: The Soul of Nature. 2016-12-14. Cambridge Scholars Publishing. 978-1-4438-5728-4. 23. en. According to Islam, the Four Arch Angels are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)..
  49. Book: Busool, Assad Nimer. The Wise Qur'an: These are the Verses of the Wise Book: These are the verses of the Wise Book. 2010-12-28. Xlibris Corporation. 978-1-4535-2526-5. 50. en. 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....
  50. Book: Shabazz, Hassan. Al Islaam, and the Transformation of Society. 2020-01-06. Lulu.com. 978-1-7948-3337-1. 15. en. There are four books in Islam: Torah, Zaboor, Injeel and Holy Qur'an....
  51. Book: Bukhari, Muohammad Ben Ismail Al. THE CORRECT TRADITIONS OF AL'BUKHARI 1-4 VOL 3: صحيح البخاري 1/4 [عربي/انكليزي] ج3]. 2007-01-01. Dar Al Kotob Al Ilmiyah دار الكتب العلمية. 840. en. For those who take an oath for abstention from their wives, awaiting for four months is ordained;.
  52. Book: Ahmad, Yusuf Al-Hajj. The Book Of Nikkah: Encyclopaedia of Islamic Law. Darussalam Publishers. en. ...for four months and ten days..
  53. Book: Mawdudi, Sayyid Abul A'la. Towards Understanding the Qur'an: English Only Edition. 2016-12-15. Kube Publishing Ltd. 978-0-86037-613-2. 59. en. Then take four birds, ....
  54. Book: Maudoodi, Syed Abul ʻAla. Sūrah al-Aʻarāf to Sūrah bani Isrāel. 2000. Islamic Publications. 177. en. The respite of four months....
  55. Book: Barazangi, Nimat Hafez. Woman's Identity and Rethinking the Hadith. 2016-03-09. Routledge. 978-1-134-77065-6. 138. en. And those who launch a charge against chaste women and do not produce four witnesses....
  56. Book: SK, Lim. Origins of Chinese Auspicious Symbols. Asiapac Books Pte Ltd. 978-981-317-026-1. 16. en. Taoism later incorporated the four symbols into its immortality system....
  57. Book: Terry, Milton Spenser. Biblical Hermeneutics: A Treatise on the Interpretation of the Old and New Testaments. 1883. Phillips & Hunt. 382. en. 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.
  58. Book: Bulletin - State Department of Education. 1955. Department of Education. 151. en. Four was a sacred number of Zia.
  59. Book: Lachenmeyer, Nathaniel. 13: The Story of the World's Most Notorious Superstition. 2005. Penguin Group (USA) Incorporated. 978-0-452-28496-8. 187. en. In Chinese, Japanese, and Korean, the word for four is, unfortunately, an exact homonym for death.
  60. Book: Maberry. Jonathan. The Cryptopedia: A Dictionary of the Weird, Strange & Downright Bizarre. Kramer. David F.. 2007. Citadel Press. 978-0-8065-2819-9. 211. en. Svetovid is portrayed as having four heads ....
  61. Web site: FDR, 'The Four Freedoms,' Speech Text. 2020-07-30. Voices of Democracy. en-US. 1 August 2020. https://web.archive.org/web/20200801005700/http://voicesofdemocracy.umd.edu/fdr-the-four-freedoms-speech-text/. live.
  62. News: Yao Wenyuan. The Economist. 2020-07-30. 0013-0613. 1 May 2018. https://web.archive.org/web/20180501231531/https://www.economist.com/node/5381892. live.
  63. Book: MCS-40 User's Manual For Logic Designers . The Functions Of A Computer: Instruction Register And Decoder . . Howard A. . Raphael . November 1974 . Santa Clara, California, USA . viii . http://bitsavers.trailing-edge.com/components/intel/MCS40/MCS-40_Users_Manual_Nov74.pdf . 2020-03-03 . live . https://web.archive.org/web/20200303024244/http://bitsavers.trailing-edge.com/components/intel/MCS40/MCS-40_Users_Manual_Nov74.pdf . 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. [...].
  64. Book: Petsko. Gregory A.. Protein Structure and Function. Ringe. Dagmar. 2004. New Science Press. 978-0-87893-663-2. 40. en. Oligomers containing two, three, four, five, six or even more subunits are known as dimers, trimers, tetramers, pentamers, hexamers, and so on..
  65. Book: Yaqoob, Tahir. Exoplanets and Alien Solar Systems. 2011. New Earth Labs. 978-0-9741689-2-0. 12. en. The four inner planets (known as terrestrial, or rocky planets.
  66. Book: Encrenaz. Therese. Thérèse Encrenaz. The Solar System. Bibring. Jean-Pierre. Blanc. M.. Barucci. Maria-Antonietta. Roques. Francoise. Zarka. Philippe. 2004-01-26. Springer Science & Business Media. 978-3-540-00241-3. 283. en. ...the gas giants (Jupiter and Saturn), and the icy giants (Uranus and Neptune). 4 November 2020. 12 March 2022. https://web.archive.org/web/20220312050454/https://books.google.com/books?id=Je61Y7UbqWgC&q=Four+giant+gas/ice+planets+in+the+Solar+system:+Jupiter,+Saturn,+Uranus,+and+Neptune&pg=PA283. live.
  67. Book: Pidwirny, Michael. Chapter 3: Matter, Energy and the Universe: Single chapter from the eBook Understanding Physical Geography. 2020-05-07. Our Planet Earth Publishing. 10. en. including the four large Galilean moons that are easily visible from a hobby telescope.
  68. Book: Pugh, Philip. Observing the Messier Objects with a Small Telescope: In the Footsteps of a Great Observer. 2011-11-02. Springer Science & Business Media. 978-0-387-85357-4. 41. en. M4 is a globular star cluster near Antares in Scorpius..
  69. Book: Bok. Bart Jan. The Milky Way. Bok. Priscilla Fairfield. 1981. Harvard University Press. 978-0-674-57503-5. 66. en. IV, subgiants.
  70. Book: Encyclopedia of Cell Biology. 2015-08-07. Academic Press. 978-0-12-394796-3. 25. en.
  71. Book: Chien. Shu. An Introductory Text to Bioengineering. Chen. Peter C. Y.. Fung. Yuan-cheng. 2008. World Scientific. 978-981-270-793-2. 54. en. The mammalian heart consists of four chambers,....
  72. Book: Creation Research Society Textbook Committee. Biology: a search for order in complexity. 1970. Zondervan Pub. House. 978-0-310-29490-0. 209. en. Except for the flies, mosquitoes, and some others, insects with wings have four wings..
  73. Book: Pittenger, Dennis. California Master Gardener Handbook, 2nd Edition. 2014-12-15. UCANR Publications. 978-1-60107-857-5. 180. en. metamorphosis is marked by four distinct stages.
  74. Book: Darpan, Pratiyogita. Pratiyogita Darpan. 2008. Pratiyogita Darpan. 85. en. In the 'ABO' system, all blood belongs one of four major groups — A, B, AB or O.
  75. Book: Daniels. Patricia. Body: The Complete Human : how it Grows, how it Works, and how to Keep it Healthy and Strong. Stein. Lisa. 2009. National Geographic Books. 978-1-4262-0449-4. 94. en. Four canines for tearing + Eight premolars for crushing +Twelve molars (including four wisdom teeth).
  76. Book: Woodward. Thompson Elwyn. Feeding Dairy Cows. Nystrom. Amer Benjamin. 1930. U.S. Department of Agriculture. 4. en. The cow's stomach is divided into four compartments..
  77. Book: Lucas, Jerry. Great unsolved mysteries of science. 1993. F & W Pubns Inc. 978-1-55870-291-2. 168. en. Of course, carbon is not the only chemical element with a valence of +4 or -4.
  78. Book: Walsh, Kenneth A.. Beryllium Chemistry and Processing. 2009-01-01. ASM International. 978-0-87170-721-5. 93. en. Beryllium has an atomic number of four.
  79. Book: Ebeling. Werner. Quantum Statistics of Dense Gases and Nonideal Plasmas. Fortov. Vladimir E.. Filinov. Vladimir. 2017-11-27. Springer. 978-3-319-66637-2. 39. en. Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas..
  80. Book: Petkov, Vesselin. Relativity and the Nature of Spacetime. 2009-06-23. Springer Science & Business Media. 978-3-642-01962-3. 124. en. should be regarded as a four-dimensional world.
  81. Book: Giordano, Nicholas. College Physics: Reasoning and Relationships. 2009-02-13. Cengage Learning. 978-0-534-42471-8. 1073. en. We have referred to the four fundamental forces in nature,....
  82. Book: Alon. Noga. The Probabilistic Method. Spencer. Joel H.. 2011-09-20. John Wiley & Sons. 978-1-118-21044-4. 6.1. en. The Four Functions Theorem of Ahlswede Daykin.
  83. Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.
  84. Book: Hennig, Boris. Aristotle's Four Causes. 2018-12-05. Peter Lang. 978-1-4331-5929-9. en. This book examines Aristotle's four causes (material, formal, efficient, and final).
  85. Book: Wilkinson, Amy. The Creator's Code: The Six Essential Skills of Extraordinary Entrepreneurs. 2015-02-17. Simon and Schuster. 978-1-4516-6609-0. 79. en. The OODA loop consists of four steps..
  86. Web site: Howard. Brian Clark. Abdelrahman. Amina Lake. Good Housekeeping Institute. 2020-02-26. You Might Be Recycling Wrong — Here's Everything You Need to Know About Recycling Symbols. 2020-07-28. Good Housekeeping. en-US. Plastic Recycling Symbol #4: LDPE. 13 March 2015. https://web.archive.org/web/20150313073239/http://www.goodhousekeeping.com/home/green-living/recycling-symbols-plastics-460321. live.
  87. Book: Conover, Charles. Designing for Print. 2011-11-08. John Wiley & Sons. 978-1-118-13088-9. 62. en. CMYK is the standard four-color model used for all full-color print jobs that will be output on an offset printing press.
  88. Book: Vermaat. Misty E.. Discovering Computers, Essentials. Sebok. Susan L.. Freund. Steven M.. Campbell. Jennifer T.. Frydenberg. Mark. 2015-01-01. Cengage Learning. 978-1-305-53402-5. 123. en. ...the 4 key (labeled with the letters g, h and i)....
  89. Book: Bunting. Steve. EnCase Computer Forensics: The Official EnCE: EnCase?Certified Examiner Study Guide. Wei. William. 2006-03-06. John Wiley & Sons. 978-0-7821-4435-2. 246. en. A byte also contains two 4-bit nibbles....
  90. Braden. R.. R. Braden. Requirements for Internet Hosts - Communication Layers. 2020-07-28. tools.ietf.org. 1989. 9–10. 10.17487/RFC1122. en. free. 28 July 2020. https://web.archive.org/web/20200728124449/https://tools.ietf.org/html/rfc1122.html. live.
  91. Assenza. Tony. Audi Quattro: Germany's 4x4 Cruise Missile. Popular Mechanics. June 1982. Hearst Magazines. en.
  92. Book: Schaller. Bob. The Everything Kids' Basketball Book: The all-time greats, legendary teams, today's superstars - and tips on playing like a pro. Harnish. Dave. 2009-09-18. Simon and Schuster. 978-1-4405-0177-7. en. Power forward Referred to as the number 4 spot.
  93. Web site: Definition of FOUR-LETTER WORD. 2020-07-28. merriam-webster.com. en. 22 August 2016. https://web.archive.org/web/20160822110121/http://www.merriam-webster.com/dictionary/four-letter+word. live.
  94. Book: Wells, J. C.. Sounds Interesting: Observations on English and General Phonetics. 2014-09-25. Cambridge University Press. 978-1-316-12385-0. 33. en. But one confused re-spelling is fower for 'four'..
  95. Book: Guttman. Ariel. Mythic Astrology: Archetypal Powers in the Horoscope. Guttman. Gail. Johnson. Kenneth. 1993. Llewellyn Worldwide. 978-0-87542-248-0. 263. en. Sign: Cancer, the fourth Zodiacal Sign.
  96. Book: Curtiss, Harriette A.. The Key to the Universe. 1996. Health Research Books. 978-0-7873-1233-6. 161. en. The 4th Tarot Card is called "The Emperor.".
  97. Book: Weller. David. Beginning .NET Game Programming in VB .NET. Lobao. Alexandre Santos. Hatton. Ellen. 2004-09-20. Apress. 978-1-4302-0724-5. 383. en. ...tetraminos (the shapes used in Tetris) are all just a collection of four blocks.
  98. Book: Bardes. Barbara. American Government and Politics Today: The Essentials 2009 - 2010 Edition. Shelley. Mack. Schmidt. Steffen. 2008-12-16. Cengage Learning. 978-0-495-57170-4. 453. en. The court will not issue a writ unless at least four justices approve of it. This is called the rule of four..
  99. Web site: 2011-02-17. Movie Projector: 'I Am Number Four' to be No. 1 at holiday weekend box office [Updated]]. 2020-07-28. LA Times Blogs - Company Town. en-US. 20 August 2020. https://web.archive.org/web/20200820084940/https://latimesblogs.latimes.com/entertainmentnewsbuzz/2011/02/movie-projector-big-mommas-i-am-number-four-unknown.html. live.
  100. Web site: fourth wall. 2021-11-29. dictionary.cambridge.org. en-US.
  101. Book: Roberts, Gareth E.. From Music to Mathematics: Exploring the Connections. 2016-02-15. JHU Press. 978-1-4214-1918-3. 3. en. ... called common time and denoted by C, which has four beats per measure.
  102. Book: Bonds, Mark Evan. Music as Thought: Listening to the Symphony in the Age of Beethoven. 2009-01-10. Princeton University Press. 978-1-4008-2739-8. 1. en. The number, character and sequence of movements in the symphony, moreover, did not stabilize until the 1770s when the familiar format of four movements....
  103. Book: Frisch, Walter. Brahms: The Four Symphonies. 2003. Yale University Press. 978-0-300-09965-2. en.
  104. Book: Brech, Lewis. Storybook Advent Carols Collection Songbook. 2010. Couples Company, Inc. 978-1-4524-7763-3. 26. en.
  105. Book: Baker, Felicity. Houses of Hogwarts: Cinematic Guide. 2017. Scholastic Incorporated. 978-1-338-12861-1. en. ...the four houses of Hogwarts School of Witchcraft and Wizardry: Gryffindor, Ravenclaw, Hufflepuff, and Slytherin.
  106. Book: Wright. Robert J.. Developing Number Knowledge: Assessment, Teaching and Intervention with 7-11 year olds. Ellemor-Collins. David. Tabor. Pamela D.. 2011-11-04. SAGE. 978-1-4462-5368-7. en.
  107. Book: Macauley, David. Elemental Philosophy: Earth, Air, Fire, and Water as Environmental Ideas. 2010-09-29. SUNY Press. 978-1-4384-3246-5. en.
  108. Book: Brooks, Edward. Normal Higher Arithmetic Designed for Advanced Classes in Common Schools, Normal Schools, and High Schools, Academics, Etc. 1876. Sower. 227. en. Every year that is divisible by four, except the Centennial years, and every Centennial year divisible by 400, is a leap year....
  109. Book: Touche. Fred. Wilderness Navigation Handbook. Price. Anne. 2005. Touche Publishing. 978-0-9732527-0-5. 48. en. Each of the familiar cardinal directions is equivalent to a particular true bearing: north (0°), east (90°), south (180°), and west (270°).
  110. Book: Roeckelein, J. E.. Elsevier's Dictionary of Psychological Theories. 2006-01-19. Elsevier. 978-0-08-046064-2. 235. en. ...four substances or humors: blood, yellow bile, black bile and phlegm.
  111. Book: Medley, H. Anthony. Bridge. 1997. Penguin. 978-0-02-861735-0. 6. en. The four playing card suits, as you probably already know, are spades, hearts, diamonds, and clubs.