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← 2 3 4 →
−1 0 1 2 3 4 5 6 7 8 9
Numeral systemternary
Divisors1, 3
Greek numeralΓ´
Roman numeralIII, iii
Greek prefixtri-
Latin prefixtre-/ter-
Arabic, Kurdish, Persian, Sindhi, Urdu٣
Bengali, Assamese
Greekγ (or Γ)
GeorgianႢ/ⴂ/გ (Gani)
Babylonian numeral𒐗
Maya numerals•••
Morse code... _ _

3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious and cultural significance in many societies.

Evolution of the Arabic digit

The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and Chinese numerals) that are still in use. That was also the original representation of 3 in the Brahmic (Indian) numerical notation, its earliest forms aligned vertically.[1] However, during the Gupta Empire the sign was modified by the addition of a curve on each line. The Nāgarī script rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a ⟨3⟩ with an additional stroke at the bottom: .

The Indian digits spread to the Caliphate in the 9th century. The bottom stroke was dropped around the 10th century in the western parts of the Caliphate, such as the Maghreb and Al-Andalus, when a distinct variant ("Western Arabic") of the digit symbols developed, including modern Western 3. In contrast, the Eastern Arabs retained and enlarged that stroke, rotating the digit once more to yield the modern ("Eastern") Arabic digit "٣".[2]

In most modern Western typefaces, the digit 3, like the other decimal digits, has the height of a capital letter, and sits on the baseline. In typefaces with text figures, on the other hand, the glyph usually has the height of a lowercase letter "x" and a descender: "". In some French text-figure typefaces, though, it has an ascender instead of a descender.

A common graphic variant of the digit three has a flat top, similar to the letter Ʒ (ezh). This form is sometimes used to prevent falsifying a 3 as an 8. It is found on UPC-A barcodes and standard 52-card decks.


According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.[3]

Divisibility rule

A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three (3 times 7) and the sum of its digits is 2 + 1 = 3. Because of this, the reverse of any number that is divisible by three (or indeed, any permutation of its digits) is also divisible by three. For instance, 1368 and its reverse 8631 are both divisible by three (and so are 1386, 3168, 3186, 3618, etc.). See also Divisibility rule. This works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one (bases 4, 7, 10, etc.).

Properties of the number

3 is the second smallest prime number and the first odd prime number. It is the first unique prime, such that the period length value of 1 of the decimal expansion of its reciprocal, 0.333..., is unique. 3 is a twin prime with 5, and a cousin prime with 7, and the only known number such that ! − 1 and ! + 1 are prime, as well as the only prime number such that − 1 yields another prime number, 2. A triangle is made of three sides. It is the smallest non-self-intersecting polygon and the only polygon not to have proper diagonals. When doing quick estimates, 3 is a rough approximation of π, 3.1415..., and a very rough approximation of e, 2.71828...

3 is the first Mersenne prime, as well as the second Mersenne prime exponent and the second double Mersenne prime exponent, for 7 and 127, respectively. 3 is also the first of five known Fermat primes, which include 5, 17, 257, and 65537. It is the second Fibonacci prime (and the second Lucas prime), the second Sophie Germain prime, the third Harshad number in base 10, and the second factorial prime, as it is equal to 2! + 1.

3 is the second and only prime triangular number, and Gauss proved that every integer is the sum of at most 3 triangular numbers.

Three is the only prime which is one less than a perfect square. Any other number which is − 1 for some integer is not prime, since it is ( − 1)( + 1). This is true for 3 as well (with = 2), but in this case the smaller factor is 1. If is greater than 2, both − 1 and + 1 are greater than 1 so their product is not prime.

Related properties

The trisection of the angle was one of the three famous problems of antiquity.

3 is the number of non-collinear points needed to determine a plane, a circle, and a parabola.

There are only three distinct 4×4 panmagic squares.

Three of the five Platonic solids have triangular faces – the tetrahedron, the octahedron, and the icosahedron. Also, three of the five Platonic solids have vertices where three faces meet – the tetrahedron, the hexahedron (cube), and the dodecahedron. Furthermore, only three different types of polygons comprise the faces of the five Platonic solids – the triangle, the square, and the pentagon.

There are three finite convex uniform polytope groups in three dimensions, aside from the infinite families of prisms and antiprisms: the tetrahedral group, the octahedral group, and the icosahedral group. In dimensions ⩾ 5, there are only three regular polytopes: the -simplexes, -cubes, and -orthoplexes. In dimensions 9, the only three uniform polytope families, aside from the numerous infinite proprismatic families, are the simplex, cubic, and demihypercubic families. For paracompact hyperbolic honeycombs, there are three groups in dimensions 6 and 9, or equivalently of ranks 7 and 10, with no other forms in higher dimensions. Of the final three groups, the largest and most important is , that is associated with an important Kac–Moody Lie algebra .[4]

Numeral systems

There is some evidence to suggest that early man may have used counting systems which consisted of "One, Two, Three" and thereafter "Many" to describe counting limits. Early peoples had a word to describe the quantities of one, two, and three but any quantity beyond was simply denoted as "Many". This is most likely based on the prevalence of this phenomenon among people in such disparate regions as the deep Amazon and Borneo jungles, where western civilization's explorers have historical records of their first encounters with these indigenous people.[5]

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 10000
3 × x 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 150 300 3000 30000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 ÷ x 3 1.5 1 0.75 0.6 0.5 0.428571 0.375 0.3 0.3 0.27 0.25 0.230769 0.2142857 0.2 0.1875 0.17647058823529411 0.16 0.157894736842105263 0.15
x ÷ 3 0.3 0.6 1 1.3 1.6 2 2.3 2.6 3 3.3 3.6 4 4.3 4.6 5 5.3 5.6 6 6.3 6.6
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3x 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721 129140163 387420489 1162261467 3486784401
x3 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000






Main article: Trichotomy (philosophy)

Nikola Tesla

Nikola Tesla was a great scientist. Despite his work on free energy etc. he stated, " If the understand the numbers 3, 6, 9, you understand the universe."


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See also: Triple deity

Symbol of the Triple Goddess showing the waxing, full and waning Moon

Many world religions contain triple deities or concepts of trinity, including the Hindu Trimurti and Tridevi, the Triglav (lit. "Three-headed one"), the chief god of the slavs, the three Jewels of Buddhism, the three Pure Ones of Taoism, the Christian Holy Trinity, and the Triple Goddess of Wicca.

The Shield of the Trinity is a diagram of the Christian doctrine of the Trinity.









Norse mythology

Three is a very significant number in Norse mythology, along with its powers 9 and 27.

Other religions

Esoteric tradition

As a lucky or unlucky number

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Three (, formal writing: , pinyin sān, Cantonese: saam1) is considered a good number in Chinese culture because it sounds like the word "alive" ( pinyin shēng, Cantonese: saang1), compared to four (, pinyin: , Cantonese: sei1), which sounds like the word "death" ( pinyin , Cantonese: sei2).

Counting to three is common in situations where a group of people wish to perform an action in synchrony: Now, on the count of three, everybody pull! Assuming the counter is proceeding at a uniform rate, the first two counts are necessary to establish the rate, and the count of "three" is predicted based on the timing of the "one" and "two" before it. Three is likely used instead of some other number because it requires the minimal amount counts while setting a rate.

There is another superstition that it is unlucky to take a third light, that is, to be the third person to light a cigarette from the same match or lighter. This superstition is sometimes asserted to have originated among soldiers in the trenches of the First World War when a sniper might see the first light, take aim on the second and fire on the third.[citation needed]

The phrase "Third time's the charm" refers to the superstition that after two failures in any endeavor, a third attempt is more likely to succeed. This is also sometimes seen in reverse, as in "third man [to do something, presumably forbidden] gets caught". [citation needed]

Luck, especially bad luck, is often said to "come in threes".[27]


For a list of films called 3, Three, etc., see 3 (disambiguation) § Film.

See also


  1. ^ Smith, David Eugene; Karpinski, Louis Charles (1911). The Hindu-Arabic numerals. Boston; London: Ginn and Company. pp. 27–29, 40–41.
  2. ^ 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): 393, Fig. 24.63
  3. ^ Priya Hemenway (2005), Divine Proportion: Phi In Art, Nature, and Science, Sterling Publishing Company Inc., pp. 53–54, ISBN 1-4027-3522-7
  4. ^ Allcock, Daniel (May 2018). "Prenilpotent Pairs in the E10 root lattice" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 164 (3): 473–483. Bibcode:2018MPCPS.164..473A. doi:10.1017/S0305004117000287. S2CID 8547735. Archived (PDF) from the original on 2022-11-03. Retrieved 2022-11-03.
    "The details of the previous section were E10-specific, but the same philosophy looks likely to apply to the other symmetrizable hyperbolic root seems valuable to give an outline of how the calculations would go", regarding E10 as a model example of symmetrizability of other root hyperbolic En systems.
  5. ^ Gribbin, Mary; Gribbin, John R.; Edney, Ralph; Halliday, Nicholas (2003). Big numbers. Cambridge: Wizard. ISBN 1840464313.
  6. ^ Zwiebach, Barton (2009). A first course in string theory (2nd ed.). Cambridge ; New York: Cambridge University Press. ISBN 978-0-521-88032-9.
  7. ^ Harari, H. (1977). "Three generations of quarks and leptons" (PDF). In van Goeler, E.; Weinstein, R. (eds.). Proceedings of the XII Rencontre de Moriond. p. 170. SLAC-PUB-1974.
  8. ^ Adair, R.K. (1989). The Great Design: Particles, Fields, and Creation. Oxford University Press. p. 214.
  9. ^ "The Rods and Cones of the Human Eye". Retrieved 2024-06-04.
  10. ^ Barrow-Green, June (2008). "The Three-Body Problem". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. pp. 726–728.
  11. ^ "Most stable shape- triangle". Maths in the city. Retrieved February 23, 2015.
  12. ^ Eric John Holmyard. Alchemy. 1995. p.153
  13. ^ Walter J. Friedlander. The golden wand of medicine: a history of the caduceus symbol in medicine. 1992. p.76-77
  14. ^ Kreidler, Marc (2017-12-14). "Ayurveda: Ancient Superstition, Not Ancient Wisdom". Skeptical Inquirer. Retrieved 2024-06-04.
  15. ^ Churchward, James (1931). "The Lost Continent of Mu – Symbols, Vignettes, Tableaux and Diagrams". Biblioteca Pleyades. Archived from the original on 2015-07-18. Retrieved 2016-03-15.
  16. ^ Windle, Bryan (2022-12-22). "Who Were the Magi?". Bible Archaeology Report. Retrieved 2024-07-05.
  17. ^ "Encyclopaedia Britannica". Lexikon des Gesamten Buchwesens Online (in German). doi:10.1163/9789004337862_lgbo_com_050367.
  18. ^ "The Encyclopaedia Britannica". Nature. XV (378): 269–271. 25 January 1877. Archived from the original on 24 July 2020. Retrieved 12 July 2019.
  19. ^ Marcus, Rabbi Yossi (2015). "Why are many things in Judaism done three times?". Ask Moses. Archived from the original on 2 April 2015. Retrieved 16 March 2015.
  20. ^ "Shabbat". Judaism 101. 2011. Archived from the original on 29 June 2009. Retrieved 16 March 2015.
  21. ^ Kitov, Eliyahu (2015). "The Three Matzot". Archived from the original on 24 March 2015. Retrieved 16 March 2015.
  22. ^ Kaplan, Rabbi Aryeh (28 August 2004). "Judaism and Martyrdom". Archived from the original on 20 March 2015. Retrieved 16 March 2015.
  23. ^ "The Basics of the Upsherin: A Boy's First Haircut". 2015. Archived from the original on 22 March 2015. Retrieved 16 March 2015.
  24. ^ "The Conversion Process". Center for Conversion to Judaism. Archived from the original on 23 February 2021. Retrieved 16 March 2015.
  25. ^ Kaplan, Aryeh. "The Soul Archived 2015-02-24 at the Wayback Machine". Aish. From The Handbook of Jewish Thought (Vol. 2, Maznaim Publishing. Reprinted with permission.) September 4, 2004. Retrieved February 24, 2015.
  26. ^ James G. Lochtefeld, Guna, in The Illustrated Encyclopedia of Hinduism: A-M, Vol. 1, Rosen Publishing, ISBN 978-0-8239-3179-8, page 265
  27. ^ See "bad Archived 2009-03-02 at the Wayback Machine" in the Oxford Dictionary of Phrase and Fable, 2006, via