3 (number)

3 (number)

−1 0 1 2 3 4 5 6 7 8 9

Cardinal 3
Ordinal 3rd
Numeral system ternary
Factorization prime
Divisors 1, 3
Roman numeral III
Roman numeral (Unicode) Ⅲ, ⅲ
Greek γ (or Γ)
Arabic ٣,3
Chinese numeral 三,弎,叁
Hebrew ג (Gimel)
prefixes tri- (from Greek)

tre-/ter- (from Latin)

Binary 11
Octal 3
Duodecimal 3
Hexadecimal 3

3 (three; play /ˈθr/) is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.


In mathematics

  • Three is the first odd prime number,[1] and the second smallest prime. It is both the first Fermat prime (22n + 1) and the first Mersenne prime (2n − 1), the only number that is both, as well as the first lucky prime. However, it is the second Sophie Germain prime, the second Mersenne prime exponent, the second factorial prime (2! + 1), the second Lucas prime, the second Stern prime.
  • Three is the first unique prime due to the properties of its reciprocal.
  • Three is the aliquot sum of 4.
  • Three is the third Heegner number.
  • Three is the second triangular number and it is the only prime triangular number. Three is the only prime which is one less than a perfect square. Any other number which is n2 − 1 for some integer n is not prime, since it is (n − 1)(n + 1). This is true for 3 as well, but in its case one of the factors is 1.
  • Three non-collinear points determine a plane and a circle.
  • Three is the fifth Fibonacci number and the third that is unique. In the Perrin sequence, however, 3 is both the zeroth and third Perrin numbers.
  • Three is the fourth open meandric number.
  • Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions, (.000..., .333..., .666...)
  • 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.).
  • A triangle is the only figure which, if all endpoints have hinges, will never change its shape unless the sides themselves are bent.
  • 3 is the smallest prime of a Mersenne prime power tower 3, 7,127,170141183460469231731687303715884105727. It is not known whether any more of the terms are prime.
  • Three of the five regular polyhedra have triangular faces — the tetrahedron, the octahedron, and the icosahedron. Also, three of the five regular polyhedra 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 regular polyhedra — the triangle, the quadrilateral, and the pentagon.
  • There are only three distinct 4×4 panmagic squares.
  • Only three tetrahedral numbers are also perfect squares.
  • The first number, according to the Pythagoreans, and the first male number.
  • The first number, according to Proclus, being the first number such that n2 is greater than 2n.
  • The trisection of the angle was one of the three famous problems of antiquity.
  • Gauss proved that every integer is the sum of at most 3 triangular numbers.
  • Gauss proved that for any prime number p (with the sole exception of 3) the product of its primitive roots is ≡ 1 (mod p).
  • Any number not in the form of 4n(8m+7) is the sum of 3 squares.

In numeral systems

It is frequently noted by historians of numbers that early counting systems often relied on the three-patterned concept of "One- Two- Many" to describe counting limits. In other words, in their own language equivalent way, early peoples had a word to describe the quantities of one and two, but any quantity beyond this point was simply denoted as "Many". As an extension to this insight, it can also be noted that early counting systems appear to have had limits at the numerals 2, 3, and 4. References to counting limits beyond these three indices do not appear to prevail as consistently in the historical record.

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
3 \times 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
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 \div x 3 1.5 1 0.75 0.6 0.5 0.\overline{428571} 0.375 0.\overline{3} 0.3 0.\overline{27} 0.25 0.\overline{230769} 0.2\overline{142857} 0.2
x \div 3 0.\overline{3} 0.\overline{6} 1 1.\overline{3} 1.\overline{6} 2 2.\overline{3} 2.\overline{6} 3 3.\overline{3} 3.\overline{6} 4 4.\overline{3} 4.\overline{6} 5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
3 ^ x\, 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323
x ^ 3\, 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197

Evolution of the glyph


Three is often the largest number written with as many lines as the number represents. The Romans tired of writing 4 as IIII, instead using IV, but to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved. The Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and evolved it to a character that looks very much like a modern 3 with an extra stroke at the bottom. It was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. (The "extra" stroke, however, was very important to the Eastern Arabs, and they made it much larger, while rotating the strokes above to lie along a horizontal axis, and to this day Eastern Arabs write a 3 that looks like a mirrored 7 with ridges on its top line): ٣[2]

While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in Text figures 036.svg. In some French text-figure typefaces, though, it has an ascender instead of a descender.

A common variant of the digit 3 has a flat top, similar to the character Ʒ (ezh), sometimes used to prevent people from falsifying a 3 into an 8.

In science


  • A human ear has three semicircular canals.
  • A human middle ear has three ossicles.
  • Most elbows consist of three bones, the only joint in the human body where three articulations are surrounded by one capsule.
  • Humans perceive white light as the mixture of the three additive primary hues: red, green, and blue.


Georges Dumézil developed the Trifunctional Hypothesis which divides prehistoric Indo-European society into three classes: priests, warriors, and commoners.


Biology (specific and general)


  • Three is the atomic number of lithium.
  • Atoms consist of three constituents: protons, neutrons, and electrons.


In religion

Many world religions contain triple deities or concepts of trinity, including:

Abrahamic religions

The Shield of the Trinity is a diagram of the Christian doctrine of the Trinity
  • Three people (including Jesus) were crucified at the Crucifixion
  • The three Theological virtues referred to 1 Corinthians 13.
  • In Roman Catholicism, a group of three martyrs, collectively known as Faith, Hope, and Charity (named after the Theological Virtues).
  • Also in Roman Catholic doctrine, there are three realms of the afterlife: Heaven, Hell and Purgatory (Limbo is regarded as hypothetical).
  • The three members of the Holy Family: Jesus, Mary, and Joseph.
  • The Wise Men who visited Jesus after His birth left Him three gifts.
  • During wudhu, the hands, arms, face and feet are each washed three times.
  • According to the prophet Muhammad, there are three holy cities of Islam (to which pilgrimage should be made): Mecca, Medina, and Jerusalem.

In Hinduism

In Buddhism

  • The Three Jewels (or Triple Gem) in which Buddhists "take refuge" are the Buddha, the Dharma (the Buddha's teachings), and the Sangha (the community of enlightened beings).
  • The Triple Bodhi (ways to understand the end of birth) are Budhu, Pasebudhu, and Mahaarahath.
  • The Buddha has three bodies.[citation needed]

Other religions

In esoteric tradition

In philosophy

As a lucky or unlucky number

Three (三, formal writing: 叁, pinyin san1, Cantonese: saam1) is considered a good number in Chinese culture because it sounds like the word "alive" (生 pinyin sheng1, Cantonese: saang1), compared to four (四, pinyin: si4, Cantonese: sei1), which sounds like the word "death" (死 pinyin si3, 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, but then everyone can predict when three" will come based on "one" and "two"; this is likely why three is used instead of some other number.

In Vietnam, there is a superstition that considers it bad luck to take a photo with three people in it; it is professed that the person in the middle will die soon.

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.

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".

Luck, especially bad luck, is often said to "come in threes".[3]

In technology

3 as a resin identification code, used in recycling.
Seven-segment 3.svg
  • The glyph "3" may be used as a substitute for yogh (Ȝ, ȝ) or Greek xi (Ξ, ξ) or ze (З, з) when those characters are not available.
  • Three is the minimum odd number of voting components for simple easy redundancy checks by direct comparison.
  • Three is approximately pi (actually closer to 3.14159) when doing rapid engineering guesses or estimates. The same is true if one wants a rough-and-ready estimate of e, which is actually approximately 2.7183.
  • "3" is the DVD region code for many East Asian countries, except for Japan (which is Region 2) and China (which is Region 6).
  • 3 is a brand of 3G mobile phones.
  • The television VHF channel most often used for hooking up VCRs and/or video game systems. If it is otherwise occupied by a local broadcaster, then channel 4 is used instead.
  • Some may use "3" as an alternate to the letter "E", often in jest or when using Leetspeak, to denote being experienced in certain technology related fields.
  • In Resource Description Framework, subject-predicate-object expressions are referred to as triples, because they contain 3 values.

In music

In geography

Flag of Trinacria with a three-legged symbol.
  • Several cities are known as Tripoli from Greek for "three cities".
  • Sicilia was known as Trinacria for its triangle-shape.
  • Three Mile Island is known for a nuclear accident.
  • Several cities are also known as Triad Winston-Salem, High Point, and Greensboro NC

In filmography

  • Three is a 1969 film starring Charlotte Rampling and Sam Waterston.
  • Three is a 1965 Yugoslav film directed by Aleksandar Petrović.
  • Three is a 2002 Asian horror movie collaboration.
  • Three is a 2010 German film.
  • Thr3e is a 2007 film adaptation of the novel of the same name by Ted Dekker.
  • Three, also known as Survival Island is a 2006 film starring Billy Zane and Kelly Brook.
  • There is a 1977 film titled 3 Women.
  • In both the film The Craft and the fantasy television series Charmed, the "power of three" has been used as part of wiccan incantations.

In sports

In games

The game rock-paper-scissors involves three hand shapes. Rock, paper, and scissors.

In literature

  • 3 is the number of witches in William Shakespeare's Macbeth.
  • 3 is the number of words or phrases in a Tripartite motto.
  • 3 is the number of novels or films in a trilogy and the number of interconnected works of art in a triptych.
  • Thr3e is a 2003 suspense novel written by Thriller author Ted Dekker.
  • Dante Alighieri's Divine Comedy has three parts each of thirty-three cantos (plus one introductory canto totaling 100). It was written in terza rima, a combination of tercets. All of this is an allusion to the Christian Trinity.
  • The number three recurs several times in Tolkien's The Lord of the Rings and also in The Silmarillion. Three Rings of Power were given to the Elves. There are three Silmarils. The unions of the Eldar (Elves) and the Edain (Men) were three in number: Beren and Lúthien, Tuor and Idril, and Aragorn and Arwen.
  • Three Blind Mice is a children's nursery rhyme and musical round.
  • The Three Musketeers is a novel by Alexandre Dumas, and is part of a trilogy.
  • Three Sisters is a play by Anton Chekhov.
  • A recurring theme in Arthur C. Clarke's Rama series is the observation that "the Ramans do everything in threes."
  • The Three Bears – children's classic literature
  • The Three Little Pigs – children's classic literature.
  • 3 is the number of wishes normally granted in most fairy tales and stories. Likewise, the protagonist in most stories faces 3 conflicts, whether mental or physical before his or her great triumph.[citation needed]
  • "Threes" is a poem by Carl Sandburg.
  • In many Czech folktales, a great beast of some sort will, if bound in some manner, usually be bound by three chains, hooks, ropes, etc., and a menial task must be repeated three times to free it.
  • The Day of the Triffids, 1951 by John Wyndham. Genetically modified plants with three legs take over the earth.
  • The number three is a recurring theme in the Series of Unfortunate Events: there are three Baudelaire orphans, three Snicket orphans, three Quagmire orphans, etc.
  • The Good, the Bad and the Ugly centers around a theme of 3

Original scholarly articles/reviews about the three

  • "The Number Three in The American Culture". A selected chapter found in the book entitled Every Man His Way (1967–68) by Alan Dundes.
  • "People in Threes Going Up in Smoke and Other Triplicities in Russian Literature and Culture" (Fall 2005, Rocky Mountain Review) by Lee B. Croft.
  • "Buckland's Third Revolution" (1997–98) and "Three Wise Men" (1984–85) posters by Herb O. Buckland.
International maritime signal flag for 3 is known as a triband, a form of the tricolour.
Travelling in a troika (three-horse sled).


  1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 39
  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. ^ See "bad" in the Oxford Dictionary of Phrase and Fable, 2006, via Encyclopedia.com.

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