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Cardinal 3

Ordinal 3rd

Numeral system ternary
Factorization prime
Divisors 1, 3
Roman numeral III
Roman numeral (Unicode) Ⅲ, ⅲ
Arabic ٣
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) 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, and the second smallest prime. It is both the first Fermat prime (22n + 1) and the first Mersenne prime (2n − 1), as well as the first lucky prime. However, it's 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 one number, the square number 4 and is the base of the 3-aliquot tree.

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 fourth 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.

A triangle is the most durable shape possible, the only "perfect" figure which if all endpoints have hinges will never change its shape unless the sides themselves are bent.

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.

In numeral systems

It is frequently noted by historians of numbers that early counting systems oftenrelied 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 todescribe the quantities of one and two, but any quantity beyond this point wassimply denoted as "Many". As an extension to this insight, it can also be noted thatearly counting systems appear to have had limits at the numerals 2, 3, and 4. Referencesto counting limits beyond these three indices do not appear to prevail as consistentlyin the historical record.

Base Numeral system
2 binary 11
3 ternary 10
over 3 (decimal, hexadecimal) 3

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, 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

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 .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 (ezh), sometimes used to prevent people from falsifying a 3 into an 8.

In science

In religion and mythology

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

Three is also a symbolic number in Judaism. King Solomon states in Ecclesiastes 4:12: "A three-ply cord is not easily severed." Threes in Judiasm include the Three Patriarchs, the Three Pilgrim Festivals, and the three sections of the Hebrew Bible.

In philosophy

   |+3-way Philosophical Distinctions
   | Aristotle's 3-in-1 idea: || Mind, Self-knowledge, Self-love
   | Aristotle's 3 Dramatic Unities: || Unity of Action, Unity of Time, Unity of Place
   | Plotinus's Philosophy: || One, One Many, One and Many
   | Lucretius's 3 Ages (see also Christian Thomsen): || Stone Age, Bronze Age, Iron Age
   | St. Augustine's 3 Laws: || Divine Law, Natural Law, Temporal, positive, or human Law
   | St. Augustine's 3 characterizations of the soul: || Memory, Understanding, Will
   | Aquinas's 3 causal principles (based in Aristotle): || Agent, Patient, Act
   | Aquinas's 3 acts of intellect (based in Aristotle): || Conception, Judgment, Reasoning
   | Aquinas's 3 transcendentals of being: || Unity, Truth, Goodness
   | Aquinas's 3 requisites for the beautiful: || Wholeness or perfection, Harmony or due proportion, Radiance
   | Albertus Magnus's 3 Universals: || Ante rem (Idea in God's mind), In re (potential or actual in things), Post rem (mentally abstracted)
   | Sir Francis Bacon's 3 Tables: || Presence, Absence, Degree
   | Thomas Hobbes's 3 Fields: || Physics, Moral Philosophy, Civil Philosophy
   | Auguste Comte's Religion of Humanity: || Great Being (humanity), Great Medium (the world-space), Great Fetish (the Earth)
   | Johannes Nikolaus Tetens's 3 powers of mind: || Feeling, Understanding, Will
   | Immanuel Kant's 3 Critiques: || Pure Reason, Practical Reason, Judgment
   | Hegel's 3 Spirits: || Subjective Spirit, Objective Spirit, Absolute Spirit
   | Ludwig Andreas Feuerbach's 3 Thoughts: || God (1st thought), Reason (2nd), Man (3rd)
   | Ferdinand de Saussure's 3 "Signs": || Sign, Signified, Signifier
   | Charles Peirce's 3 semiotic elements: || Sign (representamen), Object, Interpretant
   | Charles Peirce's 3 categories: || Quality of feeling, Reaction/resistance, Representation
   | Charles Peirce's 3 universes of experience: || Ideas, Brute fact, Habit (habit-taking)
   | Charles Peirce's 3 orders of philosophy: || Phenomenology, Normative Sciences, Metaphysics
   | Charles Peirce's 3 normatives: || The good (esthetic), The right (ethical), The true (logical)
   | Charles Peirce's 3 grades of conceptual clearness: || By familiarity, Of definition's parts, Of conceivable practical consequences
   | Charles Peirce's 3 modes of evolution: || Fortuitous variation, Mechanical necessity, Creative love
   | Darwin's essentials of biological evolution: || Variation, Heredity, Struggle for existence
   | James Joyce's 3 aesthetic stages: || Arrest (by wholeness), Fascination (by harmony), Enchantment (by radiance)
   | Louis Zukofsky's 3 aesthetic elements || Shape, Rhythm, Style
   | Pythagoras's "fusion" idea: || Monarchy, Oligarchy, Democracy (into harmonic whole)
   | Karl Marx's 3 isms: || Communism, Socialism, Capitalism
   | Woodrow Wilson's 3 isms: || Colonialism, Racism, Anti-Communism
   | Hippocrates's Mind Disorders: || Mania, Melancholia, Phrenitis
   | Émile Durkheim's 3 Suicides: || Egoistic, Altruistic, Anomic
   | David Riesman's 3 Social Characters: || Tradition-directed, Inner-directed, Other-directed
   | Erich Fromm's 3 Symbols: || The Conventional, The Accidental, The Universal
   | Søren Kierkegaard's 3 Stages: || Aesthetic, Ethical, Religious
   | Edmund Husserl's 3 Reductions: || Phenomenological, Eidetic, Religious
   | Maurice Merleau-Ponty's 3 fieldsMerleau-Ponty, Maurice (1942), La structure du comportement, and published in English as The Structure of Behavior.: || Physical, Vital, Human
   | Maurice Merleau-Ponty's 3 categories: || Quantity, Order, Meaning
   | Alan Watts's 3 world views: || Life as machine (Western), Life as organism (Chinese), Life as drama (Indian)
   | 3-monkey Philosophy: || Hear no Evil, See no Evil, Speak no Evil
   | Mark Twain's (Samuel Clemens) 3 lies: || Lies, Damned Lies, Statistics
   | Witness Stand truths: || The Truth, The whole Truth, Nothing but the Truth
   | Abraham Lincoln's 3-For-All: || Of the People, By the People, For the People
   | Martin Luther King, Jr.'s "Middle Road": || Acquiescence, Nonviolence, Violence
   | Max Weber's 3 Authorities: || Traditional, Charismatic, Legal-rational
   | John Maynard Keynes's 3 Eras: || Scarcity, Abundance, Stabilization
   | George Herbert Mead's 3 Distinctions: || Self, I, Me
   | Frederic Thrasher's 3-group Gangs: || Inner Circle, Rank & File, Fringers
   | J.W.S. Pringle's 3 intellectual problems: || Religious & Ethical, Practical, Scientific
   | Jerome Bruner's 3 cognitive processing modes: || Enactive, Iconic, Symbolic
   | Wilhelm Wundt's 3 mind elements: || Sensations, Images, Feelings
   | Ezra Pound's 3 poetic modes: || Melopoeia (sound), Phanopoeia (image), Logopoeia (meaning)
   | Robert Sternberg's 3 love components: || Passion, Intimacy, Commitment
   | Sternberg's Triarchic Intelligence: || Analytic, Creative, Practical
   | Paul D. MacLean's Triune Brain: || R-System (Reptilian), Limbic System, Neocortex
   | J.A. Fodor's mind Taxonomy: || Central Processes, Input Processes, Transducers
   | Plato's Tripartite soul: || Rational, Libidinous, Spirited (various animal qualities)
   | William Herbert Sheldon's body types: || Endomorph, Mesomorph, Ectomorph
   | Ernst Kretschmer's body types: || Pyknic, Asthenic, Athletic
   | K.J.W. Craik's 3 reasoning processes: || Translation, Reasoning, Retranslation
   | Francis Galton's 3 genius traits: || Intellect, Zeal, Power of working

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) that 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 Vietnammarker, it is bad luck to take a photo with three people in it.

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

There is a superstition that states 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 is commonly believed to date from 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.

In technology

  • The glyph "3" may be used as a substitute for yogh ( ) or ze ( ) 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.
  • Some computer users may use "3" as an alternate to the letter "E", often in jest or to prevent search engines from reading their messages. This form of code is an example of basic Leetspeak.
  • "3" is the DVD region code for many East Asian countries, except for Japan (which is Region 2) and China (which is Region 6).

In music

  • In music, the Roman numeral iii is the mediant scale degree, chord, or diatonic function, when distinguished III = major and iii = minor.
  • Three is the number of performers in a trio.
  • There are 3 notes in a triad, the most important and basic form of any chord.
  • Any diatonic chord progression's key signature is made obvious with any 3 different triads, as opposed to potential key ambiguities with any 2 chords.
  • The tritone, which divides the octave into 3 equally spaced notes (root, tritone, octave) is the rarest interval of any mode, only occurring semantically twice, and physically once. It is the only interval that, when inverted, remains unchanged functionally and harmonically.
  • The 3/4 time signature of Western classical music tradition (Three beats to a measure, with the quarter note comprising the beat.) is said to represent the Holy Trinity of Christian doctrine, Father, Son and Holy Spirit. It is for this reason that it is often utilized in compositions which were written for use in ecclesiastical rites, or that are inspired by scriptural/spiritual themes and texts.
  • In a standard jazz combo there are 3 necessary parts: bass, percussion, and chord maker.
  • In Indian classical music, three equal repetitions of a rhythmic pattern is a common device called tihai.
  • 3rd Bridge, an extended technique on string instruments.
  • 3, a song by Britney Spears

In geography

In filmography

In sports

  • In Bowling, 3 strikes in a row is called a turkey.
  • In ice hockey, a game consists of 3 periods of twenty minutes each.
  • In rugby union, 3 is the jersey number of the starting tighthead prop. It is also the number of points received for a successful drop goal or penalty kick.
  • In rugby league, 3 is the jersey number of the starting right centre threequarter.
  • In baseball, 3 is the number of strikes before the batter is out and the number of outs per side per inning. It also represents the first baseman's position. The number 3 position in the batting order is generally occupied by the team's best hitter. In high school and college, 3 is the maximum "drop" (inches of length minus ounces of weight) for a legal bat. 3 is the retired number of Baseball Hall of Fame players Babe Ruth, Joe Medwick, Bill Terry, and Harmon Killebrew. Gary Sheffield and Ken Griffey Jr wear the number three.
  • In basketball, a shot made from behind the three-point arc is worth 3 points. 3 is used to represent the small forward position. In addition, a potential "three-point play" exists when a player is fouled while successfully completing a two-point field goal, thus being awarded one additional free-throw attempt.
  • Is the number of the famous NASCAR stock car that Dale Earnhardt drove for nearly 20 years before his death in 2001. He won 6 out of his 7 championships while driving the #3 car. Although NASCAR does not officially retire numbers, no one has driven the 3 car since his death. In IROC, Hélio Castroneves had his car number changed from his standard 3 (which he drives in the Indy Racing League) to number 03.
  • Traditional number for the Tyrrell Formula One team's first car along with number 4 for the second until the end of the 1995 Formula One Season.
  • A hat-trick in sports is associated with succeeding at anything three times in three consecutive attempts, as well as when any player in ice hockey scores three goals in one game (whether or not in succession). In Cricket, 3 outs in a row is called a hat trick.
  • In volleyball, is the number of sets needed to be won to win the whole match.
  • In both American and Canadian football, the number of points received for a successful field goal. (An exception is in six-man football where the field goal is worth four points.)
  • In Canadian football, the last down before a team loses possession on downs. Usually, a team faced with a third down will punt (if far from the opponent's goal line) or attempt a field goal (if relatively close).
  • An Ironman triathlon consists of three events, a 2.4 mile (3.86 kilometer) swim, a 112 mile (180.2 kilometer) bike ride, and a 26.2 mile (42.2 kilometer) marathon run.
  • In football, number 3 is assigned in most cases to the left defender or fullback.
  • On March 24, 2006 the number 3 became the second number retired by the New Jersey Devils in honor of defenseman Ken Daneyko.

In literature

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.

In other fields

For other uses of "3", see 3 .
Three is:


  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. * Plotinus, the Fifth Ennead, Section 8. Eprint. * Plotinus and Corrigan, Kevin (2005), Reading Plotinus: a practical introduction to neoplatonism, p. 26.
  4. Augustine through the Ages (1999), p. 582.
  5. Encyclopedia of Christian Theology v. 1 (2004), p. 54.
  6. See The Pocket Aquinas (1991).
  7. "St. Albertus Magnus" in the Catholic Enclyclopedia. Eprint.
  8. "Francis Bacon, Viscount Saint Alban", Eprint
  9. Pringle-Pattison, Andrew Seth (1917), The idea of God in the light of recent philosophy, p. 149.
  10. Teo, Thomas (2005), The critique of psychology: from Kant to postcolonial theory, p. 43.
  11. Redding, Paul (1997, 2006), "Georg Wilhelm Friedrich Hegel" in the Stanford Encyclopedia of Philosophy. Eprint.
  12. Lange, Friedrich Albert and Thomas, Ernest Chester (1880), History of materialism and criticism of its present importance, v. 2, p. 247, Google Books Eprint.
  13. "Darwinism" in Britannica Concise Encyclopedia via " Darwinism" at
  14. Joyce, James (1914-1915), A Portrait of the Artist as a Young Man, see Chapter 5, especially (but not only) lines 8215-8221.
  15. Zukofsky, Louis, "A" - 12 (1966), and Prepositions (1967, 1981), p. 55.
  16. "Pythagoreanism" at Eprint
  17. McDonald, William (1996, 2009), "Søren Kierkegaard" in the Stanford Encyclopedia of Philosophy. See Section 6.
  18. King, Martin Luther, Jr. (1959), "My Trip to the Land of Ghandi", published in A Testament of Hope: The Essential Writings and Speeches of Martin Luther King by Martin Luther King, edited by James M. Washington, 6th ed. 1990, see p. 25.
  19. Mini, Peter V. (1996), "Keynes on markets: a survey of heretical views" in the American Journal of Economics and Sociology, January. Eprint.
  • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 46 - 48

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