3 (number)

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This article is about the number. For the year, see 3. For other uses, see 3 (disambiguation).
2 3 4
−1 0 1 2 3 4 5 6 7 8 9
Cardinal three
Ordinal 3rd
Numeral system ternary
Factorization prime
Divisors 1, 3
Roman numeral III
Roman numeral (unicode) Ⅲ, ⅲ
Greek prefix tri-
Latin prefix tre-/ter-
Binary 112
Ternary 103
Quaternary 34
Quinary 35
Senary 36
Octal 38
Duodecimal 312
Hexadecimal 316
Vigesimal 320
Base 36 336
Arabic ٣,3
Urdu ۳
Chinese 三,弎,叁
Devanāgarī (tin)
Greek γ (or Γ)
Hebrew ג
Korean 셋,삼

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

Evolution of the glyph[edit]


Three is the largest number still written with as many lines as the number represents. (The Ancient Romans usually wrote 4 as IIII, but this was almost entirely replaced by the subtractive notation IV in the Middle Ages.) 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[1]

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.

Flat top 3[edit]

A common variant of the digit 3 has a flat top, similar to the character Ʒ (ezh). This form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks.

In mathematics[edit]

  • Three is approximately π (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 2.71828.
  • Three is the first odd prime number,[2] 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. It is the first super-prime. However, it is the second Sophie Germain prime, the second Mersenne prime exponent, the second factorial prime (2! + 1), the second Lucas prime, and 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.
  • According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it 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]
  • 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 (with n=2), but in this case the smaller factor is 1. If n is greater than 2, both n − 1 and n + 1 are greater than 1 so their product is not prime.
  • Three non-collinear points determine a plane and a circle.
  • Three is the fourth Fibonacci number. 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: 0.000..., 0.333..., 0.666... (See also 0.999..., which = 1.)
  • 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.).
  • 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.
  • 3 is the smallest number of sides that a simple (non-self-intersecting) polygon can have.
  • 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, i.e. 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.
  • Each real number belongs to only 1 of the following 3 categories: positive, negative, and zero.

In numeral systems[edit]

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 phenomena 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.[4]

List of basic calculations[edit]

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 \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 30000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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 0.1875 0.1\overline{7647058823529411} 0.1\overline{6} 0.1\overline{57894736842105263} 0.15
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 5.\overline{3} 5.\overline{6} 6 6.\overline{3} 6.\overline{6}
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 ^ x\, 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721 129140163 387420489 1162261467 3486784401
x ^ 3\, 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000

In science[edit]

In protoscience[edit]

In astronomy[edit]

  • The Saros number of the solar eclipse series which began on April 24, 2814 BC and ended on June 1, 1534 BC. The duration of Saros series 3 was 1280.14 years, and it contained 72 solar eclipses.
  • The Saros number of the lunar eclipse series which began on December 30, 2567 BC and ended on March 21, 1214 BC. The duration of Saros series 3 was 1352.26 years, and it contained 76 lunar eclipses.

In philosophy[edit]

In religion[edit]

See also: Triple deity

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

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

In Christianity[edit]

In Judaism[edit]

In Buddhism[edit]

  • The Triple Bodhi (ways to understand the end of birth) are Budhu, Pasebudhu, and Mahaarahath.
  • The Three Jewels, the three things that Buddhists take refuge in.

In Taoism[edit]

In Hinduism[edit]

The "Om" symbol, in Devanagari is also written ओ३म् (ō̄m [õːːm]), where ३ is दीर्घ (dirgha, "three times as long")

In Zoroastrianism[edit]

  • The three virtues of Humata, Hukhta and Huvarshta (Good Thoughts, Good Words and Good Deeds) are a basic tenet in Zoroastrianism.

In Norse mythology[edit]

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

  • Prior to Ragnarök, there will be three hard winters without an intervening summer, the Fimbulwinter.
  • Odin endured three hardships upon the World Tree in his quest for the runes: he hanged himself, wounded himself with a spear, and suffered from hunger and thirst.
  • Bor had three sons, Odin, Vili, and .

In other religions[edit]

In esoteric tradition[edit]

As a lucky or unlucky number[edit]

Three (三, formal writing: 叁, pinyin sān, Cantonese: saam1) is considered a good number in Chinese culture because it sounds like the word "alive" (生 pinyin shēng, Cantonese: saang1), compared to four (四, pinyin: , Cantonese: sei1), which sounds like the word "death" (死 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.

In East and Southeast Asia, there is a widespread superstition that considers it inauspicious to take a photo with three people in it; it is professed that the person in the middle will die first.

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".[15]

In sports[edit]

  • In association football in almost all leagues, and in the group phases of most international competitions, three competition points are awarded for a win.
  • In Gaelic football, hurling and camogie, a "goal", with a scoring value of three, is awarded when the attacking team legally sends the ball into the opponent's goal.
  • In baseball, three is the number of strikes before the batter is out and the number of outs per side per inning; in scorekeeping, "3" denotes the first baseman.
  • In basketball:
    • A shot made from behind the three-point arc is worth three points (except in the 3x3 variant, in which it is worth two points).
    • 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.
    • On offense, the "three-second rule" states that an offensive player cannot remain in the opponent's free-throw lane for more than three seconds while his team is in possession of the ball and the clock is running.
    • In the NBA only, the defensive three-second violation, also known as "illegal defense", states that a defensive player cannot remain in his own free-throw lane for more than three seconds unless he is actively guarding an offensive player.
    • The "3 position" is the small forward.
  • In gridiron football, a field goal is a scoring play in which a kicker kicks the football from behind the line of scrimmage through the goalposts on the opponent's side of the field. This play, if successful, is worth three points.
  • 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 or soccer scores three goals in one game (whether or not in succession). In cricket, if a bowler takes three wickets in a row it is called a hat trick.
  • A "threepeat" is a term for winning three consecutive championships.
  • A triathlon consists of three events: swimming, bicycling, and running.
  • A pin (professional wrestling) in professional wrestling is when one's shoulders are held the opponent's shoulders against the mat for a count of three.
  • NASCAR driver Dale Earnhardt, Sr. drove under number 3 for most of his career for car owner Richard Childress.

See also[edit]


  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): 393, Fig. 24.63
  2. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 39
  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. ^ Big Numbers. ISBN 1840464313. 
  5. ^ "Most stable shape- triange". Maths in the city. Retrieved February 23, 2015.
  6. ^ Eric John Holmyard. Alchemy. 1995. p.153
  7. ^ Walter J. Friedlander. The golden wand of medicine: a history of the caduceus symbol in medicine. 1992. p.76-77
  8. ^ Marcus, Rabbi Yossi (2015). "Why are many things in Judaism done three times?". Ask Moses. Retrieved 16 March 2015. 
  9. ^ "Shabbat". Judaism 101. 2011. Retrieved 16 March 2015. 
  10. ^ Kitov, Eliyahu (2015). "The Three Matzot". Chabad.org. Retrieved 16 March 2015. 
  11. ^ Kaplan, Rabbi Aryeh (28 August 2004). "Judaism and Martyrdom". Aish.com. Retrieved 16 March 2015. 
  12. ^ "The Basics of the Upsherin: A Boy's First Haircut". Chabad.org. 2015. Retrieved 16 March 2015. 
  13. ^ "The Conversion Process". Center for Conversion to Judaism. Retrieved 16 March 2015. 
  14. ^ Kaplan, Aryeh. "The Soul". Aish. From The Handbook of Jewish Thought (Vol. 2, Maznaim Publishing. Reprinted with permission.) September 4, 2004. Retrieved February 24, 2015.
  15. ^ See "bad" in the Oxford Dictionary of Phrase and Fable, 2006, via Encyclopedia.com.

External links[edit]