3

For technical reasons, ":3" redirects here. For the keyboard symbols, see List of emoticons.

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.

← 2 3 4 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	three
Ordinal	3rd (third)
Numeral system	ternary
Factorization	prime
Prime	2nd
Divisors	1, 3
Greek numeral	Γ´
Roman numeral	III, iii
Latin prefix	tre-/ter-
Binary	112
Ternary	103
Senary	36
Octal	38
Duodecimal	312
Hexadecimal	316
Arabic, Kurdish, Persian, Sindhi, Urdu	٣
Bengali, Assamese	৩
Chinese	三，弎，叄
Devanāgarī	३
Ge'ez	፫
Greek	γ (or Γ)
Hebrew	ג
Japanese	三/参
Khmer	៣
Armenian	Գ
Malayalam	൩
Tamil	௩
Telugu	౩
Kannada	೩
Thai	๓
N'Ko	߃
Lao	໓
Georgian	Ⴂ/ⴂ/გ (Gani)
Babylonian numeral	𒐗
Maya numerals	•••
Morse code	... _ _

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. 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: ३.

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".

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.

Mathematics

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.

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 ${\displaystyle n}$ such that ${\displaystyle n}$! − 1 and ${\displaystyle n}$! + 1 are prime, as well as the only prime number ${\displaystyle p}$ such that ${\displaystyle p}$ − 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 ${\displaystyle n^{2}}$ − 1 for some integer ${\displaystyle n}$ is not prime, since it is (${\displaystyle n}$ − 1)(${\displaystyle n}$ + 1). This is true for 3 as well (with ${\displaystyle n}$ = 2), but in this case the smaller factor is 1. If ${\displaystyle n}$ is greater than 2, both ${\displaystyle n}$ − 1 and ${\displaystyle n}$ + 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 of prespecified orientation.

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 ${\displaystyle n}$ ⩾ 5, there are only three regular polytopes: the ${\displaystyle n}$-simplexes, ${\displaystyle n}$-cubes, and ${\displaystyle n}$-orthoplexes. In dimensions ${\displaystyle n}$ ⩾ 9, the only three uniform polytope families, aside from the numerous infinite proprismatic families, are the ${\displaystyle \mathrm {A} _{n}}$ simplex, ${\displaystyle \mathrm {B} _{n}}$ cubic, and ${\displaystyle \mathrm {D} _{n}}$ 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 ${\displaystyle {\bar {T}}_{9}}$, that is associated with an important Kac–Moody Lie algebra ${\displaystyle \mathrm {E} _{10}}$.

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.

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
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

Engineering

• The triangle, a polygon with three edges and three vertices, is the most stable physical shape. For this reason it is widely utilized in construction, engineering and design.

Pseudoscience

• Three is the symbolic representation for Mu, Augustus Le Plongeon's and James Churchward's lost continent.

Religion

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.

As a lucky or unlucky number

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: sì, Cantonese: sei¹), which sounds like the word "death" (死 pinyin sǐ, Cantonese: sei²).

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".

See also

• Mathematics portal

• Cube (algebra) – (3 superscript)
• Thrice
• Third
• Triad
• Trio
• Rule of three
• ɜ, U+025C ɜ LATIN SMALL LETTER REVERSED OPEN E also known as Reversed epsilon

References

1. "Merriam-Webster Dictionary". Merriam-webster.com. Retrieved December 5, 2024.
2. Smith, David Eugene; Karpinski, Louis Charles (1911). The Hindu-Arabic numerals. Boston; London: Ginn and Company. pp. 27–29, 40–41.
3. 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
4. Priya Hemenway (2005), Divine Proportion: Phi In Art, Nature, and Science, Sterling Publishing Company Inc., pp. 53–54, ISBN 1-4027-3522-7
5. "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
6. 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 systems...it seems valuable to give an outline of how the calculations would go", regarding E10 as a model example of symmetrizability of other root hyperbolic E_n systems.

7. Gribbin, Mary; Gribbin, John R.; Edney, Ralph; Halliday, Nicholas (2003). Big numbers. Cambridge: Wizard. ISBN 1840464313.
8. "Most stable shape- triangle". Maths in the city. Retrieved February 23, 2015.
9. 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.
10. "Definition of THE THIRD TIME IS THE CHARM". www.merriam-webster.com. Retrieved 2024-12-08.
11. See "bad Archived 2009-03-02 at the Wayback Machine" in the Oxford Dictionary of Phrase and Fable, 2006, via Encyclopedia.com.

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 46–48

External links

Look up three in Wiktionary, the free dictionary.

Wikimedia Commons has media related to 3 (number).

• Tricyclopedic Book of Threes by Michael Eck
• Threes in Human Anatomy by John A. McNulty
• Grime, James. "3 is everywhere". Numberphile. Brady Haran. Archived from the original on 2013-05-14. Retrieved 2013-04-13.
• The Number 3
• The Positive Integer 3
• Prime curiosities: 3