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Cardinal 4

Ordinal 4th

Numeral system quaternary
Factorization 2^2
Divisors 1, 2, 4
Roman numeral IV or IIII
Roman numeral (Unicode) Ⅳ, ⅳ
Arabic ٤
Arabic (Urdu) ۴
Chinese numeral 四,亖,肆
Hebrew ארבע (Arba, pronounced are-buh) or ד (Dalet, 4th letter of the Hebrew alphabet)
prefix tetra- (from Greek)quadri-/quadr- (from Latin)
Binary 100
Octal 4
Duodecimal 4
Hexadecimal 4
Vigesimal 4

4 (four) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5.

In mathematics

Four is the smallest composite number, its proper divisors being and .Four is also a highly composite number. The next highly composite number is .

Four is the second square number, the second centered triangular number.

4 is the smallest squared prime ( p^2 ) and the only even number in this form. It has an aliquot sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members (4,3,1,0) and is accordingly the first member of the 3-aliquot tree.

Only one number has an aliquot sum of 4 and that is squared prime 9.

The prime factorization of four is two times two.

Four is the smallest composite number that is equal to the sum of its prime factors. (As a consequence of this, it is the smallest Smith number). However, it is the largest (and only) composite number n for which (n - 1)!\ \equiv\ 0 \ ({\rm mod}\ n) is false.

It is also a Motzkin number.

In addition, 2 + 2 = 2 \times 2 = 2^2 = 4 . Continuing the pattern in Knuth's up-arrow notation, 2 \uparrow\uparrow 2 = 2 \uparrow\uparrow\uparrow 2 = 4, and so on, for any number of up arrows.

A four-sided plane figure is a quadrilateral (quadrangle) or square, sometimes also called a tetragon. A circle divided by 4 makes right angles. Because of it, four (4) is the base number of plane . Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four.

A solid figure with four faces is a tetrahedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.

Four-dimensional space has the largest number of dimensions used by more than three convex regular figures. There are infinitely many convex regular polygons (two-dimensional); five convex regular polyhedra (three-dimensional, the five Platonic Solids); six convex regular polychora (four-dimensional); and three regular convex polytopes occupying each higher-dimensional space.

Four-dimensional differential manifolds have some unique properties.There is only one differential structure on \mathbb{R}^{n} except when n = 4, in which case there are uncountably many.

The smallest non-cyclic group has four elements; it is the Klein four-group. Four is also the order of the smallest non-trivial groups that are not simple.

Four is the maximum number of dimensions of a real division algebra (the quaternions), by a theorem of Ferdinand Georg Frobenius.

The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.

Four is the first positive non-Fibonacci number.

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y^2 - z^2.

Four is an all-Harshad number and a semi-meandric number.

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
4 \times x 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000

Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 \div x 4 2 1.\overline{3} 1 0 .8 0.\overline{6} 0.\overline{5}7142\overline{8} 0.5 0.\overline{4} 0.4 0.\overline{3}\overline{6} 0.\overline{3} 0.\overline{3}0769\overline{2} 0.\overline{2}8571\overline{4} 0.2\overline{6} 0.25
x \div 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
4 ^ x\, 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
x ^ 4\, 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

Evolution of the glyph

Representing 1, 2 and 3 in as many lines as the number represents worked well. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like our modern plus sign. The Sunga would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the numeral less cursive, ending up with a glyph very close to the original Brahmin cross.

While the shape of the 4 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4 is seen with an open top. This is to distinguish 4 from the number 9.

Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4". Another form of the 4 glyph that was invented for television was the arrow 4, which combines the 4 glyph with an arrow.

In religion

Judeo-Christian symbolism

In science

  • A tetramer is a thing formed out of four sub-units.

In astronomy

In biology

In chemistry

  • Valency of carbon (that is basis of life on the Earth) is four. Thanks to its tetrahedral crystal bond structure diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth's crust.
  • The atomic number of beryllium
  • There are four basic states of matter: solid, liquid, gas, and plasma.
  • The four elements of alchemy are earth, water, air and fire.

In physics

In logic and philosophy

  • The symbolic meanings of the number four are linked to those of the cross and the square. 'almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself'. Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called 'lord of the four suns'...'lord of the four quarters of the earth'... by which is understood to the extent of their powers both territorially and in terms of total control of their subjects' doings.
  • The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R").
  • In regard to whether two given propositions can have the same truth value, there are four distinct possibilities: the propositions are subalterns (both can be true and both can be false); subcontraries (can be both of them true but cannot be both of them false); contraries (can be both of them false but cannot be both of them true); or contradictories (cannot be both of them true and cannot be both of them false).
  • Aristotle held that there are basically four causes in nature: the efficient cause, the matter, the end, and the form.
  • The Stoics held with four basic categories, all viewed as bodies (substantial and insubstantial): (1) substance in the sense of substrate, primary formless matter; (2) quality, matter's organization to differentiate and individualize something, and coming down to a physical ingredient such as pneuma, breath; (3) somehow holding (or disposed), as in a posture, state, shape, size, action, and (4) somehow holding (or disposed) toward something, as in relative location, familial relation, and so forth.
  • Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to 1. Quantity, 2. Quality, 3. Relation, 4. Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories.
  • Arthur Schopenhauer's doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason.
  • Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly progressing with scientific interest and results; then declining through the remaining phases (2) practical, (3) increasingly skeptical, and (4) literary, mystical, and scientifically worthless — until philosophy is renewed through a new period's first phase. (See Brentano's essay "The Four Phases of Philosophy and Its Current State" 1895, tr. by Mezei and Smith 1998.)
  • C.S. Peirce, usually a trichotomist, discussed four basic methods of seeking to settle questions and arrive at firm beliefs: (1) the method of tenacity (sticking to that which one is inclined to think), (2) the method of authority, (3) the a priori method, and (4) the method of science (see "The Fixation of Belief", 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is absolutely unknowable; (3) maintaining that something is absolutely inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena (See " F.R.L." (First Rule of Logic), 1899).
  • Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatio-temporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss's Modes of Being, 1958).
  • Karl Popper outlined a tetradic schema to describe the growth of theories and, via generalization, also the emergence of new behaviors and living organisms: 1. problem, 2. tentative theory, 3. (attempted) error-elimination (especially by way of critical discussion), and 4. new problem(s). (See Popper's Objective Knowledge, 1972, revised 1979.)
  • John Boyd made his key concept the decision cycle or OODA loop, consisting of four stages: 1. observation (data intake through the senses), 2. orientation (analysis and synthesis of data), 3. decision, and 4. action. Boyd held that his decision cycle has philosophical generality, though for strategists the point remains that, through swift decisions, one can disrupt an opponent's decision cycle.
  • Richard McKeon outlined four classes (each with four subclasses) of modes of philosophical inquiry: (1) Modes of Being (Being); (2) Modes of Thought (That which is); (3) Modes of Fact (Existence); (4) Modes of Simplicity (Experience) — and, corresponding to them, four classes (each with four subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See McKeon's "Philosophic Semantics and Philosophic Inquiry" in Freedom and History and Other Essays, 1989.)
  • Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: kinds (substantial universals), attributes (relational universals and property-universals), objects (substantial particulars), and modes (relational particulars and property-particulars, also known as "tropes"). (See Lowe's "Recent Advances in Metaphysics," 2001, Eprint)

In technology

  • The resin identification code used in recycling to identify low-density polyethylene.
  • Most furniture has four legs - tables, chairs, etc.
  • Four horses (quadriga) is the maximal number of horses in one row for carriage.
  • The four color process (CMYK) is used for printing.
  • Wide use of rectangles (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames).
  • In the Rich Text Format specification, language code 4 is for the Chinese language. Codes for regional variants of Chinese are congruent to 4 mod 256.
  • Credit card machines have four-twelve function keys.
  • On most phones, the 4 key is associated with the letters G, H, and I, but on the BlackBerry cellular phone, it is the key for D and F.
  • On many computer keyboards, the "4" key may also be used to type the dollar sign ($) if the shift key is held down.
  • It is the number of bits in a nibble, equivalent to half a byte
  • In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
  • In Leetspeak, "4" may be used to replace the letter "A".
  • In Expressing love, "4" may be used to replace the word "Love" (143).

In transport

In sports

  • In the sport of cricket, a four is a specific type of scoring event, whereby the ball crosses the boundary after touching the ground at least one time, scoring four runs. Taking four wickets in four consecutive balls is typically referred to as a double hat trick (two consecutive, overlapping hat tricks).
  • In rugby union, the number of the lock forward, who usually jumps at number 2 in the line-out.
  • In rugby league, the number of one of the two centres. It is also the number of points awarded for a try.
  • In baseball, 4 represents the second baseman's position.
  • In basketball,
    • 4 represents the power forward position
    • the term Final Four refers to the last four teams remaining in the NCAA playoff tournament, each of which is the winner of its respective region
  • a possible "4-point play" occurs if a player completes a three-pointer while being fouled; the player is awarded one free-throw attempt
  • In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature 4- represents a coxless four and 4+ represents a coxed four.
  • In football , number 4 is traditionally assigned to a defensive midfielder.
  • In ice hockey,
    • the "4 hole" is the space between a goaltender's glove-side arm and his glove-side leg
  • 4 is the retired jersey number of Hall of Fame hockey players Jean Béliveau and Bobby Orr
  • In gridiron football codes,
    • four points are awarded in a handful of leagues for rarely attempted types of field goals (an example is in six-man football, because of the difficulties of getting a successful kick due to the few players on the field, a field goal is worth four points)
  • the "four hole" in offense terminology is the space between the right guard and the right tackle on the offensive line
    • the "four back" is an extra running back (outside the fullback and halfback, often referred to as an H-back) in the backfield; e.g. a play call for a "44 lead" indicates the H-back will follow the fullback into the hole between the right guard and the right tackle.

In other fields

See also '4 .
  • In Claude E. Shannon's information theory, there are four stages in which a signal is properly generated, converted, or otherwise processed: the source, the encoding, the decoding, and the recipient or destination. There is also the channel between encoding and decoding, but in that stage the aim is to avoid modification, since there any modification amounts to noise.
  • The phrase "four-letter word" is used to describe most swear words in the English language, as most swear words do indeed possess four letters.
  • Four (四, formal writing: 肆, pinyin sì) is considered an unlucky number in Chinese, Korean, and Japanese cultures because it sounds like the word "death" (死, pinyin sǐ). Due to that, many numbered product lines skip the "four": e.g. Nokia cell phones (there is no series beginning with a 4), Microsoft Windows (which switched to “Windows 95” for version 4), Palm PDAs, the Leisure Suit Larry games, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture.
  • The number of characters in a canonical four-character idiom.
  • In the NATO phonetic alphabet, the digit 4 is called "fower".
  • In Astrology, Cancer is the 4th astrological sign of the Zodiac.
  • Arthur Conan Doyle wrote a book titled "The Sign of the Four"
  • In Tetris, every shape in the game is formed of 4 blocks each. Also the game was named after "tetra" the Greek word for 4.
  • In the English language, four is the only number with the same number of letters as its value.
  • 4 is one of The Numbers - 4, 8, 15, 16, 23, and 42 - featured in Lost.
  • 4 represents the number of Justices on the Supreme Court of the United Statesmarker necessary to grant a writ of certiorari (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision).
  • A word game exists in which participants trace a word back to the number four, usually by counting the letters in the word, then counting the number of letters in that number, and so on. For instance, "Wikipedia" contains nine letters. The word "nine," in turn, has four letters. The word "four" has four letters, and at this point the chain ends, or at least repeats indefinitely.

In music

Groups of four

See also


  1. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 48
  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): 394, Fig. 24.64

  • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55 - 58

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