# 4

 ← 3 4 5 →
Cardinalfour
Ordinal4th
(fourth)
Numeral systemquaternary
Factorization22
Divisors1, 2, 4
Greek numeralΔ´
Roman numeralIV, iv
Greek prefixtetra-
Binary1002
Ternary113
Senary46
Octal48
Duodecimal412
Arabic, Kurdish٤
Persian, Sindhi۴
Shahmukhi, Urdu۴
Ge'ez
Bengali, Assamese
Chinese numeral四，亖，肆
Devanagari
Telugu
Malayalam
Tamil
Hebrewד
Khmer
Thai
Burmese

4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures.

## In mathematics

Four is the smallest composite number, its proper divisors being 1 and 2. Four is the sum and product of two with itself: $2+2=4=2\times 2$ , the only number b such that $a+a=b=a\times a$ , which also makes four the smallest squared prime number $2^{2}$ . In Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so forth, for any number of up arrows. By consequence, four is the only square one more than a prime number, specifically three. The sum of the first four prime numbers two + three + five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other hand, the square of four ($4^{2}$ ), equivalently the fourth power of two ($2^{4}$ ), is sixteen; the only number that has $a^{b}=b^{a}$ as a form of factorization. Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (), multiplication (×), and division (÷); and four basic number systems, the real numbers $\mathbb {R}$ , rational numbers $\mathbb {Q}$ , integers $\mathbb {Z}$ , and natural numbers $\mathbb {N}$ .

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. $4x=y^{2}-z^{2}$ . A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because 92 = 4 × 23.

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.

There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal.

A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square.

Four is the highest degree general polynomial equation for which there is a solution in radicals.

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.

A solid figure with four faces as well as four vertices is a tetrahedron, which is the smallest possible number of faces and vertices a polyhedron can have. The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid. It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron. Tetrahedra can be inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb.

Four-dimensional space is the highest-dimensional space featuring more than three regular convex figures:

• Two-dimensional: infinitely many regular polygons.
• Three-dimensional: five regular polyhedra; the five Platonic solids which are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
• Four-dimensional: six regular polychora; the 5-cell, 8-cell or tesseract, 16-cell, 24-cell, 120-cell, and 600-cell. The 24-cell, made of regular octahedra, has no analogue in any other dimension; it is self-dual, with its 24-cell honeycomb dual to the 16-cell honeycomb.
• Five-dimensional and every higher dimension: three regular convex $n$ -polytopes, all within the infinite family of regular $n$ -simplexes, $n$ -hypercubes, and $n$ -orthoplexes.

The fourth dimension is also the highest dimension where regular self-intersecting figures exist:

• Two-dimensional: infinitely many regular star polygons.
• Three-dimensional: four regular star polyhedra, the regular Kepler-Poinsot star polyhedra.
• Four-dimensional: ten regular star polychora, the Schläfli–Hess star polychora. They contain cells of Kepler-Poinsot polyhedra alongside regular tetrahedra, icosahedra and dodecahedra.
• Five-dimensional and every higher dimension: zero regular star-polytopes; uniform star polytopes in dimensions $n$ > $4$ are the most symmetric, which mainly originate from stellations of regular $n$ -polytopes.

Altogether, sixteen (or 16 = 42) regular convex and star polychora are generated from symmetries of four (4) Coxeter Weyl groups and point groups in the fourth dimension: the $\mathrm {A} _{4}$ simplex, $\mathrm {B} _{4}$ hypercube, $\mathrm {F} _{4}$ icositetrachoric, and $\mathrm {H} _{4}$ hexacosichoric groups; with the $\mathrm {D} _{4}$ demihypercube group generating two alternative constructions.

There are also sixty-four (or 64 = 43) four-dimensional Bravais lattices, and sixty-four uniform polychora in the fourth dimension based on the same $\mathrm {A} _{4}$ , $\mathrm {B} _{4}$ , $\mathrm {F} _{4}$ and $\mathrm {H} _{4}$ Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. There are also two infinite families of duoprisms and antiprismatic prisms in the fourth dimension.

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. An alternating groups are not simple for values $n$ $4$ .

There are four Hopf fibrations of hyperspheres:

{\begin{aligned}S^{0}&\hookrightarrow S^{1}\to S^{1},\\S^{1}&\hookrightarrow S^{3}\to S^{2},\\S^{3}&\hookrightarrow S^{7}\to S^{4},\\S^{7}&\hookrightarrow S^{15}\to S^{8}.\\\end{aligned}} They are defined as locally trivial fibrations that map $f:S^{2n-1}\rightarrow S^{n}$ for values of $n=2,4,8$ (aside from the trivial fibration mapping between two points and a circle).

Further extensions of the real numbers under Hurwitz's theorem states that there are four normed division algebras: the real numbers $\mathbb {R}$ , the complex numbers $\mathbb {C}$ , the quaternions $\mathbb {H}$ , and the octonions $\mathbb {O}$ . Under Cayley–Dickson constructions, the sedenions $\mathbb {S}$ constitute a further fourth extension over $\mathbb {R}$ . The real numbers are ordered, commutative and associative algebras, as well as alternative algebras with power-associativity. The complex numbers $\mathbb {C}$ share all four multiplicative algebraic properties of the reals $\mathbb {R}$ , without being ordered. The quaternions loose a further commutative algebraic property, while holding associative, alternative, and power-associative properties. The octonions are alternative and power-associative, while the sedenions are only power-associative. The sedenions and all further extensions of these four normed division algebras are solely power-associative with non-trivial zero divisors, which makes them non-division algebras. $\mathbb {R}$ has a vector space of dimension 1, while $\mathbb {C}$ , $\mathbb {H}$ , $\mathbb {O}$ and $\mathbb {S}$ work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively.

## 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 × 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 ÷ x 4 2 1.3 1 0.8 0.6 0.571428 0.5 0.4 0.4 0.36 0.3 0.307692 0.285714 0.26 0.25
x ÷ 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 14 15 16
4x 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456 1073741824 4294967296
x4 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625 65536

## Evolution of the Hindu-Arabic digit

Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the 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 digit less cursive, ending up with a digit very close to the original Brahmin cross.

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

On 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.

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".

## In politics

• Four Freedoms: four fundamental freedoms that Franklin D. Roosevelt declared ought to be enjoyed by everyone in the world: Freedom of Speech, Freedom of Religion, Freedom from Want, Freedom from Fear.
• Gang of Four: Popular name for four Chinese Communist Party leaders who rose to prominence during China's Cultural Revolution, but were ousted in 1976 following the death of Chairman Mao Zedong. Among the four was Mao's widow, Jiang Qing. Since then, many other political factions headed by four people have been called "Gangs of Four".

## In science

### In chemistry

• Valency of carbon (that is basis of life on the Earth) is four. Also because of 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.

## 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" or "lord of the four quarters of the earth", which is understood to refer 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 separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (both may be true, but not that both are false); contraries (both may be false, but not that both are true); or contradictories (it is not possible that both are true, and it is not possible that both are false).
• Aristotle held that there are basically four causes in nature: the material, the formal, the efficient, and the final.
• 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 methods for overcoming troublesome uncertainties and achieving secure beliefs: (1) the method of tenacity (policy of sticking to initial belief), (2) the method of authority, (3) the method of congruity (following a fashionable paradigm), and (4) the fallibilistic, self-correcting 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 unknowable; (3) maintaining that something is 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, spatiotemporally 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 (military strategist) 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)
• Four opposed camps of the morality and nature of evil: moral absolutism, amoralism, moral relativism, and moral universalism.

## In sports

• In the Australian Football League, the top level of Australian rules football, each team is allowed 4 "interchanges" (substitute players), who can be freely substituted at any time, subject to a limit on the total number of substitutions.
• In baseball:
• There are four bases in the game: first base, second base, third base, and home plate; to score a run, an offensive player must complete, in the sequence shown, a circuit of those four bases.
• When a batter receives four pitches that the umpire declares to be "balls" in a single at-bat, a base on balls, informally known as a "walk", is awarded, with the batter sent to first base.
• For scoring, number 4 is assigned to the second baseman.
• Four is the most runs that can be scored on any single at bat, whereby all three baserunners and the batter score (the most common being via a grand slam).
• The fourth batter in the batting lineup is called the cleanup hitter.
• In basketball, the number four is used to designate the power forward position, often referred to as "the four spot" or "the four".
• In 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 American Football teams get four downs to reach the line of gain.
• 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 rugby league:
• A try is worth 4 points.
• One of the two starting centres wears the jersey number 4. (An exception to this rule is the Super League, which uses static squad numbering.)
• In rugby union:
• One of the two starting locks wears the jersey number 4.
• In the standard bonus points system, a point is awarded in the league standings to a team that scores at least 4 tries in a match, regardless of the match result.