# 4 (number)

{{#invoke:Hatnote|hatnote}}Template:Main other Template:Example farm Template:Infobox number 4 (four; Template:IPAc-en) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. Four is the only number that has the same number of characters as its value in the English language.[1]

## In mathematics

Four is the smallest composite number, its proper divisors being 1 and 2. Four is also a highly composite number. The next highly composite number is 6.

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

4 is the smallest squared prime (p2) 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.

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.

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 only composite number n for which ${\displaystyle (n-1)!\ \equiv \ 0\ ({\rm {mod}}\ n)}$ is false.

It is also a Motzkin number.

In bases 6 and 12, 4 is a 1-automorphic number.

In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, ${\displaystyle 2\uparrow \uparrow 2=2\uparrow \uparrow \uparrow 2=4}$, 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 (mathematics). 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 is the highest-dimensional space featuring more than three convex regular figures:

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on ${\displaystyle \mathbb {R} ^{n}}$ 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.[2] 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 = y2z2.

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

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

## 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
${\displaystyle 4\times x}$ 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
${\displaystyle 4\div x}$ 4 2 ${\displaystyle 1.{\overline {3}}}$ 1 0.8 ${\displaystyle 0.{\overline {6}}}$ ${\displaystyle 0.{\overline {571428}}}$ 0.5 ${\displaystyle 0.{\overline {4}}}$ 0.4 ${\displaystyle 0.{\overline {36}}}$ ${\displaystyle 0.{\overline {3}}}$ ${\displaystyle 0.{\overline {307692}}}$ ${\displaystyle 0.{\overline {285714}}}$ ${\displaystyle 0.2{\overline {6}}}$ 0.25
${\displaystyle x\div 4}$ 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
${\displaystyle 4^{x}\,}$ 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
${\displaystyle x^{4}\,}$ 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

## Origins

Representing 1, 2 and 3 in as many lines as the number represented worked well{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}. 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.[3] 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 . 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 religion Buddhism Judeo-Christian symbolism Hinduism Islam • Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja. • There are four Rashidun or Rightly Guided Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib. • The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael) • There are four Sacred Months in Islam: Muharram, Rajab, Dhu al-Qi'dah and Dhu al-Hijjah. • There are four Sunni schools of fiqh: Hanafi, Shafii, Maliki and Hanbali. • There are four major sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafii, Malik ibn Anas and Ahmad ibn Hanbal. • There are four books in Islam: Torah, Zaboor, Injeel, Quran. • Waiting for four months is ordained for those who take an oath for abstention from their wives. • The waiting period of the woman whose husband dies, is four months and ten days. • When Abraham said: “My Lord, show me how You give life to the dead,” Allah said: “Why! Do you have no faith?” Abraham replied: “Yes, but in order that my heart be at rest.” He said: “Then take four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will come to you flying. • The respite of four months was granted to give time to the mushriks in Surah Taubah so that they should consider their position carefully and decide whether to make preparation for war or to emigrate from the country or to accept Islam. • Those who accuse honourable women (of unchastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after. They are indeed transgressors. Other ## 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. 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 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"...[4] 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 separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (it is possible that both are true, but not that both are false); contraries (it is possible that both are 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 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 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 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 (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) ## In technology {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}. • 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".