Definitions

# 4 (number)

4
Cardinal 4
four
Ordinal 4th
fourth
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
Ge'ez
Bengali
Chinese numeral 四，亖，肆
Devanagari
Hebrew ארבע (Arba, pronounced are-buh) or ד (Dalet, 4th letter of the Hebrew alphabet)
Khmer
Thai
Binary 100
Octal 4
Duodecimal 4
Vigesimal 4
This article discusses the number Four. For the year 4 AD, see 4. For other uses of 4, see 4 (disambiguation)

4 (four) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. Four is the only number in the English language for which the number of letters in its name is equal to the number itself. This is also true in several other languages, such as German, Afrikaans, Dutch, and Flemish, as "vier".

## 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 ($p^2$) 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 $\left(n - 1\right)! equiv 0 \left(\left\{rm mod\right\} n\right)$ 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, $2 uparrowuparrow 2 = 2 uparrowuparrowuparrow 2 = 4$, 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 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.

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
$4 div x$ 4 2 $1.overline\left\{3\right\}$ 1 0.8 $0.overline\left\{6\right\}$ $0.overline\left\{5\right\}7142overline\left\{8\right\}$ 0.5 $0.overline\left\{4\right\}$ 0.4 $0.overline\left\{3\right\}overline\left\{6\right\}$ $0.overline\left\{3\right\}$ $0.overline\left\{3\right\}0769overline\left\{2\right\}$ $0.overline\left\{2\right\}8571overline\left\{4\right\}$ $0.2overline\left\{6\right\}$
$x div 4$ 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

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 enough, but writing four lines proved tiresome. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like our modern plus sign. The Sunga and other Indians 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 at best a secondary concern. The Arabs didn't have time for cursive fancy: their 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 off 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 .

On the seven-segment displays of pocket calculators and digital watches, 4 is seen with an open top.

## In religion

Judeo-Christian symbolism

## In science

### 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.

## 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").
• Aristotle held that there are basically four causes in nature: the efficient cause, the matter, the end, and the form.
• 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.
• C.S. Peirce, usually a trichotomist, discussed four basic methods of seeking to settle questions and arrive at firm beliefs: the method of tenacity (sticking to that which one is inclined to think), the method of authority, the a priori method, and the method of science. (See "The Fixation of Belief," 1877, Eprint)
• Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: objects, attributes, kinds, and modes (by "modes" he means 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 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 sports

• In the sport of cricket, a four is a specific type of scoring event, 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. Also, The term Final Four refers to the last four teams remaining in the NCAA playoff tournament. If a player completes a three-pointer while being fouled, the player is awarded one free-throw for a possible 4-point-play.
• In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature 4- reprsents a coxless four and 4+ represents a coxed four.
• In football (soccer), number 4 is often assigned to a centre back.

## In Music

In Western Music, common time is constructed of four beats.

## References

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