4 (number)

 4 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 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 ፬ Bengali ৪ Chinese numeral 四，亖，肆 Devanagari ४ Hebrew ארבע (Arba, pronounced are-buh) or ד (Dalet, 4th letter of the Hebrew alphabet) Khmer ៤ Thai ๔ prefixes 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 that represents the number. It is the natural number[1] that follows 3 and precedes 5. It is an integer and a cardinal number, that is, a number that is used for counting.[2] In addition, it is classified as a real number,[3] distinguishing it from imaginary numbers.

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 changed the numeral to a point where speed of writing was at best a secondary concern. The Arabs did not 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.[4]

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, for example, in .

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

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 (a "square number" is an integer that can be written as the square of some other integer) and the second centered triangular number. (A "centered triangular number" is the total number of dots involved when a triangle is constructed with a dot in the center and all other dots surrounding the center in successive triangular layers.)

Four is the smallest squared prime ($p^2$). It has an aliquot sum (the aliquot sum of a number n is the sum of the proper divisors, that is, the divisors excluding the number n itself) of 3 which is itself prime. The aliquot sequence (an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term) of 4 has 4 members (4,3,1,0).

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.

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

In science

Astronomy

• Four terrestrial (or rocky) planets in the Solar System: Mercury, Venus, Earth, Mars.
• The orbits of four planets in the solar system lie within that of the asteroid belt.
• Four giant gas planets in the Solar system: Jupiter, Saturn, Uranus, Neptune.
• Messier object M4, a magnitude 7.5 globular cluster in the constellation Scorpius.
• The Saros number[5] of the solar eclipse series that began on May 6, 2731 B.C.E. and ended on June 13, 1451 B.C.E. The duration of Saros series 4 was 1280.1 years, and it contained 72 solar eclipses.
• The Saros number[6] of the lunar eclipse series that began on September 3, 2700 B.C.E. and ended on April 2, 1131 B.C.E. The duration of Saros series 4 was 1568.6 years, and it contained 88 lunar eclipses.
• The Roman numeral IV (usually) stands for the fourth-discovered satellite of a planet or minor planet (for example, Jupiter IV).
• The Roman numeral IV also stands for subgiant in the Yerkes spectral classification scheme.

Chemistry

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

In technology

• Four is the resin identification code used in recycling to identify low-density polyethylene.
• Usually, each item of furniture has four legs. Examples include tables and chairs.
• Rectangles (with four angles and four sides) are widely used in making things because they have effective form and capability for close adjacency to each other. Examples include houses, rooms, tables, bricks, sheets of paper, screens, film frames.
• The four color process (CMYK) is used for printing.
• 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.
• Four is the number of bits in a nibble, equivalent to half a byte.

Transport

• Most vehicles, including motor vehicles, and particularly cars/automobiles and light commercial vehicles have four road wheels.
• "Quattro," meaning four in the Italian language, is used by Audi as a trademark to indicate that four-wheel drive (4WD) technologies are used on Audi-branded cars. The word "Quattro" was initially used by Audi, in 1980, in its original 4WD coupé, the Audi Quattro. Audi also has a privately held subsidiary company called quattro GmbH.

In religion

Islam

• The Four Archangels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)

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," 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.[7]
• 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").[8]

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 other fields

International maritime signal flag for 4

• Four is the only number in the English language for which the number of letters in its name is equal to the number itself.
• Four (四, formal writing: 肆, pinyin sì) is considered an unlucky number in Chinese and Japanese cultures because it sounds like the word "death" (死, pinyin sǐ). Due to that, many numbered product lines skip the "four:" For example, Nokia cell phones (there is no series beginning with a 4), Palm PDAs, the Leisure Suit Larry games. Some buildings skip floor 4 or replace the number with the letter "F," particularly in heavily Asian areas.
• In the NATO phonetic alphabet, the digit 4 is called "fower."
• The number of characters in a canonical four-character idiom.
• In Astrology, Cancer is the 4th astrological sign of the Zodiac.
• 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 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).
• A four-letter word is used to describe most swear words in the English language, as most such terms do indeed possess four letters.

Groups of four

• Four rules in arithmetic: Addition, subtraction, multiplication, division.
• Classical Greek elements: Fire, air, water, earth.
• Four seasons: Spring, summer, autumn, winter.
• Four parts of a day: Morning, afternoon, evening, night.
• Four cardinal directions: North, south, east, west.
• Four Temperaments: Sanguine, choleric, melancholic, phlegmatic.
• Four suits of playing cards: Hearts, diamonds, clubs, spades.
• Four Corners is the only location in the United States where four states come together at a single point: Colorado, Utah, New Mexico, and Arizona.

Notes

1. A natural number is any number that is a positive integer, such as 1, 2, 3, 4, and so forth. Often, the number 0 is also called a natural number.
2. A cardinal number indicates the quantity of things, but not the order in which they occur. By contrast, ordinal numbers are first, second, third, and so on, indicating their positions in a series.
3. A real number is a number that can be given by a finite or infinite decimal representation. The term "real number" was coined to distinguish it from an "imaginary number." The set of real numbers includes rational and irrational numbers, which can be positive, negative, or zero.
4. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated by David Bellos et al. (London: The Harvill Press), 394.
5. NASA, Solar Eclipses of Saros 0 to 180. Retrieved August 19, 2008.
6. Fred Espenak, Lunar Eclipses of Saros Series 1 to 175, NASA. Retrieved August 19, 2008.
7. Charles S. Peirce, The Fixation of Belief, Popular Science Monthly. Retrieved August 20, 2008.
8. E.J. Lowe, Recent Advances in Metaphysics, Dept. of Computer Science, Vassar College. Retrieved August 20, 2008.

References

• Flegg, Graham. 2002. Numbers: Their History and Meaning. Mineola, NY: Dover Publications. ISBN 0486421651.
• Ifrah, Georges. 2000. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos et al. New York: Wiley. ISBN 0471393401.
• McLeish, John. 1994. The Story of Numbers: How Mathematics Has Shaped Civilization. New York: Fawcett Columbine. ISBN 0449909387.
• Menninger, Karl. 1992. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover Publications. ISBN 0486270963.
• Wells, D. G. 1998. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books. ISBN 0140261494.