| ||||
---|---|---|---|---|
Cardinal | four | |||
Ordinal | 4th (fourth) | |||
Numeral system | quaternary | |||
Factorization | 22 | |||
Divisors | 1, 2, 4 | |||
Greek numeral | Δ´ | |||
Roman numeral |
| |||
Greek prefix | tetra- | |||
Latin prefix | quadri-/quadr- | |||
Binary | 1002 | |||
Ternary | 113 | |||
Senary | 46 | |||
Octal | 48 | |||
Duodecimal | 412 | |||
Hexadecimal | 416 | |||
Armenian | Դ | |||
Arabic, Kurdish | ٤ | |||
Persian, Sindhi | ۴ | |||
Shahmukhi, Urdu | ۴ | |||
Ge'ez | ፬ | |||
Bengali, Assamese | ৪ | |||
Chinese numeral | 四,亖,肆 | |||
Devanagari | ४ | |||
Telugu | ౪ | |||
Malayalam | ൪ | |||
Tamil | ௪ | |||
Hebrew | ד | |||
Khmer | ៤ | |||
Thai | ๔ | |||
Kannada | ೪ | |||
Burmese | ၄ | |||
Babylonian numeral | 𒐘 | |||
Egyptian hieroglyph, Chinese counting rod | |||| | |||
Maya numerals | •••• | |||
Morse code | .... _ |
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.
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. [1]
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: . [2]
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". [3]
Four is the smallest composite number, its proper divisors being 1 and 2. [4] Four is the sum and product of two with itself: , the only non-zero number such that , which also makes four the smallest and only even squared prime number and hence the first squared prime of the form , where is a prime. Four, as the first composite number, has a prime aliquot sum of 3; and as such it is part of the first aliquot sequence with a single composite member, expressly (4, 3, 1, 0). It is the smallest non-unitary tetrahedral number. [5]
Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (−), multiplication (×), and division (÷); and four basic number systems, the real numbers , rational numbers , integers , and natural numbers .
Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. . 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). [7]
Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. [8] Three are not always sufficient; 7 for instance cannot be written as the sum of three squares. [9]
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. [10]
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. [11] Three colors are not, in general, sufficient to guarantee this. [12] The largest planar complete graph has four vertices. [13]
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. [14] [15] The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid. [16] It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron. [17] Tetrahedra can be inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb.
The third dimension holds a total of four Coxeter groups that generate convex uniform polyhedra: the tetrahedral group, the octahedral group, the icosahedral group, and a dihedral group (of orders 24, 48, 120, and 4, respectively). There are also four general Coxeter groups of generalized uniform prisms, where two are hosoderal and dihedral groups that form spherical tilings, with another two general prismatic and antiprismatic groups that represent truncated hosohedra (or simply, prisms) and snub antiprisms, respectively.
Four-dimensional space is the highest-dimensional space featuring more than three regular convex figures:
The fourth dimension is also the highest dimension where regular self-intersecting figures exist:
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 simplex, hypercube, icositetrachoric, and hexacosichoric groups; with the demihypercube group generating two alternative constructions. There are also sixty-four (or 64 = 43) four-dimensional Bravais lattices, alongside sixty-four uniform polychora in the fourth dimension based on the same , , and Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. Two infinite families of duoprisms and antiprismatic prisms exist in the fourth dimension.
There are only four polytopes with radial equilateral symmetry: the hexagon, the cuboctahedron, the tesseract, and the 24-cell.
Four-dimensional differential manifolds have some unique properties. There is only one differential structure on except when = , in which case there are uncountably many.
The smallest non-cyclic group has four elements; it is the Klein four-group. [18] An alternating groups are not simple for values ≤ .
There are four Hopf fibrations of hyperspheres:
They are defined as locally trivial fibrations that map for values of (aside from the trivial fibration mapping between two points and a circle). [19]
Further extensions of the real numbers under Hurwitz's theorem states that there are four normed division algebras: the real numbers , the complex numbers , the quaternions , and the octonions . Under Cayley–Dickson constructions, the sedenions constitute a further fourth extension over . The real numbers are ordered, commutative and associative algebras, as well as alternative algebras with power-associativity. The complex numbers share all four multiplicative algebraic properties of the reals , 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. has a vector space of dimension 1, while , , and work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively.
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 |
In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of the Platonic solid and of the deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, .
In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative and has an identity element, and every element of the set has an inverse element.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other.
23 (twenty-three) is the natural number following 22 and preceding 24.
63 (sixty-three) is the natural number following 62 and preceding 64.
In mathematics, the circle group, denoted by or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
14 (fourteen) is a natural number following 13 and preceding 15.
The smallest composite number is 4.
2 ↑↑ ... ↑↑ 2 is always 4
An integer is divisible by 4 if the last two digits form a multiple of 4.
7 is an example of an integer that can't be written as the sum of three squares.
There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher.
(i.e. That there are maps for which three colors are not sufficient)
... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4.
...the smallest possible number of faces that a polyhedron may have is four
...face of the platonic solid. The simplest of these shapes is the tetrahedron...
...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs...
The Klein four-group is the smallest noncyclic group,...
The four main pilgrimages sites are: Lumbini, Bodh Gaya, Sarnath and Kusinara....four Noble Truths of Buddhism
He first observed the suffering of the world in the Four Passing Sites
The four great elements, earth, water, fire and wind...
The Buddhists adopted him as one of the four Devarajas or Heavenly Kings
The four right exertions are...
these four bases of psychic power
This book is about the four jhanas
...the states of the four arupajhanas.
There are four of them: loving-kindness, metta, compassion, karuna, sympathetic joy, mudita and equanimity, upekkha.
...four types of shravaka (stream enterer, oncereturner, nonreturner, and arhat)
We have already mentioned the four living creatures—the man, the lion, the ox and the eagle
The four horsemen of the Apocalypse are one of the most familiar images of Revelation
...as well as to the palm ( lulav ), myrtle ( hadas ), willow ( aravah ) and citron ( etrog ), the four species of plants
...be like Sarah, Rachel, Rebecca, and Leah, the foremothers of Judaism
The Passover Seder is particularly structured around fours: the Four Questions, the Four Sons, and four cups of wine.
There are four expressions of redemption in the Torah
The four holy cities of Judaism are Jerusalem, Hebron, Safed, and Tiberius.
There are four Vedas
that these four proper aims and objects
The Four Stages of Life
The four primary castes or strata of society:...
Brahma has four faces,...
...Eid al-Adha (Feast of Sacrifice) lasts four days ...
... four Rightly Guided Caliphs, Abu-Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib,...
According to Islam, the Four Arch Angels are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael).
The sacred months are four, Rajab, Dhu al-Qi'dah, Dhu al-Hijjah, and al-Muharram. During those four sacred months there were no war...
There are four books in Islam: Torah, Zaboor, Injeel and Holy Qur'an...
For those who take an oath for abstention from their wives, awaiting for four months is ordained;
...for four months and ten days.
Then take four birds, ...
The respite of four months...
And those who launch a charge against chaste women and do not produce four witnesses...
Taoism later incorporated the four symbols into its immortality system...
the four corners or extremities of the earth (Isa. xi, 12; Ezek. vii, 2.; Rev. vii, 1; xx, 8), corresponding, doubtless, with the four points of the compass
Four was a sacred number of Zia
In Chinese, Japanese, and Korean, the word for four is, unfortunately, an exact homonym for death
Svetovid is portrayed as having four heads ...
[...] The characteristic eight bit field is sometimes referred to as a byte, a four bit field can be referred to as a nibble. [...]
Oligomers containing two, three, four, five, six or even more subunits are known as dimers, trimers, tetramers, pentamers, hexamers, and so on.
The four inner planets (known as terrestrial, or rocky planets
...the gas giants (Jupiter and Saturn), and the icy giants (Uranus and Neptune)
including the four large Galilean moons that are easily visible from a hobby telescope
M4 is a globular star cluster near Antares in Scorpius.
IV, subgiants
The mammalian heart consists of four chambers,...
Except for the flies, mosquitoes, and some others, insects with wings have four wings.
metamorphosis is marked by four distinct stages
In the 'ABO' system, all blood belongs one of four major groups — A, B, AB or O
Four canines for tearing + Eight premolars for crushing +Twelve molars (including four wisdom teeth)
The cow's stomach is divided into four compartments.
Of course, carbon is not the only chemical element with a valence of +4 or -4
Beryllium has an atomic number of four
Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas.
should be regarded as a four-dimensional world
We have referred to the four fundamental forces in nature,...
The Four Functions Theorem of Ahlswede Daykin
This book examines Aristotle's four causes (material, formal, efficient, and final)
The OODA loop consists of four steps.
Plastic Recycling Symbol #4: LDPE
CMYK is the standard four-color model used for all full-color print jobs that will be output on an offset printing press
...the 4 key (labeled with the letters g, h and i)...
A byte also contains two 4-bit nibbles...
Power forward Referred to as the number 4 spot
But one confused re-spelling is fower for 'four'.
Sign: Cancer, the fourth Zodiacal Sign
The 4th Tarot Card is called "The Emperor."
...tetraminos (the shapes used in Tetris) are all just a collection of four blocks
The court will not issue a writ unless at least four justices approve of it. This is called the rule of four.
... called common time and denoted by C, which has four beats per measure
The number, character and sequence of movements in the symphony, moreover, did not stabilize until the 1770s when the familiar format of four movements...
...the four houses of Hogwarts School of Witchcraft and Wizardry: Gryffindor, Ravenclaw, Hufflepuff, and Slytherin
Every year that is divisible by four, except the Centennial years, and every Centennial year divisible by 400, is a leap year...
Each of the familiar cardinal directions is equivalent to a particular true bearing: north (0°), east (90°), south (180°), and west (270°)
...four substances or humors: blood, yellow bile, black bile and phlegm
The four playing card suits, as you probably already know, are spades, hearts, diamonds, and clubs