4 (four) is a
number,
numeral and
digit. It is the
natural number following
3 and preceding
5. It is 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 and . Four is the sum and product of
two
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultu ...
with itself:
+
=
=
x
, the only number
such that
+
=
=
x
, which also makes four the smallest squared
prime number . In
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperat ...
, , 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
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultu ...
+
three +
five
5 is a number, numeral, and glyph.
5, five or number 5 may also refer to:
* AD 5, the fifth year of the AD era
* 5 BC, the fifth year before the AD era
Literature
* ''5'' (visual novel), a 2008 visual novel by Ram
* ''5'' (comics), an awa ...
+
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
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
,
three and
five
5 is a number, numeral, and glyph.
5, five or number 5 may also refer to:
* AD 5, the fifth year of the AD era
* 5 BC, the fifth year before the AD era
Literature
* ''5'' (visual novel), a 2008 visual novel by Ram
* ''5'' (comics), an awa ...
, 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
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
of two 2
4, is
sixteen; the only number that has
=
as a form of
factorization. Holistically, there are four elementary arithmetic
operations in mathematics:
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
(+),
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 .
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; for instance cannot be written as the sum of three squares.
There are four
all-Harshad number
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base.
Harshad numbers in base are also known as -harshad (or -Niven) numbers.
Harshad number ...
s:
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
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
,
rhombus, and
square.
Four is the highest degree general
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic 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
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
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
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
Other names i ...
.
Four-dimensional space is the highest-dimensional space featuring more than three
regular convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polyto ...
figures:
*Two-dimensional: infinitely many
regular polygons.
*Three-dimensional: five
regular polyhedra
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equiv ...
; the five
Platonic solids which are the
tetrahedron,
cube,
octahedron,
dodecahedron, and
icosahedron.
*Four-dimensional: six
regular polychora; the
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It ...
, 8-cell or
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
,
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the m ...
,
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octa ...
,
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, he ...
, and
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from " ...
. The
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octa ...
, made of regular
octahedra
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, has no analogue in any other dimension; it is
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
, with its
24-cell honeycomb dual to the
16-cell honeycomb.
*Five-dimensional and every higher dimension: three regular convex
-
polytopes, all within the infinite family of regular
-
simplexes,
-
hypercubes, and
-
orthoplexes.
The fourth dimension is also the highest dimension where regular
self-intersecting figures exist:
*Two-dimensional: infinitaly many regular
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
s.
*Three-dimensional: ''four'' regular
star polyhedra
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
There are two general kinds of star polyhedron:
*Polyhedra which self-intersect in a repetitive way.
*Concave ...
, the
regular Kepler-Poinsot star polyhedra.
*Four-dimensional: ten regular
star polychora, the
Schläfli–Hess star polychora. They contain
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
of Kepler-Poinsot polyhedra alongside regular tetrahedra,
icosahedra
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
and
dodecahedra
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
.
*Five-dimensional and every higher dimension: zero regular
star-polytopes;
uniform star polytopes in dimensions
>
are the most symmetric, which mainly originate from
stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...
s of regular
-polytopes.
Altogether,
sixteen (or 16 = 4
2) regular convex and star polychora are generated from symmetries of ''four'' (4)
Coxeter Weyl groups and
point groups
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
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 = 4
3) four-dimensional
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s, ''and'' sixty-four
uniform polychora in the fourth dimension based on the same
,
,
and
Coxeter groups
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
, and extending to
prismatic groups of
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also f ...
, 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
except when
=
, in which case there are uncountably many.
The smallest non-
cyclic group has four elements; it is the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
. ''A''
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s are not
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
for values
≤
.
Further extensions of the real numbers under
Hurwitz's theorem states that there are four
normed division algebras: the real numbers
, the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, the
quaternions
, and the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
s
. Under
Cayley–Dickson construction
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by t ...
s, the
sedenion
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic t ...
s
constitute a further fourth extension over
. The real numbers are
ordered,
commutative and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, as well as
alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is ...
s with
power-associativity In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra ge ...
. 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 field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s of dimensions 2, 4, 8, and 16, respectively.
List of basic calculations
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
is a type of Japanese erotic art typically executed as a kind of ukiyo-e, often in woodblock print format. While rare, there are also extant erotic painted handscrolls which predate ukiyo-e. Translated literally, the Japanese word ''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
Arab
The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, No ...
s' 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
Optical character recognition or optical character reader (OCR) is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a sc ...
fonts, 4 is seen with an open top.
Television station
A television station is a set of equipment managed by a business, organisation or other entity, such as an amateur television (ATV) operator, that transmits video content and audio content via radio waves directly from a transmitter on the earth ...
s that operate on
channel 4
Channel 4 is a British free-to-air public broadcast television network operated by the state-owned Channel Four Television Corporation. It began its transmission on 2 November 1982 and was established to provide a fourth television service in ...
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
Canadian syllabic writing, or simply syllabics, is a family of writing systems used in a number of Indigenous Canadian languages of the Algonquian, Inuit, and (formerly) Athabaskan language families. These languages had no formal writing sy ...
letter ᔦ. The
magnetic ink character recognition
Magnetic ink character recognition code, known in short as MICR code, is a character recognition technology used mainly by the banking industry to streamline the processing and clearance of cheques and other documents. MICR encoding, called the ' ...
"CMC-7" font also uses this variety of "4".
In religion
Buddhism
*
Four Noble Truths
In Buddhism, the Four Noble Truths (Sanskrit: ; pi, cattāri ariyasaccāni; "The four Arya satyas") are "the truths of the Noble Ones", the truths or realities for the "spiritually worthy ones". _–_Dukkha">Four_Noble_Truths:_BUDDHIST_PHILOSOPHY_Encycl_...
_–_Dukkha">Four_Noble_Truths:_BUDDHIST_PHILOSOPHY_Encycl_...