4
Four is the smallest composite natural number following 1, 2, and 3, and it is the square of the smallest even prime number, 2 (i.e., $2^2 = 4).[1] As an even integer, it represents the first even perfect square and appears as the smallest even leg in a primitive Pythagorean triple (3, 4, 5).[1] In number theory, 4 is the highest degree for which general algebraic equations are solvable by radicals, and it is the smallest order of a finite field that is not a prime field, as well as the smallest order possessing two non-isomorphic groups: the Klein four-group (C_2 \times C_2) and the cyclic group (C_4).[1] Beyond mathematics, 4 holds fundamental significance in physics and chemistry. In relativity, spacetime is modeled as a four-dimensional continuum, combining three spatial dimensions with one time dimension, as conceptualized by Albert Einstein.[2] In quantum mechanics, electrons in atoms are described by four quantum numbers: the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s), which fully specify an electron's state.[3] Additionally, beryllium (Be) is the chemical element with atomic number 4, a lightweight alkaline earth metal used in alloys and nuclear applications. Culturally and biologically, 4 symbolizes completeness and stability across civilizations. In ancient Pythagorean tradition, it represented justice as the fourth of the first ten numbers, forming the sacred tetraktys (1+2+3+4=10), a triangular figure embodying cosmic harmony.[1] Biologically, humans and many animals can subitize—instinctively recognize—up to four objects without counting, a cognitive limit observed in species like honeybees and linked to evolutionary survival advantages in threat assessment.[4] In religion, Hinduism depicts the creator god Brahma with four faces symbolizing the four Vedas, while Christianity's four canonical Gospels (Matthew, Mark, Luke, and John) evoke the four corners of the heavenly Jerusalem; Islam permits up to four wives in polygamous marriages to support vulnerable women historically.[1] Geometrically, 4 defines the tetrahedron, the simplest Platonic solid with four triangular faces, and prefixes like "tetra-" (Greek) and "quadri-" (Latin) denote multiplicity of four in terms such as tetrominoes or quadrilaterals.[1]Notation and representation
Hindu-Arabic numeral 4
The Hindu-Arabic numeral 4 traces its origins to the Brahmi script used in ancient India, emerging around the mid-3rd century BCE as evidenced in inscriptions from regions near Poona, Bombay, and Uttar Pradesh.[5] This early form served as a distinct symbol for the value 4, possibly adapted for writing on materials like palm leaves or tree bark, and represented a simplification of counting-based notations where lower numbers used stacked lines.[5] Over centuries, it evolved through intermediate scripts: from Brahmi to the Gupta numerals of the 4th–6th centuries CE in northeastern India, and subsequently to the Nagari (or Devanagari) numerals by the 7th century CE, which featured more refined and angular shapes.[5] The transmission to the Islamic world occurred between the 7th and 9th centuries CE, with Persian mathematician Al-Khwarizmi playing a pivotal role in its dissemination through his treatise On the Calculation with Hindu Numerals (circa 825 CE), the earliest known Arabic explanation of the place-value system including the digit 4.[6] Al-Khwarizmi's work, later translated into Latin as Algoritmi de numero Indorum in the 12th century, helped bridge Indian innovations to Arabic scholarship, where the numeral's form began to standardize while retaining regional variations, such as in al-Sijzi's 969 CE treatise from the eastern Arabic empire.[6] By the 11th century, scholar al-Biruni noted the regularity of these "Nagari" numerals in his accounts of Indian mathematics, facilitating their further spread.[5] Upon reaching Europe via Spain around 976 CE—as seen in the Codex Vigilanus—the glyph for 4 exhibited significant variations in medieval manuscripts, often rendered with an open top resembling a cursive cross or angular L-shape without a enclosing triangle, a form prevalent until the 15th century.[7] This open-top style, documented in 10th-century St. Gall manuscripts and 14th-century Vienna Codex 4770, contrasted with the closed-top version that emerged later, featuring a triangular enclosure at the top for greater distinctiveness from similar digits like 9.[7] The advent of printing in the 15th century, exemplified by Johannes Gutenberg's press, accelerated standardization, fixing the closed-top form as the global norm during the Renaissance and enabling widespread adoption in mathematical texts and commerce.[5][8] In modern digital representation, the digit 4 is encoded as U+0034 (DIGIT FOUR) in Unicode, ensuring consistent rendering across systems as a basic Latin block character with decimal value 52. Typographically, it appears in varied styles across fonts, such as old-style figures with descending elements in serif typefaces or proportional lining figures in sans-serif designs; certain fonts, like those supporting OpenType features, include contextual alternates for the 4 to improve legibility when adjacent to curved glyphs, though true ligatures involving digits remain uncommon.[9]Representations in other numeral systems
In ancient Rome, the number 4 was denoted by the subtractive notation IV, where the value of I (1) is subtracted from V (5), a convention that emerged during the late Republic period around the 1st century BCE and coexisted with the additive form IIII. This subtractive principle simplified writing larger numbers and is evident in inscriptions from the classical era, though both forms were used interchangeably until the Middle Ages when IV became more standardized in formal texts. Today, IV appears commonly in outlines, chapter headings, and dates, while IIII persists on clock faces for aesthetic symmetry and tradition, as seen in horological designs since the Renaissance.[10] In traditional Chinese numeration, the standard character for 4 is 四 (sì), but a formal variant 肆 (sì) is employed in financial and official documents to deter forgery due to its structural complexity. This distinction emerged during the Tang dynasty (618–907 CE) or later periods.[11] This system traces back to the Han dynasty (206 BCE–220 CE), when counting rods—bamboo or metal sticks arranged on a board—facilitated calculations in a positional decimal system; the rod representation for 4 consisted of four vertical sticks for units or horizontal ones for tens, enabling merchants and astronomers to perform arithmetic without permanent writing.[11] The ancient Maya employed a vigesimal (base-20) positional system originating in the Preclassic period around the 4th century BCE, where numbers were inscribed using dots for units (1), bars for fives (5), and a shell for zero. Specifically, 4 was represented by four dots aligned horizontally, stacked to form higher values like 9 (four dots over a bar); this iconographic approach allowed for compact notation on stelae and codices, reflecting the culture's astronomical and calendrical precision.[12] In binary, a base-2 system fundamental to digital computing, 4 is expressed as 100, corresponding to $2^2 + 2^1 \cdot 0 + 2^0 \cdot 0 = 4. Hexadecimal, or base-16, denotes 4 simply as 4, utilizing the digits 0–9 followed by A–F for 10–15; the letter digits derive from the Latin alphabet's initial letters, a convention formalized in the mid-20th century for efficient binary data representation in programming and engineering.[13] Among the earliest numeral systems, Sumerian cuneiform from the 3rd millennium BCE used repeated impressions of a wedge-shaped stylus on clay tablets; the sign for 1 was a vertical wedge (𒁹), so 4 appeared as four such wedges aligned vertically (𒁹𒁹𒁹𒁹), part of a sexagesimal (base-60) framework adapted for accounting and administration in Mesopotamian city-states. Similarly, ancient Egyptian hieroglyphs, developed by the 3rd millennium BCE, employed a decimal system where 4 was four vertical strokes (||||), grouped additively for larger quantities in administrative papyri and tomb inscriptions, emphasizing multiples of powers of ten.[14][15]Mathematical properties
As a natural number
Four is the smallest composite natural number, defined as a positive integer greater than 1 that is not prime, having factors other than 1 and itself, specifically as the product of the prime 2 with itself. It is even, meaning it is divisible by 2 without remainder, and it is the square of the smallest prime number, satisfying $2^2 = 4. The positive divisors of 4 are 1, 2, and 4. The proper divisors, excluding 4 itself, are 1 and 2, and the aliquot sum—the sum of these proper divisors—is 3. The prime factorization of 4 is $2^2, indicating it is a power of the prime 2. Euler's totient function evaluates to \phi(4) = 2, counting the positive integers up to but not including 4 that are coprime to 4 (namely, 1 and 3). In group theory, 4 is the smallest positive integer for which there exist two non-isomorphic groups of that order: the cyclic group C_4 and the Klein four-group C_2 \times C_2.[16] In field theory, 4 is the order of the smallest finite field that is not a prime field, namely \mathbb{F}_4, the quadratic extension of the prime field \mathbb{F}_2.[17] In the sequence of prime powers ordered by magnitude, 4 is the third term, following 2 and 3, and preceding 5; it is also the first square of a prime (the square of 2). Regarding parity, 4 is even. In modular arithmetic, $4 \equiv 0 \pmod{4} since 4 divides evenly into itself, and $4 \equiv 1 \pmod{3} as the remainder when 4 is divided by 3 is 1.In geometry and algebra
In geometry, the number 4 prominently features in basic polygonal and polyhedral structures. A square is a quadrilateral with four equal sides and four vertices, forming a regular polygon where all interior angles measure 90 degrees.[18] Similarly, the tetrahedron, one of the five Platonic solids, is a polyhedron with four triangular faces, four vertices, and six edges, representing the simplest regular polyhedron where each face is an equilateral triangle.[19] For all Platonic solids, including the tetrahedron, the Euler characteristic—defined as the number of vertices (V) minus the number of edges (E) plus the number of faces (F)—equals 2, satisfying the relation V - E + F = 2. This topological invariant underscores the spherical topology of these convex polyhedra.[19] In coordinate geometry, the Cartesian plane is divided into four quadrants by the intersecting x- and y-axes, originating at the point (0,0). These quadrants—I (positive x and y), II (negative x, positive y), III (negative x and y), and IV (positive x, negative y)—facilitate the localization and analysis of points in two-dimensional space.[20] A related geometric equation involving 4 is the standard form of a circle centered at the origin with radius 2: x^2 + y^2 = 4, which describes all points at a distance of 2 units from (0,0).[21] Algebraically, 4 appears as the dimension of the quaternion number system, a non-commutative extension of complex numbers discovered by William Rowan Hamilton in 1843. Quaternions, denoted as q = a + bi + cj + dk where i^2 = j^2 = k^2 = ijk = -1, form a four-dimensional division algebra over the real numbers, with applications in three-dimensional rotations.[22] Additionally, the fourth roots of unity are the solutions to z^4 = 1 in the complex plane, given by $1, -1, i, -i, which lie equally spaced on the unit circle at angles of 0°, 90°, 180°, and 270°. These roots generate the cyclic group of order 4 and play a key role in solving quartic equations and Fourier analysis.[23] In algebra, the general polynomial equation of degree 4 (quartic) is solvable by radicals, whereas those of degree 5 or higher are not solvable by radicals in general, according to the Abel–Ruffini theorem; thus, 4 is the largest such degree.[24] The symmetries of geometric figures involving 4 are captured by group theory, particularly the dihedral group D_4, which describes the rotations and reflections of a square. This group has 8 elements: four rotations (0°, 90°, 180°, 270°) and four reflections across the axes of symmetry and diagonals, forming a non-abelian group of order 8 that preserves the square's structure.[25]Theorems and identities involving 4
Lagrange's four-square theorem, proved by Joseph-Louis Lagrange in 1770, states that every natural number can be represented as the sum of four integer squares.[26] The proof relies on the Euler four-square identity, which allows the product of two sums of four squares to be expressed as another sum of four squares, combined with the fact that the prime 2 and all primes congruent to 1 modulo 4 are sums of two squares (extending Fermat's theorem on sums of two squares), while primes congruent to 3 modulo 4 require an additional square.[26] For example, the number 7, which is congruent to 3 modulo 4 and thus not a sum of two squares, can be written as $7 = 2^2 + 1^2 + 1^2 + 1^2.[26] Fermat's theorem on sums of two squares, stated by Pierre de Fermat around 1640 and proved by Leonhard Euler in 1749, asserts that an odd prime p can be expressed as p = x^2 + y^2 with integers x and y if and only if p \equiv 1 \pmod{4}. This result forms a foundation for Lagrange's extension to four squares, as numbers requiring more than two squares under Fermat's criterion can be completed to four using additional 1's or 0's, ensuring universality.[26] The four-color theorem, conjectured by Francis Guthrie in 1852, states that the regions of any planar map can be colored with at most four colors such that no two adjacent regions share the same color. The theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken using a computer-assisted approach that reduced the problem to verifying 1,936 reducible configurations, marking the first major theorem proved with extensive computational verification. A key algebraic identity involving four variables is the expansion of the square of their sum: (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd), which follows directly from the multinomial theorem for exponent 2. The binomial coefficients for n=4, given by \binom{4}{k} for k = [0](/page/0), [1](/page/1), 2, 3, 4, are 1, 4, 6, 4, 1, respectively, representing the number of ways to choose k items from 4 and appearing in the expansion of ([x + y](/page/X+Y))^4.Basic calculations with 4
Addition
Addition involving the number 4 combines it with other integers to find their sum. The basic addition facts for 4 added to numbers from 0 to 10 are presented in the following table, which illustrates how each increment builds sequentially by 4 from the starting value.| n | 4 + n | Result |
|---|---|---|
| 0 | 4 + 0 | 4 |
| 1 | 4 + 1 | 5 |
| 2 | 4 + 2 | 6 |
| 3 | 4 + 3 | 7 |
| 4 | 4 + 4 | 8 |
| 5 | 4 + 5 | 9 |
| 6 | 4 + 6 | 10 |
| 7 | 4 + 7 | 11 |
| 8 | 4 + 8 | 12 |
| 9 | 4 + 9 | 13 |
| 10 | 4 + 10 | 14 |
Subtraction
Subtraction with 4 can involve subtracting from 4 or subtracting 4 from other numbers, often resulting in positive differences within small ranges but introducing negatives when the minuend is smaller than 4. For 4 - n where n ranges from 0 to 5, the results are as follows, highlighting the transition to negative values: Conversely, for n - 4 where n ranges from 5 to 10, the positive differences are: These operations underscore the inverse relationship to addition and the concept of negative numbers in basic arithmetic.[28]Multiplication
Multiplication by 4 repeatedly adds 4 to itself, producing even products that follow a pattern of doubling the corresponding entries in the 2 times table. The standard multiplication table for 4 × n, where n ranges from 1 to 12, is:| n | 4 × n | Result |
|---|---|---|
| 1 | 4 × 1 | 4 |
| 2 | 4 × 2 | 8 |
| 3 | 4 × 3 | 12 |
| 4 | 4 × 4 | 16 |
| 5 | 4 × 5 | 20 |
| 6 | 4 × 6 | 24 |
| 7 | 4 × 7 | 28 |
| 8 | 4 × 8 | 32 |
| 9 | 4 × 9 | 36 |
| 10 | 4 × 10 | 40 |
| 11 | 4 × 11 | 44 |
| 12 | 4 × 12 | 48 |
Division
Division involving 4 typically means dividing a number by 4 to find how many groups of 4 fit into it, yielding integers for multiples of 4 and fractions otherwise. Integer division facts, where the dividend is a multiple of 4 up to 48, include examples like 8 ÷ 4 = 2 and 12 ÷ 4 = 3. For non-multiples, results are fractions; for instance, 4 ÷ 3 = \frac{4}{3} = 1 \frac{1}{3}, and 4 ÷ 2 = 2. The basic division table for multiples divided by 4 (up to 48 ÷ 4) is:| Dividend | Dividend ÷ 4 | Quotient |
|---|---|---|
| 4 | 4 ÷ 4 | 1 |
| 8 | 8 ÷ 4 | 2 |
| 12 | 12 ÷ 4 | 3 |
| 16 | 16 ÷ 4 | 4 |
| 20 | 20 ÷ 4 | 5 |
| 24 | 24 ÷ 4 | 6 |
| 28 | 28 ÷ 4 | 7 |
| 32 | 32 ÷ 4 | 8 |
| 36 | 36 ÷ 4 | 9 |
| 40 | 40 ÷ 4 | 10 |
| 44 | 44 ÷ 4 | 11 |
| 48 | 48 ÷ 4 | 12 |