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5

Five (5) is the natural number following 4 and preceding 6 in the sequence of positive integers. It is the third prime number, an odd integer, and a digit in the decimal system.^[1] In mathematics, 5 exhibits numerous notable properties. It is the second Fermat prime (of the form $2^{2^n} + 1 for n=1), the third Sophie Germain prime (where both 5 and $2 \times 5 + 1 = 11 are prime), and the fifth number in the Fibonacci sequence (after 1, 1, 2, 3).^[1] Geometrically, 5 forms the hypotenuse of the smallest primitive Pythagorean triple (3-4-5), and it underlies the regular pentagon and pentagram, which are connected to the golden ratio \phi = (1 + \sqrt{5})/2.^[1] Algebraically, 5 is the smallest degree for which general polynomial equations are unsolvable by radicals, as established by Abel's theorem, and it marks the largest order where all finite groups are abelian.^[1] There are exactly five Platonic solids, underscoring 5's role in classical geometry.^[1] Historically, the representation of 5 emerged in ancient civilizations as a response to the human cognitive limit in subitizing quantities beyond four, prompting the invention of distinct symbols.^[2] In pre-classical Rome, it was denoted by "V"; in archaic China by "X"; in South Arabian antiquity by a shape resembling "U"; among the Maya by a bar "|"; and by the Vikings with ">".^[2] Ancient Egyptians often avoided a dedicated symbol, instead using grouped patterns like three and two for recognition.^[2] To the Pythagoreans, 5 symbolized marriage as the sum of the first even number (2, female) and odd number beyond 1 (3, male).^[1]

Basic Properties

Definition and Notation

Five is the positive integer that follows 4 and precedes 6 in the sequence of natural numbers, representing a quantity one more than four.^[3] As a natural number, it belongs to the set of positive integers starting from 1 and extending to infinity, used fundamentally for counting.^[4] It is an odd number, not divisible by 2, and the third prime number after 2 and 3, meaning it has no positive divisors other than 1 and itself.^[5] In standard decimal notation, the predominant numeral system in modern mathematics and everyday use, five is represented by the symbol 5, derived from the Hindu-Arabic numeral system.^[6] In Roman numerals, it is denoted by V, symbolizing a value equivalent to five units, as part of an additive and subtractive system originating in ancient Rome.^[7] The binary representation of five, used in computing and digital systems, is 101, where each digit corresponds to a power of 2 (1×2² + 0×2¹ + 1×2⁰ = 5).^[8] The English word "five" traces its etymology to Old English fīf, from Proto-Germanic *fimf, ultimately deriving from the Proto-Indo-European root *penkʷe, which denoted the number following four.^[9] This root appears in cognates across Indo-European languages, such as Latin quinque and Greek pente, reflecting a shared linguistic heritage for numeral terms.^[10] Visually, five has been represented through tally marks since prehistoric times, typically as five vertical strokes (|||||) grouped for ease of counting, a unary numeral system predating positional notations.^[11] Evidence of such markings dates back over 40,000 years, found on notched bones used as primitive counting tools by early humans.^[11]

Arithmetic Characteristics

A key arithmetic property of 5 is its divisibility rule: an integer is divisible by 5 if and only if its units digit is 0 or 5.^[12]^[13] In basic arithmetic operations, addition yields $5 + 0 = 5, multiplication gives $5 \times 1 = 5, squaring results in $5^2 = 25, and the reciprocal is \frac{1}{5} = 0.2, which has a terminating decimal expansion of a single digit after the decimal point due to 5's factors aligning with the base-10 system's denominator powers.^[14] The number 5 serves as the base for the quinary (base-5) numeral system, where digits range from 0 to 4, and numbers are represented using powers of 5, such as $10_5 = 5_{10}.^[15] Additionally, in modular arithmetic modulo 5, the residues form the set \{0, 1, 2, 3, 4\}, enabling operations that "wrap around" every five integers, such as $7 \equiv 2 \pmod{5}. Its prime factorization is 5.^[16]^[17]

Mathematical Applications

Geometry and Topology

The number 5 plays a central role in plane geometry through the regular pentagon, a convex polygon with five equal sides and equal interior angles of 108° each. This angle arises from the formula for the interior angle of a regular n-gon, ((n-2) × 180°)/n, yielding (3 × 180°)/5 = 108° for n=5.^[18] The diagonals of a regular pentagon exhibit a profound connection to the golden ratio φ ≈ 1.618, defined as φ = (1 + √5)/2, where the ratio of a diagonal to a side equals φ; this relationship satisfies the equation φ² = φ + 1 and underscores the pentagon's aesthetic and mathematical significance in constructions like the Parthenon.^[18]^[19] Closely related is the pentagram, a self-intersecting star polygon denoted by the Schläfli symbol {5/2}, formed by extending the sides of a regular pentagon until they intersect. This figure can be constructed using only a compass and straightedge: begin with a regular pentagon, then connect every second vertex to form the star, resulting in five intersecting line segments that enclose a smaller pentagon at the center. The pentagram's geometry is rich with golden ratio proportions, as the ratios of its segment lengths (from tip to intersection points) equal φ, and it decomposes into golden triangles with angles of 72°, 72°, and 36°. Its self-intersecting nature creates a compound structure with five acute points, making it a fundamental element in classical Euclidean constructions.^[20]^[21] In higher-dimensional geometry, 5 characterizes the 4-simplex (or pentatope), the four-dimensional analog of a tetrahedron, defined as the convex hull of five equidistant vertices with all edges of length a. This regular polytope generalizes the simplex to n dimensions and has a 4-dimensional volume given by

V = \frac{\sqrt{5}}{96} a^4,

derived via the Cayley-Menger determinant, which encodes the volume from edge lengths; the √5 factor highlights 5's intrinsic role in the structure.^[22]^[23] Topologically, 5 manifests in the rotational symmetries of polyhedra, notably the regular icosahedron, a Platonic solid with 20 equilateral triangular faces, 12 vertices, and 30 edges, yielding an Euler characteristic χ = V - E + F = 12 - 30 + 20 = 2, characteristic of a spherical topology. The icosahedron's symmetry group includes six 5-fold rotation axes passing through opposite vertices, enabling 72° rotations that preserve the structure and relate to quasicrystalline tilings and viral capsids. This 5-fold symmetry, incompatible with periodic lattices in Euclidean space, exemplifies 5's influence on non-crystallographic topological configurations.^[24]^[25]

Algebra and Number Theory

In number theory, 5 is recognized as a Fermat prime, specifically the second in the sequence of Fermat numbers defined by F_n = 2^{2^n} + 1, where F_1 = 2^{2^1} + 1 = 5.^[26] These primes hold significance due to their role in constructions like the regular pentagon and their rarity among known primes.^[26] As a prime, 5 exemplifies Fermat's Little Theorem, which states that if p is prime and a is an integer not divisible by p, then a^{p-1} \equiv 1 \pmod{p}; for p = 5, this yields a^4 \equiv 1 \pmod{5}.^[27] The quadratic residues modulo 5 are the squares taken modulo 5, namely 0, 1, and 4, since $0^2 \equiv 0, $1^2 \equiv 1,

[2^2](/page/2_+_2_=_?) \equiv 4

, [3^2](/page/3-2) \equiv 4, and $4^2 \equiv 1 \pmod{5}.^[28] Consequently, non-residues like 2 and 3 exist, and the Legendre symbol provides a measure: for example, \left( \frac{2}{5} \right) = [-1](/page/−1), indicating 2 is not a quadratic residue modulo 5.^[29] This follows from the explicit evaluation or quadratic reciprocity law, highlighting 5's behavior in quadratic fields.^[29] In algebra, the equation x^5 - 1 = 0 factors over the integers as (x - 1) \Phi_5(x) = 0, where \Phi_5(x) is the 5th cyclotomic polynomial, given by

\Phi_5(x) = x^4 + x^3 + x^2 + x + 1.

^[30] The roots of \Phi_5(x) are the primitive 5th roots of unity, which are irreducible over the rationals and generate the cyclotomic field \mathbb{Q}(\zeta_5) of degree 4.^[30] This polynomial's symmetry underscores 5's primality in the context of Galois theory. Within the ring of Gaussian integers \mathbb{Z}, 5 factors non-trivially as $5 = (1 + 2i)(1 - 2i), where each factor has norm N(1 + 2i) = 1^2 + 2^2 = 5 and N(1 - 2i) = 5.^[31] Since 5 is a prime in \mathbb{Z} and congruent to 1 modulo 4, it splits in \mathbb{Z}, and the elements $1 + 2i and $1 - 2i (up to units) are Gaussian primes of norm 5.^[31] This decomposition illustrates the unique factorization property of \mathbb{Z} as a Euclidean domain.^[31]

Advanced Structures

In group theory, the cyclic group \mathbb{Z}_5 of order 5 is the unique abelian group of that order up to isomorphism, generated by any non-identity element under addition modulo 5.^[32] This group is isomorphic to the rotational symmetry group of a regular pentagon, consisting of rotations by multiples of $72^\circ around its center.^[33] In contrast, the alternating group A_5 is the simple non-abelian group of even permutations on 5 elements, with order $60 = 5!/2, notable as the smallest such simple group.^[34]^[35] Combinatorics highlights the role of 5 through binomial coefficients \binom{5}{k}, which count the ways to choose k elements from a 5-element set; for instance, \binom{5}{2} = 10.^[36] These coefficients form the fifth row of Pascal's triangle: $1, 5, 10, 10, 5, 1, illustrating the recursive relation \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}.^[37] In higher-dimensional geometry, the 5-cell, or pentachoron, is the regular 4-polytope with Schläfli symbol \{3,3,3\}, bounded by 5 regular tetrahedral cells meeting at each of its 5 vertices.^[38] This simplex in 4-space exemplifies the progression from lower-dimensional polytopes, with 10 triangular faces and 10 edges. Topologically, lens spaces provide examples where the fundamental group has order 5; specifically, the lens space L(5,1) is a 3-manifold obtained as the quotient of S^3 by the cyclic group \mathbb{Z}_5 acting freely, yielding \pi_1(L(5,1)) \cong \mathbb{Z}/5\mathbb{Z}.^[39]

Historical Development

Origins in Ancient Numerals

One of the earliest indications of the number 5 as a conceptual unit appears in prehistoric tally systems, where groupings of five notches or marks facilitated counting, likely inspired by the five fingers of a hand. A prominent example is the Ishango bone, a baboon fibula discovered in the Democratic Republic of the Congo and dated to circa 20,000 BCE, featuring three columns of incisions that include sequences interpretable as quinary groupings, such as sets of 5, 10 (two fives), and multiples thereof, suggesting an early base-5 organizational method for enumeration or possibly lunar tracking.^[40]^[41] In Mesopotamian civilization, around 3000 BCE, the Sumerians incorporated 5 into their sexagesimal numeral system, which combined base-60 structure with sub-base-10 elements for practicality. The value 5 was denoted by five vertical cuneiform wedges impressed into clay tablets, functioning as a core building block in administrative records, astronomical calculations, and trade accounts across Sumerian city-states like Uruk.^[42]^[43] Ancient Egyptian mathematics, from approximately 3000 BCE onward, utilized a decimal additive hieroglyphic system without unique symbols for intermediate values like 5, instead repeating the unit glyph (a single vertical stroke) five times to represent it. This approach integrated seamlessly into broader notations, such as combining with the coil-of-rope symbol for 100 or the pointing-finger glyph for 10,000, emphasizing accumulation in pyramid inscriptions, tomb inventories, and the Rhind Mathematical Papyrus.^[44]^[45] Early Indo-European linguistic evidence reflects 5's association with manual anatomy and universal harmony, rooted in the Proto-Indo-European term *pénkʷe, reconstructed as deriving from concepts of "fist" or "whole hand" encompassing five digits. This manifests in Vedic Sanskrit as "pañca," appearing in the Rigveda (circa 1500 BCE) to denote not only numerical quantity but also symbolic wholeness, as in hymns invoking fivefold cosmic divisions—such as the five layers of the universe or ritual offerings mirroring natural order.^[46]^[47]

Evolution of the Arabic Digit

The modern digit 5 traces its origins to the Brahmi numeral system of ancient India, dating back to around the 3rd century BCE, where it was represented by a simple vertical stroke, often with a short horizontal extension to the right, reflecting early additive counting methods on materials like palm leaves.^[48] This form evolved during the Gupta period (c. 4th–6th centuries CE) into a more curved and compact shape, resembling a gentle arc or hook opening downward, as the numerals became more fluid for inscription on metal and stone, facilitating the integration of positional notation.^[48] By the 8th and 9th centuries CE, Arabic scholars adopted and refined these Indian forms during the Islamic Golden Age, with the digit 5 appearing as a pronounced downward hook in early manuscripts. Al-Khwarizmi's treatise On the Calculation with Hindu Numerals (c. 825 CE) described the system, including this hooked variant of 5, emphasizing its use in calculation on dust boards before paper's widespread adoption.^[49] Transmission was furthered by scholars like Al-Kindi, who in works such as On the Use of the Hindu Numerals (c. 830 CE) integrated the digits into Arabic mathematical texts, promoting their practical application in astronomy and commerce across the Islamic world.^[50] The digit reached Europe through Leonardo of Pisa (Fibonacci)'s Liber Abaci in 1202 CE, where 5 was depicted as an angular, open-topped form similar to the modern shape but with a more elongated stem, introduced to Western merchants via North African trade routes.^[51] Initial resistance persisted due to familiarity with Roman numerals, but adoption accelerated in the 15th century with the advent of printing presses, which standardized the glyph's proportions—closing the upper curve and straightening the base—for consistency in printed texts like arithmetic books.^[52] Regional variations persist between Eastern Arabic (٥), featuring a sharp, rightward-extending hook reminiscent of earlier Gupta curves and used in much of the Arab world, and Western Arabic (5), the open, looped form that evolved in the Maghreb and influenced global typography through European printing.^[49] Over centuries, typographic refinements in the West refined the digit's serifs and curves for readability in typefaces, solidifying its current universal appearance by the 16th century.^[48]

Cultural and Symbolic Significance

In Religion and Mythology

In Judaism, the number five holds profound significance through the Torah, also known as the Pentateuch or Five Books of Moses—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which form the foundational text of Jewish law, ethics, and narrative history, revealed to Moses at Mount Sinai and central to religious study and observance.^[53] The numeral five also corresponds to the Hebrew letter Heh (ה), with a gematria value of 5, symbolizing divine breath, revelation, and the presence of God, as seen in its appearance in the sacred name of God (YHWH). Additionally, there are five Megillot (scrolls) recited during Jewish holidays, underscoring five's role in liturgical and seasonal cycles. In Buddhism, five symbolizes ethical and spiritual foundations through the Five Precepts (Pañca Sīla), voluntary guidelines for lay practitioners to cultivate moral conduct: abstaining from killing, stealing, sexual misconduct, false speech, and intoxicants that cloud the mind.^[54] These precepts promote harmony, mindfulness, and progress toward enlightenment, forming the basis of right action in the Noble Eightfold Path. The number also appears in the five aggregates (skandhas) that constitute the self—form, sensation, perception, mental formations, and consciousness—and the five wisdoms of the Buddha, representing aspects of awakened mind. In Islam, the number five holds central symbolic importance through the Five Pillars, which form the foundational acts of worship and ethical practice for Muslims: the Shahada (profession of faith), Salah (ritual prayer), Zakat (almsgiving), Sawm (fasting during Ramadan), and Hajj (pilgrimage to Mecca).^[55] These pillars represent the core obligations that structure Muslim life and devotion to Allah. Additionally, Salah is performed five times daily—at dawn (Fajr), noon (Dhuhr), afternoon (Asr), sunset (Maghrib), and night (Isha)—symbolizing constant remembrance of the divine and rhythmic alignment with the day's cycles.^[56] The Hand of Fatima, also known as the Hamsa or Khamsa, emerges as a prominent protective symbol in Islamic tradition, particularly among Shia Muslims, depicting an open hand to ward off the evil eye and invoke blessings of safety, strength, and divine providence.^[57] In Hinduism, five symbolizes the fundamental building blocks of the universe through the Pancha Bhoota, or five great elements: earth (Prithvi), water (Apas), fire (Agni), air (Vayu), and ether or space (Akasha), which together constitute all matter and cosmic creation.^[58] These elements are revered in rituals, yoga, and philosophy as interconnected forces sustaining life and spiritual harmony. Mythologically, Lord Shiva manifests as Panchamukha or Sadashiva with five faces—Sadyojata (creation, facing west), Vamadeva (preservation, facing north), Aghora (destruction, facing south), Tatpurusha (concealment, facing east), and Ishana (revelation, facing upward)—embodying the multifaceted aspects of divine power and the transcendence over the material world.^[59] Within Christianity, five signifies the Five Holy Wounds inflicted on Jesus Christ during the Crucifixion: the nail marks in each hand and foot, and the spear wound in his side, which collectively represent his sacrificial suffering and redemptive love for humanity.^[60] Devotion to these wounds developed prominently from the 12th century onward, inspiring prayers, feasts, and liturgical practices across Europe, such as the Golden Mass with five candles symbolizing each wound.^[60] Early Christians also employed the pentagram, or pentalpha, as a symbol of these five wounds before its later associations with other traditions, using it to evoke Christ's passion and the unity of faith.^[61] In other mythologies, five appears in Chinese philosophy through the Wu Xing, or five phases—wood, fire, earth, metal, and water—which dynamically interact in cycles of generation and conquest to explain cosmic order, seasonal changes, and moral harmony in ancient texts like the Zuo Zhuan.^[62] Among some Native American tribes, such as certain Lakota and Anishinaabe groups, directional symbolism extends to five aspects: the four cardinal directions (east, south, west, north) plus a central or zenith point, representing interconnectedness, balance, and the sacred center of creation in ceremonies and storytelling.^[63]

In Science and Everyday Use

In chemistry, the number 5 denotes the atomic number of boron (B), a metalloid element essential for various biological and industrial processes. Boron plays a critical role in plant nutrition by facilitating cell wall formation, pollen tube growth, and sugar transport, with deficiencies leading to impaired reproduction and stunted development in crops.^[64] In humans and animals, boron supports bone health, wound healing, and hormone metabolism, though it is required only in trace amounts.^[65] Industrially, boron compounds are used in glass production, detergents, and semiconductors due to their heat resistance and doping properties.^[66] In biology, the number 5 is prominent in human anatomy and molecular structures. Humans exhibit pentadactyly, the condition of having five digits on each hand and foot, a trait inherited from early tetrapod ancestors that enabled versatile limb adaptations across vertebrates. This pentadactyl limb plan, while modified in species like birds and whales, underscores evolutionary conservation for manipulation and locomotion.^[67] Additionally, DNA and RNA strands are directional, with a 5' (five-prime) end featuring a phosphate group on the fifth carbon of the sugar backbone and a 3' (three-prime) end with a hydroxyl group, dictating replication and transcription polarity.^[68] Traditionally, humans are described as possessing five primary senses—sight, hearing, smell, taste, and touch—mediated by specialized organs that transmit sensory data to the brain, though modern neuroscience identifies additional senses like balance and proprioception.^[69] In physics, particularly particle physics, a significance level of five sigma (5σ) serves as the conventional threshold for claiming a discovery, corresponding to a probability of about 1 in 3.5 million that the observed effect is due to random fluctuation rather than a new phenomenon.^[70] This standard was pivotal in the 2012 announcement of the Higgs boson at CERN, where experiments achieved 5σ excess in data confirming the particle's existence.^[71] In everyday use, the number 5 reflects human anatomy and societal conventions. The pentadactyl structure of hands and feet facilitates fine motor skills essential for tool use, writing, and gesture-based communication. Many daily routines incorporate groupings of five, such as the five-day workweek in modern economies, which structures labor around a cycle of productivity followed by rest, originating from industrial reforms in the early 20th century. Sensory experiences also permeate routine activities, with the five traditional senses enabling navigation, nutrition, and social interaction in environments like kitchens (taste and smell) or commutes (sight and hearing).^[69]

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What are the Five Pillars? - Islamic Relief
The five pillars of Islam are Shahada, Salah, Zakat, Sawm, and Hajj, which are key practices forming the foundation of Muslim life.
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Salah (Prayer) - The Second Pillar of Islam | Islamic Relief UK
What are the 5 daily prayers of Islam? · Fajr – this takes place at pre-dawn. · Dhuhr – this takes place just after the sun reaches its zenith. · Asr – this takes ...<|separator|>
[54]
Khamsa, the hand of Fatima - FUNCI - Fundación de Cultura Islámica
Jun 1, 2020 · This icon was a symbol of divine providence, generosity, hospitality, strength, or power among the Medieval Muslims, especially among the Shia ...<|separator|>
[55]
What are the Five Elements or Pancha Bhutas? - Isha Foundation
Essentially, they are made of five elements or the pancha bhutas – earth, water, fire, air and akasha.
[56]
Five Powers of Siva
This form of Siva has five faces and is called Sadasiva. In Sanskrit the five faces are referred to as Panchabrahma, meaning “five great Lords.” The term ...
[57]
CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds - New Advent
The Five Wounds of Christ are honoured by a Mass and an Office, and on account of these wounds we venerate also the feet, hands and side of the most loving ...
[58]
What is the origin and meaning of the pentagram? | GotQuestions.org
Sep 25, 2023 · In Christian symbolism, the basic pentagram (without a circle) was originally used to represent the five wounds of Jesus Christ. It was soon ...
[59]
Wuxing (Wu-hsing) | Internet Encyclopedia of Philosophy
These five phases are wood (mu), fire (huo), earth (tu), metal (jin), and water (shui); they are regarded as dynamic, interdependent modes or aspects of the ...Meanings of Wuxing · Origins of Wuxing · Wuxing Before the Han Dynasty
[60]
Of Earth and Sky - A Native American Perspective - John Two-Hawks
Dec 28, 2014 · The seven directions are: 1. The East, 2. The South, 3 The West, 4. The North, 5. Mother/Grandmother Earth, 6. Father Sky, and ...
[61]
Importance of Boron in Plant Growth | Mosaic Crop Nutrition
Boron (B) is a micronutrient critical to the growth and health of all crops. It is a component of plant cell walls and reproductive structures.
[62]
Nothing Boring About Boron - PMC - NIH
Boron has been proven to be an important trace mineral because it (1) is essential for the growth and maintenance of bone; (2) greatly improves wound healing.
[63]
An introduction to boron: history, sources, uses, and chemistry - PMC
Modern uses of borate-mineral concentrates, borax, boric acid, and other refined products include glass, fiberglass, washing products, alloys and metals, ...
[64]
Recent Post Doc Tom Stewart has Paper Published in Nature ...
Pentadactyl amniote limbs have disparate patterns of genetic differentiation of digits. Photographs of right forelimbs. July 19, 2019. Recent Post Doc Tom ...<|separator|>
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[PDF] A nucleic acid strand is inherently directional, and the "5 prime end ...
The 5' end of a nucleic acid has a free hydroxyl (or phosphate) on a 5' carbon, and the 3' end has a free hydroxyl (or phosphate) on a 3' carbon.
[66]
Human senses and sensors from Aristotle to the present - PMC
Jul 3, 2024 · Aristotle's list of the five senses sight, hearing, touch, taste, and smell is still in use among non-scientific lay persons.
[67]
Five sigma revisited - CERN Courier
Jul 3, 2023 · Louis Lyons traces the origins of the “five sigma” criterion in particle physics, and asks whether it remains a relevant marker for claiming the discovery of ...
[68]
Standard deviation / Sigma | ATLAS Experiment at CERN
Five sigma significance is traditionally required to claim a discovery of a new particle; this was the threshold passed by the Higgs boson when its discovery ...