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3

3 is a , the immediately following 2 and preceding 4, and it is the smallest odd , meaning it is greater than 1 and has no positive divisors other than 1 and itself. As the first odd , 3 plays a foundational role in , appearing in the prime factorization of many . It is also the only that is one less than a (since $4 - 1 = 3) and the unique positive equal to the sum of the preceding positive ($1 + 2 = 3) as well as the sum of the factorials of the preceding ($1! + 2! = 3). In , 3 defines the , the simplest with three sides and three , whose properties underpin and , such as the sum of interior angles being $180^\circ. Algebraically, raising a quantity to the power of 3 results in cubing it, leading to cubic equations whose solutions involve the cubic formula, and 3 is the dimension of ordinary . Beyond , 3 appears in physics as the number of spatial dimensions in and the number of quarks in protons and neutrons according to the . In chemistry, it denotes the of , the lightest metal. Culturally, 3 holds symbolic significance across societies, often representing completeness or wholeness.

History and Notation

Origins in Ancient Civilizations

The earliest known representations of the number 3 appear in prehistoric , where groups of three parallel lines or incisions on bones and stones served as basic counting devices, often symbolizing triads in early human cognition and possibly ritual contexts. Archaeological evidence from sites like the in , dated to around 20,000 years ago, demonstrates simple groupings of marks used for , predating formal writing systems and reflecting an innate tendency to cluster counts in sets of three for practical and symbolic purposes. In ancient , the number 3 emerged prominently in around 3350–3200 BCE, where it was depicted through repetitions of the unit sign—a circular impression or conical wedge mark pressed into clay tablets for accounting and tallies. These numerical notations, found in administrative records from sites like , used three such impressions to denote the quantity 3, forming a foundational element in the shift from token-based counting to scripted in one of the world's earliest civilizations. Egyptian hieroglyphs from around 3000 BCE represented 3 through three horizontal strokes (|||), a straightforward additive system employed in monumental inscriptions, papyri, and tomb records for recording quantities in and daily administration. This stroke-based notation for units 1 through 9 underscored the decimal structure of , with examples appearing in texts like the , where 3 facilitated calculations in and fractions. While higher values like 100 used a coil of rope symbol, the strokes for 3 highlighted the system's reliance on repetition for small integers. The Babylonians, inheriting and refining traditions by the 2nd millennium BCE, incorporated 3 into their (base-60) positional system, representing it as three vertical wedges (the unit symbol repeated) in tablets for astronomical, economic, and mathematical purposes. This notation, evident in artifacts like the tablet, positioned 3 as a key component in place-value calculations, enabling complex computations such as those for Pythagorean without a zero placeholder. In , from the BCE onward, the acrophonic assigned the letter gamma (Γ), the initial of the word for "three" (τρία), to represent 3, used in inscriptions, coins, and early texts before the adoption of alphabetic numerals. This symbolic approach, seen on Attic pottery and public monuments, reflected the cultural emphasis on linguistic in quantification, bridging oral traditions with written records.

Development of the Arabic Digit

The digit 3 originated in the of ancient , which emerged around the BCE as part of the earliest known positional notation. In its initial Brahmi form, the digit was represented by three horizontal strokes, reflecting a simple additive principle before the adoption of place value. By the period (4th to 6th centuries CE), the evolved into a more fluid script under the , where the digit 3 took on a curved form resembling a 'C' with an additional bend, facilitating writing and broader dissemination across . This variant appears in mathematical texts and inscriptions, marking a transitional stage toward the modern . The , a birch-bark document dated to the 3rd to 4th centuries CE and discovered in present-day , provides one of the earliest surviving examples of these numerals in use, integrated with calculations. During the , numerals, including the digit 3, were adopted and refined by Persian and Arab scholars in the 8th and 9th centuries. Muhammad ibn Musa al-Khwarizmi's treatise On the Calculation with Hindu Numerals (c. 825 ) systematically described the positional system, as concepts were translated and integrated into mathematical works, such as those at the in . This adaptation occurred as concepts were translated and integrated into mathematical works, such as those at the in . The shapes of the numerals continued to evolve in subsequent scribal traditions. The transmission of these numerals to began in the medieval period through and scholarly exchange, culminating in the mathematician 's Liber Abaci (1202 CE), which introduced the system—including the digit 3 in an open-top form resembling the modern Western variant—to Western audiences via examples from North African merchants. 's text emphasized practical applications in commerce, accelerating adoption despite resistance from traditional Roman numeral users. By the 13th century, this open '3' had taken root in European manuscripts. Variations persisted between Eastern and Western Arabic numerals: the Eastern form (٣), used in the and , features an open curve facing right with angular lines, while the Western form (3), prevalent in and later , closes the top curve for a more compact shape. These differences arose from regional scribal traditions, with the Western variant influencing European styles. Standardization occurred in the , driven by the invention of the around 1450 by , which fixed the open-top '3' in printed texts and mathematical works across , establishing the glyph as we know it today.

Etymology and Linguistic Representation

The word for the number three in English derives from the Old English þrīe (masculine) or þreo (neuter and feminine), which itself stems from the Proto-Germanic \þrīz, ultimately tracing back to the Proto-Indo-European (PIE) root \tréyes or \trei- meaning "three." This PIE root exhibits remarkable stability across Indo-European languages, evolving into forms such as Latin trēs, Ancient Greek treĩs, and Sanskrit trayas or tri-, reflecting a shared phonetic core with initial tr- and a final -s or -i in nominative cases. In these descendant languages, the root often appears as a prefix or stem, as in Greek trias ("triad") or Sanskrit trika ("group of three"), highlighting its role in compounding for numerical concepts. Outside the Indo-European family, the word for three shows distinct origins. In , it is thalāthah (ثَلَاثَة), derived from the Proto-Semitic \θalāθ- , a triliteral form typical of where encode core meanings through patterns. Similarly, in , the term is sān (三), originating from the Proto-Sino-Tibetan \sum or \sɨm, part of a reconstructed through across Sinitic and . These non-Indo-European examples illustrate divergent etymological paths, with emphasizing consonantal and Sino-Tibetan favoring monosyllabic forms. Phonetically, words for three across languages often feature initial dental or alveolar consonants, such as the voiceless dental fricative /θ/ in English three and thalāthah, or the stop-trill cluster /tr/ in PIE-derived forms like Latin trēs. Linguistic analyses suggest this pattern may arise from historical sound shifts rather than universal symbolism, though some cross-linguistic studies note a tendency for numerals to retain occlusive or onsets for perceptual salience in early systems. In Indo-European branches, aspirated variants like the Germanic /θ/ (from PIE *t via ) appear in English, drie, and drei, contrasting with unaspirated forms in . Orthographically, the linguistic representation of three varies by writing system, reflecting both phonetic and symbolic adaptations. In the Latin alphabet, it is spelled three in English, with the digraph th denoting /θ/, a convention inherited from Old English manuscripts using the thorn (þ). Roman numeral notation uses III, a repetitive stroke system predating alphabetic forms. In Chinese script, sān is written as 三, an ideogram historically depicting three horizontal lines, evolving from oracle bone inscriptions around 1200 BCE. Devanagari, used for Hindi and Sanskrit, represents the digit as ३ (a curved form derived from Brahmi script) and the word tīn (तीन) with conjunct consonants capturing the /tri-/ onset from Sanskrit tri. Historical shifts in English illustrate phonetic evolution: þrīe (/θriːe/), pronounced with a long /iː/ and final , transitioned through thre amid the (c. 1400–1700 CE), which raised the vowel to modern /θriː/, while the was replaced by th under influence. This change preserved the core consonants but simplified the ending, aligning with broader Germanic patterns where final vowels weakened.

Mathematical Properties

Fundamental Characteristics

The integer 3 is defined as the successor of 2 in the s, obtained by applying the S where S(2) = 3, according to the . It follows 2 and precedes 4 in the sequence of positive s. In conventions where s start from 1 (1, 2, 3, ...), 3 is the third ; in starting from 0, it is the fourth (0, 1, 2, 3). It serves as a foundational element in . Classified as an odd number because it is not divisible by 2, 3 ends in the digit 3 and fits the form 2k + 1 for k = 1. It is the smallest odd , greater than the even prime 2 and having no divisors other than 1 and itself. Additionally, 3 is a with the decimal representation 3.0, equivalent to its integer form without fractional parts. Basic arithmetic operations involving 3 demonstrate fundamental identities: addition with the additive identity yields $3 + 0 = 3, preserving the value. Multiplication by the multiplicative identity gives $3 \times 1 = 3, leaving the number unchanged. The square root of 3 is irrational, approximated as \sqrt{3} \approx 1.732. The factorial of 3, denoted $3!, equals 6, computed as 3 \times 2 \times 1. Notably, 3 is the unique positive integer that equals the sum of the preceding positive integers (1 + 2 = 3) and the sum of the factorials of the preceding natural numbers (1! + 2! = 3$). In , 3 is represented as III, formed by three instances of the symbol I for one. A distinctive property of 3 is that it is the only one less than a , specifically $4 = 2^2.

Divisibility and Primality

3 is a , defined as a greater than 1 that has no positive divisors other than 1 and itself. To prove its primality, note that the only possible divisors to check are primes less than or equal to the , which is approximately 1.732. Since there are no primes between 1 and 1.732 (the smallest is 2, which exceeds this bound), and 3 is not divisible by 2, 3 has no divisors other than 1 and itself. Wilson's Theorem provides another confirmation of 3's primality: for a prime p, (p-1)! \equiv -1 \pmod{p}. For p=3, (3-1)! = 2! = 2, and $2 \equiv -1 \pmod{3} since $2 + 1 = 3, which is divisible by 3. This theorem, proposed by John Wilson and published by Edward Waring in 1770, holds for all primes, including 3. A key property related to divisibility by 3 is the rule that an integer is divisible by 3 if and only if the sum of its digits is divisible by 3. For example, 123 is divisible by 3 because $1 + 2 + 3 = 6, and 6 is divisible by 3. This rule arises from : since $10 \equiv 1 \pmod{3}, any power of 10 satisfies $10^k \equiv 1 \pmod{3}. For a number n = d_k 10^k + \cdots + d_1 10 + d_0, it follows that n \equiv d_k + \cdots + d_1 + d_0 \pmod{3}, so n is divisible by 3 precisely when the is. As a , the for 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9, derived similarly because $10 \equiv 1 \pmod{9}. This shared foundation in makes the rules efficient for checking multiples of 3 and 9 without full division.

in and

In geometry, the number 3 defines the as the simplest , consisting of three sides and three vertices, which forms the foundational building for more polygonal structures. The tetrahedron, as the simplest polyhedron in three-dimensional space, features four vertices connected by six edges and bounded by four equilateral triangular faces, each with three sides, embodying the minimal in . Furthermore, three-dimensional , denoted \mathbb{R}^3, is characterized by three mutually axes corresponding to , width, and , enabling the description of all points via ordered triples of real coordinates. Triangular numbers, which count the objects in an arranged in rows, highlight 3 as the second such number, given by the formula T_n = \frac{n(n+1)}{2}, where T_2 = 3. In , the primitive (3, 4, 5) exemplifies a satisfying $3^2 + 4^2 = 5^2, providing a fundamental integer-sided configuration for geometric constructions and proofs. Algebraically, the of order 3, denoted \mathbb{Q}(\zeta_3) where \zeta_3 is a third satisfying \zeta_3^3 = 1 and \zeta_3 \neq 1, represents the minimal extension of containing these and plays a key role in and class number computations. Cubic equations often yield three , as illustrated by x^3 - 3x + 1 = 0, which has three distinct real roots derivable via trigonometric identities or Cardano's formula. In group theory, the A_3, comprising the even permutations of three elements, is cyclic of 3 and isomorphic to \mathbb{Z}/3\mathbb{Z}.

Applications in Science and Technology

In Physics and Dimensions

In physics, the number 3 manifests prominently in the structure of spatial dimensions that govern everyday experience and the behavior of . Humans perceive and navigate the within three spatial dimensions, conventionally described by Cartesian coordinates (x, y, z), which allow for the positioning and movement of objects in length, width, and height. While in encompasses four dimensions—including time as the fourth—physical and forces operate within these three observable spatial dimensions, enabling the formulation of most classical and quantum mechanical laws. Classical mechanics underscores 3's foundational role through Isaac Newton's three laws of motion, first articulated in his 1687 work . The first law states that an object at rest remains at rest, and an object in uniform motion continues in a straight line unless acted upon by an external , embodying the principle of . The second law relates (F) to (m) and (a) via the equation F = ma, quantifying how forces alter motion. The third law asserts that for every action, there is an equal and opposite reaction, describing interactions between bodies. These laws collectively form the bedrock of , explaining planetary motion, trajectories, and principles. In and , 3 appears in the trichromatic theory of and mixing. Human vision relies on three types of cells sensitive to , , and wavelengths, allowing the perception of a full through their combinations. This underpins models, where , , and lights mix to produce or other hues, as seen in displays and lighting systems; for instance, equal intensities of these primaries yield . This three-color basis stems from the physiological limits of the , where signals from the cones are processed to interpret color. Quantum mechanics highlights 3 in the composition of baryons, the building blocks of atomic nuclei, under (QCD). Baryons, such as protons and neutrons, consist of three bound by the mediated by gluons. A proton, for example, comprises two up quarks and one (uud), while a neutron has one up and two down (udd); these configurations determine their charge, mass, and stability. QCD, the theory describing quark interactions, mandates this triquark structure for color-neutral hadrons, ensuring confinement and explaining . Special relativity extends 3's significance to momentum conservation in three-dimensional space. The three-momentum vector, denoted as p = (p_x, p_y, p_z), represents the spatial components of the four-momentum, where each p_i = γ m v_i (with γ as the , m as rest mass, and v_i as velocity components). This vector transforms under Lorentz boosts to preserve the invariance of the interval, distinguishing relativistic dynamics from Newtonian cases while maintaining three spatial directions for momentum transfer in collisions and particle interactions.

In Chemistry and Matter

In chemistry, the number 3 holds fundamental significance as the atomic number of (), the lightest and the third in the periodic table. Lithium consists of three protons in its , with a standard of [He] 2s¹, or more explicitly 1s² 2s¹, featuring two electrons in the 1s orbital and one in the 2s orbital. This configuration imparts lithium with high reactivity, as the single is loosely bound, enabling it to readily form +1 ions in chemical reactions./Descriptive_Chemistry/Elements_Organized_by_Block/1_s-Block_Elements/Group__1%3A_The_Alkali_Metals/Z003_Chemistry_of_Lithium_(Z3)) The classical understanding of states of matter recognizes three primary phases—, , and gas—based on the arrangement and mobility of particles under varying temperatures and pressures. In , particles are tightly packed in a fixed ; allow particles to while maintaining ; and gases exhibit particles in random motion with no fixed shape or . While represents a fourth , characterized by ionized particles, the trio of , , and gas forms the foundational framework for describing everyday matter transformations, such as , freezing, , and . In molecular bonding, the number 3 appears prominently in the valence shell repulsion (, particularly for AX₃ molecules where a central atom (A) is bonded to three surrounding atoms (X) with no lone pairs on the central atom. This configuration results in a trigonal planar geometry, minimizing electron pair repulsion with bond angles of 120°. (BF₃) exemplifies this structure, where the atom forms three equivalent B–F bonds in a flat, symmetrical plane, influencing its nonpolar nature despite the difference. Group 3 of the periodic table marks the onset of the d-block, comprising transition metals such as and yttrium (Y), with lanthanum (La) or lutetium (Lu) often debated for the subsequent position. These elements feature partially filled d orbitals in their atoms or ions, enabling variable oxidation states, colorful compounds, and catalytic properties typical of transition metals across groups 3 through 12. According to IUPAC conventions, group 3 elements initiate the filling of the (n-1)d subshell, distinguishing them from s-block alkali metals. Isotopes involving mass number 3 further highlight the number's role in nuclear chemistry. Lithium-3 (^3Li) is an extremely unstable isotope with a half-life on the order of 10^{-21} seconds, decaying rapidly via proton emission and not occurring naturally. In contrast, tritium (^3H), the mass-3 isotope of hydrogen, plays a crucial role in nuclear fusion reactions, serving as a key fuel alongside deuterium in processes that mimic stellar energy production, such as the deuterium-tritium (D-T) reaction releasing high-energy neutrons.

Engineering and Computational Uses

In , the tripod principle leverages the geometric fact that three non-collinear points define a unique , providing inherent for structures on uneven surfaces without the wobbling common in four-point supports. This concept is applied in bridge design, where three supports can distribute loads evenly and enhance during construction, as seen in techniques using three points of vertical support to prevent excessive rotation in curved girders. Three-phase power systems are a cornerstone of , enabling efficient (AC) transmission and distribution by using three sinusoidal voltage waveforms offset by 120 degrees. The voltages are expressed as V = V_0 \sin(\omega t + \phi), where \phi = 0^\circ, 120^\circ, 240^\circ, allowing for constant power delivery with reduced material compared to single-phase systems. This configuration powers most industrial motors and grids, minimizing vibrations and improving efficiency. In , balanced ternary systems represent numbers using digits -1, 0, and +1 in base-3, offering advantages in arithmetic operations without a separate , as demonstrated in the Soviet computer from 1958. This ternary logic simplifies multiplication and division compared to , potentially reducing complexity for certain computations. The RGB color model, foundational to , employs three channels—red, green, and blue—each ranging from 0 to 255, to produce over 16 million colors additively for displays. Robotics and utilize three-axis control in CNC machines, enabling precise linear movements along , and Z directions to fabricate complex parts from materials like metal and . This allows simultaneous axis operation for efficient shaping, forming the basis for many processes. In , the three-point bending test evaluates by supporting a at two ends and applying a load at the midpoint, measuring deflection to determine properties like modulus of elasticity. This standardized method, often performed on universal testing machines, provides critical data for assessing material durability in applications from to .

Cultural and Symbolic Significance

In Religion and Mythology

In , the number 3 holds profound theological significance, most notably through the doctrine of the Holy Trinity, which describes God as three co-equal, co-eternal persons: the Father, the Son (Jesus Christ), and the Holy Spirit. This triune nature underscores the completeness and unity of the divine essence, as articulated in key scriptural passages like Matthew 28:19. Additionally, the Gospel of recounts the visit of the wise men (), who presented three gifts and are traditionally numbered as three, to the infant Jesus, symbolizing Gentile recognition of the and emphasizing themes of divine revelation and kingship. The on the third day further exemplifies 3's association with renewal and victory over death, as affirmed in 1 Corinthians 15:4, marking a pivotal event in Christian . In , the represents the threefold manifestation of the supreme divine reality, comprising as the creator, as the preserver, and as the destroyer, embodying the cyclical processes of cosmic existence. This triad illustrates the interconnectedness of creation, maintenance, and dissolution in . Complementing this, the three gunas— (purity and harmony), (activity and passion), and tamas (inertia and darkness)—are the fundamental qualities of prakriti () described in the (Chapter 14), influencing all beings and phenomena while binding the soul to the material world until transcendence is achieved. Norse mythology features the three Norns—Urd (past), Verdandi (present), and Skuld (future)—as powerful female beings who reside at the base of , the , and weave the for gods, humans, and all existence, determining destinies with irrevocable judgment. itself is sustained by three roots extending into distinct realms: one to the Well of Urd in , another to the Well of Mimir in Jotunheim, and the third to Hvergelmir in , symbolizing the tree's role as the connecting the nine worlds. Ancient Egyptian religion prominently incorporated triads of deities, such as the archetypal (god of the underworld and resurrection), (goddess of magic and motherhood), and (falcon-headed sky god and avenger), which exemplified familial and cosmic harmony while influencing royal ideology and funerary practices. The soul's journey after death involved three essential components—ka (life force), ba (personality and mobility), and akh (transfigured spirit)—navigating the through judgment and rituals to achieve eternal union with the divine, as detailed in texts like the . In Buddhism, the three marks of existence—anicca (impermanence), dukkha (suffering or unsatisfactoriness), and anatta (non-self)—form the foundational insights into the nature of reality, teaching that all conditioned phenomena lack inherent stability, lead to dissatisfaction, and have no permanent essence, as expounded in the Pali Canon. These principles guide practitioners toward enlightenment by fostering detachment. The Three Jewels, or Triratna—Buddha (the awakened teacher), Dharma (the teachings), and Sangha (the community)—serve as the core refuges for Buddhists, providing the path to liberation from samsara through ethical conduct, meditation, and wisdom.

As a Lucky or Unlucky Number

In , the number 3, pronounced "san," is regarded as lucky because its sound resembles the word for "birth" or "life," symbolizing growth, vitality, and new beginnings. This phonetic association extends to positive connotations in and daily life, where multiples of 3 often appear in rituals or preferences for harmony and prosperity. In Italian traditions, the number 3 holds lucky significance tied to the Holy Trinity, influencing customs like repeating toasts or wishes three times to invoke good fortune and protection. Conversely, the number 3 carries unlucky implications in various superstitions, often suggesting failure or peril after initial attempts. The idiom "third time's the charm" acknowledges that the first two efforts may fail, but the third brings success, rooted in ancient beliefs in the magical potency of threes dating to the early . Similarly, the "" in popular superstitions posits that misfortunes, such as celebrity deaths, occur in trios, a pattern observed in pop culture since at least the mid-20th century and attributed to human tendency to perceive clusters in random events. Folklore amplifies these omens, particularly in traditions where sighting signals an impending death, drawing from medieval associations of corvids with the and . A related warns against lighting three cigarettes from one match, believed to invite danger as the prolonged flame—enough time for a to spot and target the third smoker—originated among soldiers and persisted into later conflicts. In and , 3 embodies creativity, self-expression, and joyful communication, yet it can also signify excess or scattered energy if not balanced, encouraging individuals to channel its vibrant force productively. This dual nature echoes Pythagorean philosophy, where the (3) represents perfect as the of and duality, symbolizing wholeness, , and ethical completeness in the . Modern idioms reflect 3's social tensions, such as "three's a crowd," which implies that a third person's presence disrupts the intimacy of a pair, originating in 19th-century English expressions of relational dynamics. In sports, baseball's "three strikes and you're out" rule marks the third strike as definitive failure, leading to elimination and underscoring themes of finality in trial-and-error scenarios.

In Arts, Literature, and Language

In literature, the serves as a foundational principle, structuring stories around sets of three elements to enhance , memorability, and emotional impact. This device appears prominently in fairy tales, such as "," where the successive encounters between the wolf and each pig build escalating tension toward a satisfying climax. The motif of three wishes, granted to protagonists in folk tales like those collected by the , similarly employs this pattern to frame moral dilemmas and resolutions, reinforcing the tale's didactic purpose. In the visual and performing arts, three-part divisions enable layered storytelling. The three-act dramatic structure, originating in Aristotle's Poetics with its emphasis on a beginning, middle, and end, organizes plays into setup, confrontation, and resolution, influencing countless theatrical works from ancient Greek tragedy to modern drama. Triptychs, multi-paneled artworks hinged together, extend this triadic form in painting; Hieronymus Bosch's The Garden of Earthly Delights (c. 1490–1510), housed in the Prado Museum, unfolds a moral allegory across three oak panels depicting Eden, human indulgence, and infernal punishment. Music frequently harnesses three as a structural cornerstone for harmony and rhythm. Triads, the simplest chords, consist of a root note, a third above it, and a fifth, forming the basis of tonal harmony in Western music from classical compositions to contemporary genres. The waltz exemplifies triadic rhythm through its 3/4 time signature, where three quarter-note beats per measure create a lilting, rotational pulse essential to the dance's elegance, as seen in works by composers like Johann Strauss II. Linguistic expressions often embed three for idiomatic vividness and cultural resonance. The phrase "three sheets to the wind" idiomatically signifies a state of drunkenness, evoking unsteady imbalance. Similarly, "third wheel" describes an individual who feels extraneous in a paired social context, such as accompanying a romantic couple. In literary form, a refers to three interconnected works developing a unified , exemplified by J.R.R. Tolkien's (1954–1955), which spans , , and to chronicle an epic quest in . Film screenwriting adapts the to propel cinematic narratives, allocating roughly the first quarter to world-building and inciting incidents, the middle half to rising conflicts and complications, and the final quarter to climax and denouement. ' A Christmas Carol (1843) illustrates this in adaptation, with its three sequential ghosts—Past, Present, and Yet to Come—visiting to prompt his redemption, a of visions mirroring the novella's transformative arc.