84 (eighty-four /ˈeɪti.fɔːr/) is the natural number following 83 and preceding 85 in the sequence of positive integers.[1]As an even composite number, 84 has the prime factorization $2^2 \times 3 \times [7](/page/+7), making it the product of these prime factors.[2][1] Its positive divisors are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84, totaling twelve divisors.[3][1] The sum of its proper divisors (excluding 84 itself) is 140, which exceeds 84, classifying it as an abundant number.[1]In the sciences, 84 holds significance as the atomic number of polonium (Po), a rare, highly radioactive metalloidelement first isolated in 1898 by Pierre and Marie Curie through radiochemical analysis of pitchblende ore.[4] The isotopepolonium-210 (half-life of 138 days) is used in static eliminators and was employed as a heat source in early space probes, such as the SNAP-1 generator launched in 1959.[4][5]Polonium's scarcity, the short half-life of Po-210, and its toxicity limit broader applications. The most stable isotope, Po-209, has a half-life of about 125 years.[6]Beyond mathematics and chemistry, 84 serves as the international direct dialing country code for Vietnam, assigned by the International Telecommunication Union to facilitate global telephone connectivity to the nation.[7][8] This code, prefixed as +84, supports Vietnam's extensive landline and mobile networks across its 63 provinces.[8]In various cultural and historical contexts, 84 appears sporadically, such as in ancient Egyptian mathematics where it featured in practical calculations documented on papyri like the Rhind Mathematical Papyrus, though not as a uniquely pivotal figure.[9] More prominently in modern numerology and symbolism, 84 is sometimes interpreted as representing stability and introspection due to its factors summing to patterns of balance (e.g., 8+4=12, linked to completeness in some traditions), but these associations lack empirical foundation and vary widely across cultures.
Mathematics
Basic properties
84 is the natural number following 83 and preceding 85 in the sequence of positive integers.[10] It is also equivalent to seven dozen, since one dozen consists of 12 units and $7 \times 12 = 84.[11]In the decimal system, 84 is represented as 84. Its Roman numeral equivalent is LXXXIV, where L denotes 50, XXX denotes 30, and IV denotes 4.[12] In binary, it is 1010100₂, corresponding to $64 + 16 + 4 = 84 in powers of 2.[13] In hexadecimal, it is 54₁₆, as $5 \times 16 + 4 = 84.[14]The English cardinal name for 84 is "eighty-four," used to denote quantity.[15] The corresponding ordinal name is "eighty-fourth," indicating position or order.[16]As a basic arithmetic characteristic, 84 is an even composite number, meaning it is divisible by 2 and has more than two positive divisors.[17] Its prime factorization is $2^2 \times 3 \times 7.[18]
Factorization and divisors
The prime factorization of 84 can be derived through successive division by the smallest prime numbers: 84 ÷ 2 = 42, 42 ÷ 2 = 21, and 21 ÷ 3 = 7, with 7 being prime, yielding $84 = 2^2 \times 3 \times 7.[19] This representation highlights 84's composite nature, as it is the product of multiple distinct primes raised to positive powers.[3]The positive divisors of 84 are all products of these prime factors within their exponent limits: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.[20] The total number of divisors, denoted d(84) or \sigma_0(84), is calculated using the divisor function formula for a number with prime factorization n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}: d(n) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1). For 84, this gives d(84) = (2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12.[21][1]Euler's totient function \phi(84), which counts the integers up to 84 that are coprime to it, is derived from the formula \phi(n) = n \prod_{p \mid n} (1 - 1/p), where the product is over distinct prime factors of n. Substituting the primes 2, 3, and 7 yields \phi(84) = 84 \times (1 - 1/2) \times (1 - 1/3) \times (1 - 1/7) = 84 \times 1/2 \times 2/3 \times 6/7 = 24.[22][23]
Geometric interpretations
In geometry, the number 84 appears as the seventh tetrahedral number, which represents the number of spheres in a tetrahedral pyramid with seven layers. Tetrahedral numbers are figurate numbers formed by stacking triangular numbers, given by the formula T_n = \frac{n(n+1)(n+2)}{6}. For n = 7, this yields T_7 = \frac{7 \times 8 \times 9}{6} = 84, corresponding to the sum of the first seven triangular numbers: $1 + 3 + 6 + 10 + 15 + 21 + 28 = 84.[24][25]Similarly, 84 is the third dodecahedral number, a figurate number modeling the spheres in a dodecahedral arrangement with three layers. The formula for the n-th dodecahedral number is D_n = \frac{n(3n-1)(3n-2)}{2}. Substituting n = 3 gives D_3 = \frac{3 \times 8 \times 7}{2} = 84. This sequence begins 0, 1, 20, 84, ..., illustrating the cumulative layering in a regular dodecahedron structure.[26]Geometrically, 84 can also be expressed as the sum of the twin prime pair 41 and 43, which are the 13th and 14th prime numbers, respectively, highlighting a connection between arithmetic progressions and spatial summations in number theory contexts.Furthermore, 84 qualifies as a largely composite number because it has 12 positive divisors, and no smaller positive integer has more than 12 divisors, establishing it as a milestone in the growth of the divisor function. This property underscores 84's abundance in factorization patterns relative to preceding integers.[27]
Advanced mathematical contexts
84 is a semiperfect number, also known as a pseudoperfect number, which means it equals the sum of a proper subset of its proper divisors, distinguishing it from perfect numbers that use all proper divisors and from merely abundant numbers whose total proper divisor sum exceeds the number itself.[28][29] Specifically, the proper divisors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, and 42, and one such subset is {14, 28, 42}, since $14 + 28 + 42 = 84.[30] This property places 84 in the sequence of semiperfect numbers starting with 20, 24, 28, and so on.[30]In higher-dimensional geometry, 84 appears as the number of 5-dimensional faces (penteracts) in a hepteract, which is the seven-dimensional hypercube or 7-cube. The general formula for the number of k-dimensional faces in an n-cube is \binom{n}{k} 2^{n-k}; for n=7 and k=5, this yields \binom{7}{5} \cdot 2^{2} = 21 \cdot 4 = 84.[31]In the context of hypercomplex numbers, the sedenions form a 16-dimensional nonassociative algebra over the real numbers, constructed via the Cayley-Dickson process from the octonions, and they contain exactly 84 zero divisors. These zero divisors arise from the loss of the division algebra property beyond the octonions, with the 84 pairs identifiable through specific cyclic structures and modes in the algebra's basis elements.[32]The number 84 also features in algebraic number theory through cyclotomic fields, where the Leech lattice—a 24-dimensional even unimodular lattice—can be realized as an ideal lattice in the cyclotomic field \mathbb{Q}(\zeta_{84}), with \phi(84)=24 matching the lattice's dimension. This construction leverages the ring of integers \mathbb{Z}[\zeta_{84}] to embed the lattice's points as algebraic integers of even norm.[33]In group theory, particularly the study of surface automorphisms, Hurwitz's automorphisms theorem states that for a compact Riemann surface of genus g \geq 2, the order of its automorphism group G satisfies |G| \leq 84(g-1). This bound is sharp, achieved by certain Hurwitz surfaces, and implies that the limit superior of |G|/(g-1) for finite subgroups of mapping class groups of such surfaces is exactly 84.[34]
Science
Chemistry
Element 84 in the periodic table is polonium (Po), a chalcogen in group 16.[35] Discovered in 1898 by Marie and Pierre Curie through the chemical analysis of pitchblende ore, polonium was isolated as a highly radioactive substance emitting alpha particles.[36] The Curies named the element after Poland, Marie's homeland, to highlight the country's then-partitioned status. Polonium's discovery marked a key advancement in understanding radioactivity, as it was the first element identified primarily through its radioactive properties rather than chemical ones.Polonium is classified as a metalloid, exhibiting properties intermediate between metals and nonmetals, such as poor electrical conductivity that increases with temperature, contrary to typical metals.[36] It is a rare, silvery-gray semi-metal with a density of approximately 9.32 g/cm³ and a low melting point of 254°C, making it volatile at moderate temperatures.[37] All 33 known isotopes of polonium are radioactive, with polonium-209 being the longest-lived, having a half-life of 125.2 ± 1.8 years and decaying primarily via alpha emission to stable lead-205.[38] The most common isotope, polonium-210 (historically known as radium F), has a shorter half-life of 138.4 days and occurs naturally in the uranium-238decay chain following lead-210 (radium D).Due to its intense alpha radiation, polonium is extremely toxic, with even microgram quantities posing lethal risks through internal exposure, as alpha particles cause severe cellular damage despite low skin penetration.[39] Its primary applications leverage this radioactivity: polonium-210 is used in antistatic devices, such as those for eliminating static charges in textile manufacturing and photographic film processing, by ionizing air to neutralize charges.[36] Additionally, polonium-beryllium mixtures serve as compact neutron sources, as alpha particles from polonium induce neutron emission from beryllium via (α, n) reactions, historically applied in research and calibration.[36] These uses are limited by polonium's scarcity and hazards, with production relying on neutron irradiation of bismuth or extraction from uranium ores.
Physics
In nuclear physics, polonium, with atomic number 84, serves as a key source of alpha particles due to the intense radioactivity of its most stable isotope, polonium-210, which undergoes alpha decay with a half-life of 138.4 days.[40] This property makes polonium valuable for calibrating detectors and conducting experiments on particle interactions, as alpha particles from polonium provide a clean, monoenergetic beam at approximately 5.3 MeV.[41]Polonium-210 finds practical applications in physics instrumentation, including as a component in neutron sources when alloyed with beryllium, where alpha particles induce (α,n) reactions to produce neutrons for scattering studies and reactor calibration.[42] Additionally, its alpha emissions ionize air to neutralize static charges in devices like photographic film cleaners and textile machinery, demonstrating a direct physical application of radioactive decay in electrostatic control. In space physics, polonium-210 powers radioisotope thermoelectric generators (RTGs) by converting decay heat into electricity, though less commonly than plutonium-238 due to its shorter half-life.[41]In quantum information theory, the BB84protocol, developed in 1984, represents a foundational advancement in quantum cryptography, enabling secure key distribution through the preparation and measurement of photon polarization states in two non-orthogonal bases.[43] This protocol leverages the no-cloning theorem and Heisenberg's uncertainty principle to detect eavesdropping, ensuring information-theoretic security for quantum key distribution systems.[43]BB84 has influenced subsequent quantum communication technologies, including satellite-based implementations for global secure networks.[44]Strontium-84, an isotope with mass number 84, has been utilized in ultracold atomic physics to achieve Bose-Einstein condensation, facilitating studies of quantum degenerate gases and many-body interactions near absolute zero.[45]
Astronomy and biology
In astronomy, the number 84 is notably associated with Messier 84 (M84), a giant elliptical galaxy located in the constellation Virgo and a prominent member of the Virgo Cluster. Discovered by Charles Messier in 1781, M84 lies approximately 60 million light-years from Earth and features an active galactic nucleus powered by a supermassive black hole with a mass estimated at around 1.5 billion solar masses.[46][47] This galaxy spans about 80,000 light-years in diameter and is known for its smooth, featureless appearance typical of elliptical galaxies, with dust lanes and shells indicating past mergers.[48]Another significant astronomical object bearing the designation 84 is Caldwell 84 (NGC 5286), a globular cluster in the constellation Centaurus. Discovered in 1826 by Scottish astronomer James Dunlop, this nearly spherical cluster is situated roughly 35,900 light-years from Earth and contains hundreds of thousands of ancient stars, making it one of the oldest structures in the Milky Way galaxy.[49] With an apparent magnitude of 7.6, Caldwell 84 is visible to amateur astronomers under dark skies and serves as a key example of stellar evolution in dense environments.[50]In biology, the number 84 appears in the context of polonium, the chemical element with atomic number 84, which occurs in trace amounts in the human body primarily through environmental exposure via the uranium-238decay chain, including radon inhalation and dietary intake. The average human body burden of polonium-210, the most common isotope, is approximately 15–20 becquerels, with roughly 86% originating indirectly from lead-210 and the rest from direct ingestion, concentrating in organs like the spleen, liver, and kidneys.[36][51]Polonium has no known essential biological role and is highly toxic due to its intense alpha radiation, which can damage tissues at low doses, particularly affecting reproductive and hematopoietic systems in studies of low-level exposure.[52][53]Additionally, 84 denotes the diploid chromosome number (2n=84) in the black rhinoceros (Diceros bicornis), a critically endangeredmammal whose karyotype consists of 41 pairs of mostly acrocentric autosomes, a metacentric X chromosome, and an acrocentric Y chromosome. This chromosomal configuration has been conserved across perissodactyl species and aids in genetic studies of biodiversity and conservation.[54][55] In cellular biology, proteins such as Golgin-84, an integral membrane component of the Golgi apparatus, play a role in vesicle tethering and intra-Golgi transport, highlighting 84's incidental occurrence in molecular nomenclature.[56]
Culture and religion
Hinduism and yoga
In Hindu traditions, particularly within Hatha yoga, the number 84 is revered for denoting the 84 principal asanas, or postures, originally taught by Lord Shiva as the Adiyogi, the primordial yogi who imparted yogic sciences to humanity.[57] These asanas, detailed in classical texts such as the Hatha Yoga Pradipika and Gheranda Samhita, form the foundational practices for aligning the physical body with pranic energies and cosmic rhythms.[58] They symbolize 84 distinct configurations or alignments of human existence, enabling practitioners to harmonize the sun (ha) and moon (tha) energies within for heightened perception and inner transformation.[57]This yogic framework ties into broader Hindu cosmology, where 84 represents the 84th cycle of creation, initiated by Shiva's cosmic roar that birthed successive universes, marking the current epoch as one of completeness in material manifestation before potential dissolution.[59] In the context of samsara, the cycle of birth and rebirth, Hindu scriptures describe 84 lakh (8.4 million) species of life forms—from aquatic beings and plants to insects, birds, animals, and humans—through which the soul (jiva) journeys according to accumulated karma.[60] This enumeration, outlined in texts like the Bhagavata Purana, underscores the exhaustive diversity of embodied existence, with human birth positioned as a rare opportunity for spiritual evolution amid these myriad forms.[60]The spiritual essence of 84 thus embodies the totality of the material world, encapsulating all permutations of life and cosmic cycles that bind the soul to illusion (maya), while yogic disciplines offer pathways to transcend this wholeness toward moksha, or liberation.[57] Practices rooted in these 84 asanas and the awareness of samsara's vastness encourage detachment from sensory attachments, fostering union with the divine and ultimate freedom from rebirth.[60]
Buddhism and Tantra
In Vajrayana Buddhism, the number 84 holds profound symbolic significance through the tradition of the 84 Mahasiddhas, archetypal enlightened masters who attained realization in a single lifetime and represent the totality of the tantric path to non-dual awakening.[61] These figures, drawn from diverse social backgrounds including kings, monks, laborers, and outcastes, embody the accessibility of enlightenment beyond conventional hierarchies, emphasizing spontaneous liberation through tantric practices such as meditation, mantra, and ritual.[62] The number 84 itself is regarded as a "perfect" or "whole" number in tantric cosmology, symbolizing completeness and serving as an archetype for the countless adepts who realized the Buddha's teachings in medieval India.[61]The 84 Mahasiddhas primarily flourished between the 8th and 12th centuries CE in eastern India, during the Pala Dynasty, a period marked by the maturation of tantric Buddhism amid cultural and political upheavals.[61] Prominent among them are Tilopa, a renunciate who achieved enlightenment through visionary encounters, and his disciple Naropa, renowned for enduring extreme trials to realize mahamudra, the great seal of non-dual awareness.[63] Lists of these masters vary across texts, but they collectively authored key tantric works and established lineages that integrated yogic postures—sharing roots with broader Indian spiritual traditions—into practices aimed at transcending duality.[61]Their enduring influence lies in the transmission of Vajrayana teachings to Tibet, where they preserved and propagated tantric methods for direct realization, founding major lineages like Mahamudra and Dzogchen that emphasize the innate purity of mind.[61] This legacy is visually manifested in temple iconography, such as the 84 life-sized statues of the Mahasiddhas seated in yogic postures within a circular hall at the Gorakhnath Temple in Gorakhpur, India, highlighting their role as exemplars of tantric accomplishment.[62] Through their dohas (spontaneous songs of realization) and unconventional lives, the Mahasiddhas challenged orthodox norms, fostering a vibrant tradition that continues to shape Vajrayana's emphasis on embodied enlightenment.[61]
Abrahamic traditions
In the Hebrew Bible, the number 84 appears in connection with significant events involving the patriarch Jacob. During the seven-year famine in Egypt described in Genesis, Jacob and his family entered the land after two years had passed, with the total duration of the famine equaling 84 months (7 years × 12 months).[64] Additionally, calculations based on biblical timelines indicate that Jacob was 84 years old when he married both Leah and Rachel, following his agreement to work for Laban for seven years to wed Rachel, only to be given Leah first and then marrying Rachel a week later.[65]Aaron, the brother of Moses and first high priest, was approximately 84 years old at the time of the first Passover, as he was 83 when confronting Pharaoh (Exodus 7:7) and the plagues culminating in the Exodus occurred over roughly one year.[66]In the New Testament, 84 is associated with Anna the prophetess, a widow from the tribe of Asher who recognized the infant Jesus as the Messiah in the temple. Luke records that she was 84 years old at the time, having devoted her life to fasting, prayer, and worship without leaving the temple.[67] This event underscores themes of devotion and prophetic witness in early Christianity.Within Jewish and Christian traditions, 84 carries symbolic weight as the product of 12 (representing the completeness of God's people, such as the twelve tribes of Israel) and 7 (denoting spiritual perfection or divine completeness).[66] This multiplication evokes wholeness in covenantal contexts, though direct scriptural emphasis on the number itself remains sparse beyond literal references.In Islam, direct mentions of 84 are limited, but the number designates Surah Al-Inshiqaq, the 84th chapter of the Quran, revealed in Mecca and consisting of 25 verses. This surah vividly depicts the cosmic upheavals of the Day of Judgment, including the sky splitting asunder, the earth flattening, and souls reckoning their deeds before Allah.
Other symbolic meanings
In modern esoteric and numerological traditions, the number 84 is interpreted as an angel number signifying achievement through persistent effort, abundance as a reward for past actions, and the karmic principles of cause and effect that govern spiritual growth.[68] This symbolism encourages individuals to maintain focus and diligence in pursuing goals, assuring divine support for success in practical and creative endeavors.[68] The emphasis on karma here reflects universal laws where actions yield corresponding outcomes, fostering a sense of balance and motivation without direct ties to scriptural religious contexts.[68]In broader numerology, 84 reduces to the root number 3 (via 8 + 4 = 12, then 1 + 2 = 3), which embodies creativity, self-expression, and harmony in social interactions.[69] This reduction highlights 84's resonance with optimistic energy that promotes artistic or business-related innovation while building stable foundations for the future, often inspiring others through balanced and uplifting pursuits.[69]Within Chinese cultural numerology, 84 evokes mixed fortunes due to its composition: the digit 8 symbolizes prosperity and wealth (pronounced "ba," akin to "fa" for fortune), while 4 is inauspicious, evoking death (pronounced "si").[70] This combination can represent progress tempered by caution, as the auspicious 8 may offset 4's negativity in contexts like phone numbers or addresses, though interpretations vary regionally.[70] In some Eastern esoteric views influenced by such numerology, 84 thus aligns with nurturing relationships and steady advancement, emphasizing harmony amid potential challenges.[69]
Society and technology
Computing and measurement
In computing, the decimal value 84 corresponds to the uppercase letter 'T' in the ASCII character set, which is a foundational standard for encoding text in digital systems.[71] This representation is used in various programming languages and data processing tasks to handle character data. In binary, 84 is expressed as 01010100, facilitating low-level operations in binary arithmetic and memory allocation.[13]Additionally, port number 84 is assigned by the Internet Assigned Numbers Authority (IANA) for the Common Trace Facility (CTF) protocol, supporting both TCP and UDP for diagnostic and tracing functions in network communications.[72]In measurement contexts, 84 inches equals exactly 7 feet, a conversion commonly referenced in engineering and construction standards for length equivalences in the imperial system.[73] The 84 mm caliber is notably used in the Carl Gustaf recoilless rifle, a man-portable weapon system developed in Sweden and adopted by numerous militaries since the mid-20th century for anti-tank and support roles.[74]
Sports and games
In American football, the number 84 has been worn by several Hall of Fame inductees, including wide receiverRandy Moss, who donned it during his rookie season with the Minnesota Vikings in 1998 and later with the New England Patriots, where he set single-season touchdown records, and tight endShannon Sharpe, who primarily wore it with the Denver Broncos from 1990 to 1999, contributing to three Super Bowl victories.[75] Defensive end Andy Robustelli also wore 84 prominently with the Los Angeles Rams and New York Giants in the 1950s and 1960s, earning six Pro Bowl selections and two NFL championships. In ice hockey, 84 was the last unused jersey number in NHL history until forward Guillaume Latendresse became the first to wear it with the Montreal Canadiens in the 2006–07 season, scoring 16 goals that year.[76]Notable records involving 84 include Boston Red Sox outfielder Ted Williams' MLB-record streak of reaching base safely in 84 consecutive games from July 1 to September 27, 1949, achieved through a combination of hits and walks during a season where he batted .343.[77] In basketball, Oklahoma City Thunder guard Shai Gilgeous-Alexander extended his streak to 84 consecutive games with at least 20 points scored on November 11, 2025, in a 126–102 win over the Golden State Warriors, surpassing previous benchmarks for consistent scoring output.[78]In games, the domino variant known as Eighty-Four is played by four to eight players using double-six sets, with the objective of the bidding team to capture exactly 84 points across tricks, incorporating both pip counts and trick values to reach the total.[79] This partnership game emphasizes strategic bidding and following suit, often without trumps unless specified, and remains popular in social settings for its balance of luck and skill.
Transportation and media
In the United States, Interstate 84 comprises two separate segments totaling over 1,000 miles. The western portion extends approximately 770 miles from Portland, Oregon, eastward through Idaho to a junction with Interstate 80 near Echo, Utah, facilitating travel across the Pacific Northwest and serving key urban centers like Boise.[80] The eastern segment covers about 233 miles from Dunmore, Pennsylvania, across New York and Connecticut to Sturbridge, Massachusetts, providing a vital east-west corridor in the Northeast.[81]European route E84 is an east-west road entirely within Turkey, spanning 163 kilometers from the İpsala border crossing with Greece through towns like Keşan and Tekirdağ to Silivri near Istanbul.[82] In aviation, flight numbers incorporating 84 are common; for example, United Airlines flight UA84 operates daily non-stop service from Newark Liberty International Airport to Ben Gurion Airport in Tel Aviv, covering roughly 5,600 miles.[83]The number 84 features in various media, particularly as a title element evoking military or chronological themes. The 1989 independent film 84C MoPic, written and directed by Patrick Sheane Duncan, depicts a Vietnam War reconnaissance patrol through the lens of an embedded army cameraman, using a found-footage style to immerse viewers in combat realities.[84] In music, albums titled with 84 often reference the year 1984, such as Van Halen's 1984, a blockbuster release that topped charts with synth-driven tracks like "Jump" and marked a commercial peak for the band.Literature on numerology frequently interprets 84 as a vibration symbolizing abundance, inner vision, and leadership toward societal improvement, blending the practical energies of 4 (stability) with 8 (achievement).[85] Classic texts like Cheiro's Book of Numbers (1920) include 84 in its sequential analysis of compound numbers, associating it with karmic lessons in perseverance and material success.Historically, the year 84 AD in the Roman era saw Emperor Domitian's forces, under Governor Gnaeus Julius Agricola, achieve a decisive victory over Caledonian tribes at the Battle of Mons Graupius in northern Britain, extending Roman control northward during the Flavian dynasty.[86]