1
1 (one) is the first and smallest positive integer, serving as the multiplicative identity in arithmetic and algebra, such that for any number a, a \times 1 = 1 \times a = a.[1] It is neither a prime number nor a composite number, as it has exactly one positive divisor, itself.[2] In the natural number system, 1 is the initial element from which all other positive integers are generated by successive addition.[3] In ancient mathematics, the concept of 1 as the fundamental unit traces back to Euclid's Elements, where it is defined as "that by virtue of which each of the things that exist is called one," distinguishing it from greater numbers composed of multiple units.[4] The Peano axioms formalize 1 (or sometimes 0, depending on the convention) as the starting point of the natural numbers, with the successor function defining 2 as the next after 1.[5] As a cardinal number in set theory, 1 represents the cardinality of any singleton set, such as \{ \emptyset \}.[6] Key properties of 1 include its role as the identity element of the multiplicative group of nonzero rationals and reals, and its uniqueness as the only perfect nth power for every positive integer n.[7] In ring theory, 1 is the required multiplicative identity element in unital rings, ensuring compatibility with addition and multiplication operations.[8] These attributes make 1 foundational to number systems, influencing fields from abstract algebra to computational mathematics.[9]Mathematics
Arithmetic and algebraic properties
In arithmetic, 1 is defined as the smallest positive integer, serving as the multiplicative identity in the ring of integers, where for any integer x, $1 \cdot x = x.[10][11] Additively, 1 acts as the successor of 0 in the natural numbers under Peano axioms, satisfying $1 + 0 = 1 and establishing the inductive structure of integers.[12][11] In division, dividing any number x by 1 yields x / 1 = x with no remainder, preserving the value across the operation.[11] Algebraically, 1 raised to any integer power equals itself: $1^n = 1 for all integers n, a consequence of the multiplicative identity property extended through exponentiation rules.[13] Similarly, the principal square root of 1 is 1, as \sqrt{1} = 1, and more generally, $1^{1/n} = 1 for positive integers n.[14] In any field, the multiplicative identity 1 is self-inverse, meaning its reciprocal is itself: $1^{-1} = 1, distinguishing it as the unique element satisfying this while serving as the identity.[15][16]Role in number theory and sets
In number theory, the number 1 holds a unique position as neither a prime nor a composite number, a classification that distinguishes it from all other positive integers greater than 1. Primes are defined as positive integers greater than 1 that have no positive divisors other than 1 and themselves, while composites have additional divisors; 1 fails both criteria since its only positive divisor is itself. This exclusion traces back to ancient Greek mathematics, where Euclid in his Elements (circa 300 BCE) implicitly defined primes as numbers greater than the unit, treating 1 as a foundational "unit" rather than a number in the sequence of primes, thereby omitting it from his proof of the infinitude of primes. Historical debates on 1's primality persisted into the medieval and early modern periods, with some mathematicians occasionally listing 1 among primes in factorization contexts, but by the 19th century, the modern consensus solidified against it to preserve the uniqueness of prime factorization in the fundamental theorem of arithmetic.[17][18][19] Within the ring of integers \mathbb{Z}, 1 serves as the multiplicative identity element, satisfying $1 \cdot n = n for all n \in \mathbb{Z}, and it is the generator of the trivial subgroup \{1\} within the multiplicative group of units U(\mathbb{Z}) = \{\pm 1\}, where the full group consists of elements invertible under multiplication. This role underscores 1's foundational status in algebraic number theory, as the ring \mathbb{Z} is the integral domain where unique factorization into primes holds, with 1 acting as the neutral element that does not contribute to non-trivial factorizations.[20] In set theory, 1 represents the cardinality of any singleton set, such as \{\emptyset\}, which has exactly one element. Under the von Neumann construction of natural numbers, where ordinals are built iteratively from the empty set, 0 is defined as \emptyset, and 1 is the successor ordinal \{0\} = \{\emptyset\}, establishing 1 as the smallest non-empty ordinal and providing a set-theoretic foundation for the natural numbers that aligns with their arithmetic properties.[21][22] Specific results in number theory highlight 1's exceptional behavior. The divisor function \sigma(n), which sums the positive divisors of n, evaluates to \sigma(1) = 1, as 1 has only itself as a divisor, contrasting with \sigma(p) = 1 + p for primes p > 1. Wilson's theorem states that for a prime p, (p-1)! \equiv -1 \pmod{p}; for n=1, however, $1! = 1, and the theorem does not apply since modular arithmetic modulo 1 is degenerate, with all integers congruent to 0, illustrating 1's exclusion from prime-related congruences.[23][24] The Peano axioms formalize 1's role in the construction of natural numbers, defining the successor function S such that 0 is a natural number, S(0) = 1, and every natural number except 0 is a successor of some natural number, with 1 thus being the first successor and the basis for induction. This axiomatic approach, originally formulated by Giuseppe Peano in 1889, ensures that 1 initiates the inductive structure of the positives without being the starting point itself in versions including 0.[25][26]Applications in advanced mathematics
In calculus, the number 1 emerges as the limit \lim_{x \to 0} \frac{\sin x}{x} = 1, a cornerstone result used to establish the differentiability of the sine function and its derivatives at zero, facilitating the development of trigonometric calculus. This limit, proven via the squeeze theorem or geometric arguments, underpins series expansions and integration techniques involving periodic functions.[27] Similarly, the exponential function e^x evaluates to e^0 = 1, reflecting its role as the multiplicative identity in the context of power series \sum_{n=0}^{\infty} \frac{x^n}{n!} and solving differential equations where the constant solution aligns with this base case. In geometry and linear algebra, 1 defines the unit length in Euclidean space \mathbb{R}^n, where the norm \|\mathbf{e}_i\| = 1 for standard basis vectors establishes the metric structure for distances and angles. This unit scaling ensures isometries preserve lengths, as seen in the orthogonal group. The determinant of the n \times n identity matrix I_n is \det(I_n) = 1, signifying that the identity transformation has no scaling effect on volumes or orientations in multilinear algebra.[28][29] In abstract algebraic structures, 1 serves as the identity element in groups under multiplication, satisfying g \cdot 1 = 1 \cdot g = g for all g, and forms the trivial group \{1\} with the operation yielding the identity. In vector spaces over a field, the scalar 1 acts as the multiplicative identity, ensuring $1 \cdot \mathbf{v} = \mathbf{v} for any vector \mathbf{v}, which is axiomatic for preserving linear combinations.[30][31][32] Within complex analysis, 1 denotes the real unit on the complex plane, located at (1, 0) and functioning as the multiplicative identity z \cdot 1 = 1 \cdot z = z for any complex z. In polar representation, numbers on the unit circle have modulus 1 and argument \theta, with the form e^{i\theta} yielding e^{i \cdot 0} = 1 at the positive real axis. This culminates in Euler's identity, e^{i\pi} + 1 = 0, which interconnects 1 with e, i, \pi, and 0, illustrating profound unity in transcendental functions and Fourier analysis.[33][34][35]Linguistics
Etymology and word forms
The word "one" in English derives from Old English ān, a masculine adjective, pronoun, and noun meaning "one, single, or alone," which traces back to Proto-Germanic ainaz. This, in turn, originates from the Proto-Indo-European (PIE) root oi-no-, signifying "one" or "unique."[36] The PIE root oi-no- is one of two primary reconstructions for the concept of "one," the other being sem-, which emphasized unity or togetherness and appears in related forms across branches. Cognates of English "one" from the oi-no- root include Latin ūnus (neuter ūnum), which evolved from Old Latin oinos and carried the sense of singularity.[37] In Sanskrit, the equivalent eka ("one") stems from a variant of oi-no-, specifically óynos or óykos, reflecting shared Indo-Iranian heritage.[38] Greek, however, primarily uses heis (masculine) or hen (neuter) for "one," derived from the alternative PIE root sḗm (full grade) or sm̥- (zero grade), denoting oneness or a single unit. Beyond Indo-European languages, equivalents for "one" show distinct origins. In Chinese, yī (一) is represented by a simple horizontal stroke in oracle bone script, symbolizing a single line or unity, with no direct connection to PIE roots; its pronunciation has remained stable since Middle Chinese.[39] Arabic wāḥid (واحد), meaning "one" or "unique," comes from the Proto-Semitic root waḥad-, implying singularity or unification, as seen in Semitic family reconstructions. In terms of grammatical gender, English "one" functions as the source of the gender-neutral indefinite article "a/an," which lost explicit gender marking over time from its Old English origins. Romance languages retain gender inflection in their words for "one," such as Spanish uno (masculine), una (feminine), and a neuter uno for abstract or impersonal references, reflecting Latin ūnus adaptations. Historical linguistic shifts often repurposed cardinal "one" forms into ordinal "first" constructions across languages. For instance, in Germanic tongues, derivatives of ainaz influenced ordinal expressions like Old High German ērist (blending "one" with superlative elements), while in Romance, Latin ūnus contributed to ordinal uses in compounds denoting primacy. In Indo-Iranian, Sanskrit eka evolved into ordinal contexts for "first" in Vedic texts, marking a transition from counting to sequencing.Pronunciation across languages
In English, the cardinal number "one" is pronounced as /wʌn/, a monosyllabic word with a central unrounded vowel in the stressed syllable. The ordinal form "first" features a variant /fɜːst/ in Received Pronunciation or /fɜrst/ in General American, reflecting a shift from the base form due to morphological changes. In Romance languages, French renders "un" as a nasal vowel /œ̃/, characterized by a rounded front mid vowel with nasalization, often elided in connected speech.[40] Spanish uses "uno," pronounced /ˈu.no/, with a clear bilabial glide and open-mid back vowel, maintaining stress on the first syllable across standard varieties.[41] Asian languages exhibit diverse phonetic structures for the number one. Japanese expresses it as "ichi," transcribed in IPA as /itɕi/, featuring an affricate consonant and high front vowel in a pitch-accent system. Hindi uses "ek," pronounced /eːk/, with a long close-mid front unrounded vowel and a voiceless velar stop.[42] In Mandarin Chinese, "yī" is base-pronounced /i/ with a high level first tone, but tonal variations occur: it shifts to a rising second tone /í/ before fourth-tone words and a falling fourth tone /ì/ before first-, second-, or third-tone words, aiding prosodic clarity. Pronunciation of "one" in Germanic languages, including English, has been shaped by historical vowel shifts such as the Germanic umlaut and lengthening, where Proto-Germanic *ainaz evolved into forms with centralized vowels like /ʌ/ through compensatory adjustments.[43] In Arabic, the term "waḥid" is standardly /wa.ħid/, with a pharyngeal fricative and emphatic realization, though regional dialects introduce variations: Levantine Arabic often softens it to /waːħid/ with vowel lengthening, while Maghrebi forms like Moroccan Arabic reduce it to /waħəd/ with schwa insertion.[44] Dialectal differences in English are subtle for "one," with British Received Pronunciation using a more open /ʌ/ compared to the slightly raised variant in some American dialects, though both converge on /wʌn/ in isolation.[45]Notation and representation
Historical origins
The earliest known representations of the numeral 1 emerged in the context of proto-writing and early numerical notations in ancient civilizations around the fourth and third millennia BCE. In Mesopotamia, cuneiform script, developed by the Sumerians in southern Iraq circa 3000 BCE, used a simple vertical wedge (𒁹) to denote the value 1 within a sexagesimal (base-60) system for counting and accounting on clay tablets.[46] This symbol, impressed with a reed stylus on wet clay, formed the basis for higher numbers by repetition or combination with other wedges, reflecting the practical needs of urban administration in early city-states like Uruk.[47] In ancient Egypt, hieroglyphic numerals also appeared around 3000 BCE during the Early Dynastic Period, with the value 1 represented by a single vertical stroke (Gardiner's sign Z1, 𓏤), often used in monumental inscriptions and administrative records on papyrus or stone.[48] This stroke-based notation was part of a decimal system that facilitated taxation, measurement, and pyramid construction, as evidenced in texts like the Palermo Stone.[49] While the reed leaf hieroglyph (M17, 𓇋) primarily served as a phonetic sign for the sound /i/, the numeral 1 relied on the straightforward stroke for its ideographic simplicity.[48] The Indus Valley Civilization, flourishing circa 2500 BCE in present-day Pakistan and northwest India, featured undeciphered script on seals and tablets that includes presumed numeral signs, with a horizontal bar or stroke often interpreted as a precursor to the value 1 in a potential additive system.[50] These markings, found at sites like Mohenjo-daro and Harappa, appear in short sequences alongside pictographic elements, suggesting use in trade and inventory without a confirmed phonetic component. In East Asia, Chinese oracle bone script from the late Shang Dynasty (circa 1200 BCE) inscribed on animal bones and turtle shells for divination employed a single horizontal stroke to represent 1, a form that has persisted with minimal variation into modern Chinese characters.[51] This pictographic simplicity underscored early numerical recording in ritual and calendrical contexts at sites like Anyang.[52] The Roman numeral system, originating from Etruscan notations around the 8th century BCE, represented 1 with I, a vertical stroke adapted from tally marks in early Italic scripts.[53] Similarly, the Greek acrophonic numeral system, used from the 7th century BCE for monetary and civic records, denoted 1 with a vertical stroke (Ι), an obvious tally mark rather than an alphabetic form like alpha (Α), distinguishing it from the later alphabetic system where alpha represented 1.[54]Evolution in writing systems
The symbol for the number 1, often a simple vertical stroke, underwent significant transformations across alphabetic and positional numeral systems from antiquity through the medieval period. In ancient Near Eastern writing, such as cuneiform around 3000 BCE, it appeared as a vertical wedge representing unity, laying early groundwork for linear representations. By circa 1000 BCE, the Phoenician letter yodh (𐤉), depicting a hand or arm as a vertical line with short horizontal extensions, evolved into the Greek iota (Ι), a plain vertical stroke that served as both a letter and influenced numerical notation. This Phoenician form also shaped the Hebrew yod (י), a compact vertical mark used in Semitic scripts for phonetic and numeric purposes.[55][56] In the Roman numeral system, which developed from Etruscan influences around the 7th century BCE, the symbol I emerged directly from tally marks—simple incisions or strokes for counting—and was standardized as the representation for one by the 1st century CE, appearing consistently in inscriptions, coins, and legal documents. Parallel to this, in the Indian subcontinent, Brahmi numerals from around the 3rd century BCE featured 1 as a basic vertical line in an additive system, distinct from later positional developments. This form evolved through regional scripts such as Gupta and Siddham into the Eastern Arabic numeral ١ by the 8th–9th centuries CE, featuring a vertical stem with a subtle serif-like flag at the top for clarity in cursive styles.[57][58] The transmission of the positional 1 to Europe marked a pivotal shift in the medieval era. Italian mathematician Leonardo of Pisa, known as Fibonacci, popularized the Hindu-Arabic numeral system—including the straight vertical 1—in his 1202 treatise Liber Abaci, drawing from North African and Arabic sources to demonstrate its superiority for calculation over Roman tallies. This adoption facilitated the integration of the symbol into Latin manuscripts, where scribal styles further adapted it. In Gothic scripts prevalent from the 12th century, the 1 took on angular, pointed forms to align with the script's compressed, vertical aesthetic, enhancing density on the page. Conversely, during the Carolingian revival from the 8th–9th centuries, the numeral began incorporating small horizontal serifs at the top and bottom of the vertical stroke, improving readability in the rounded minuscule hand that influenced later European typography.[59]Modern typographic variations
In the Gutenberg era, the numeral 1 in printed works like the 42-line Bible (circa 1450s) featured a straight vertical stem, often accented with serifs to align with the Textura blackletter typeface's angular, gothic style.[60] This design emphasized legibility in dense religious texts, where numerals appeared sparingly for chapter or line numbering.[61] Modern typography distinguishes between serif and sans-serif forms of the numeral 1, with serifs adding decorative strokes for enhanced readability in print. In sans-serif fonts like Helvetica (designed 1957), the 1 consists of a simple, uniform stroke with a subtle hook at the top and a flat base, promoting a clean, modernist aesthetic. Conversely, serif fonts such as Times New Roman (designed 1931) incorporate a flagged base—a small, angled serif at the bottom—alongside a top serif, creating subtle visual cues that guide the eye in body text. These variations reflect evolving priorities: sans-serif for screen clarity and minimalism, serif for traditional printed elegance.[62] The Unicode standard encodes the basic numeral 1 as U+0031 (DIGIT ONE), a half-width Arabic digit essential for global text processing and compatible with ASCII.[63] For East Asian typography, where proportional spacing is common, the full-width variant U+FF11 (FULLWIDTH DIGIT ONE, １) provides a wider glyph to harmonize with ideographic characters in languages like Japanese and Chinese.[64] These encodings ensure consistent rendering across digital platforms while supporting cultural adaptations.[65] In digital environments, particularly monospace code fonts, the numeral 1 is stylized to distinguish it from similar characters like lowercase 'l' and uppercase 'I', preventing errors in programming. Fonts such as Consolas (designed 2004 for Microsoft) achieve this through a tapered top serif on the 1, a curved base on 'l', and serifs on 'I', ensuring clarity at small sizes on screens.[66] This deliberate differentiation has become standard in developer tools, enhancing usability in technical contexts. Recent trends in typography leverage variable fonts, which allow dynamic adjustment of glyph attributes like width via the 'wdth' axis, enabling the numeral 1 to condense or expand seamlessly within a single file for responsive design.[67] For instance, fonts like Roboto Flex vary the 1's stroke width and proportions across weights, optimizing for web and mobile layouts. Additionally, Unicode provides stylized variants such as U+1D7CF (MATHEMATICAL BOLD DIGIT ONE, 𝟏), resembling an emoji-like bold form for mathematical or decorative use in digital media.[68] These innovations support fluid, context-aware rendering in contemporary applications.[69]Broader contexts
In science and computing
In physics, the number 1 plays a fundamental role in defining fundamental constants and units. The speed of light in vacuum, denoted as c, is exactly 299792458 m/s, serving as the basis for the meter in the International System of Units (SI). In natural units commonly used in particle physics and quantum field theory, c = 1 and the reduced Planck constant \hbar = 1, simplifying equations by setting these constants to unity and expressing quantities in terms of energy or inverse length scales. This convention underscores 1 as the normalized value for dimensionless unity in relativistic and quantum contexts. In chemistry, 1 denotes the atomic number of hydrogen, the simplest element, which consists of a single proton in its nucleus. On the pH scale, which measures the acidity of aqueous solutions logarithmically based on hydrogen ion concentration, a pH of 1 indicates a strongly acidic solution, corresponding to a hydrogen ion concentration of 0.1 mol/L. In computing, binary representation uses 1 to signify the "on" or true state, while 0 represents "off" or false, forming the basis of Boolean logic in digital circuits and programming. The ASCII standard assigns code 49 (decimal) to the digit '1', enabling its encoding in text and data processing. In the IEEE 754 floating-point standard, the value 1.0 is represented exactly without rounding error, as its binary form aligns perfectly with the format's significand and exponent fields. Programmers often initialize loop counters to 1 in languages like C or Java when using 1-based indexing for arrays or sequences, as seen in common iterative constructs such asfor (int i = [1](/page/1); i <= n; i++). In quantum computing, the state |1⟩ serves as one of the two computational basis states for a qubit, alongside |0⟩, representing the excited or spin-up configuration in physical implementations like superconducting circuits or trapped ions.
In biology, 1 characterizes unicellular organisms, which consist of a single cell capable of performing all life functions independently, such as bacteria and protozoa. Human chromosome 1 is the largest, containing approximately 2,000 to 2,100 genes that provide instructions for making proteins and spanning about 249 million base pairs, making it critical for genetic studies of diseases like cancer.[70]