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← 1 2 3 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinaltwo
Ordinal2nd (second)
Numeral systembinary
Factorizationprime
Gaussian integer factorization
Prime1st
Divisors1, 2
Greek numeralΒ´
Roman numeralII, ii
Greek prefixdi-
Latin prefixduo-/bi-
Old English prefixtwi-
Binary102
Ternary23
Senary26
Octal28
Duodecimal212
Hexadecimal216
Greek numeralβ'
Arabic, Kurdish, Persian, Sindhi, Urdu٢
Ge'ez፪
Bengali২
Chinese numeral二，弍，貳
Devanāgarī२
Santali᱒
Tamil௨
Kannada೨
Hebrewב
ArmenianԲ
Khmer២
Maya numerals••
Thai๒
Georgian Ⴁ/ⴁ/ბ(Bani)
Malayalam൨
Babylonian numeral𒐖
Egyptian hieroglyph, Aegean numeral, Chinese counting rod||
Morse code.._ _ _

2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number.

Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.

Mathematics

[edit]

The number 2 is the second natural number, after 1. Each natural number, including 2, is constructed by succession, that is, by adding 1 to the previous natural number.[1] 2 is the smallest and the only even prime number, and the first Ramanujan prime.[2] It is also the first superior highly composite number,[3] and the first colossally abundant number.[4]

An integer is determined to be even if it is divisible by two. When written in base 10, all multiples of 2 will end in 0, 2, 4, 6, or 8;[5] more generally, in any even base, even numbers will end with an even digit.

A digon is a polygon with two sides (or edges) and two vertices.[6]: 52  Two distinct points in a plane are always sufficient to define a unique line in a nontrivial Euclidean space.[7]

A set that is a field has a minimum of two elements.[8]

Binary is a number system with a base of two, it is used extensively in computing.[9]

List of basic calculations

[edit]
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 25 50 100 1000
2 * x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 40 50 100 200 2000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 ÷ x 2 1 0.6 0.5 0.4 0.3 0.285714 0.25 0.2 0.2 0.18 0.16 0.153846 0.142857 0.13 0.125 0.1176470588235294 0.1 0.105263157894736842 0.1
x ÷ 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2x 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576
x2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400

As a word

[edit]

Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two.[10] Two is a noun when it refers to the number two as in two plus two is four.

The word two is derived from the Old English words twā (feminine), tū (neuter), and twēġen (masculine, which survives today in the form twain).[11]

Evolution of the Arabic digit

[edit]

The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages (and Korean Hanja) still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern digit.[12]

In science

[edit]
  • The first magic number - number of electrons in the innermost electron shell of an atom.[13]

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

2

View on Grokipedia
from Grokipedia
2 (two) is the immediately following 1 and preceding 3. It is the smallest , the only , the smallest greater than 1, and the base of the . As a , 2 has exactly two distinct : 1 and itself. It is unique among primes because it is the only —all other even numbers are divisible by 2 and therefore . This property makes 2 foundational in number theory, distinguishing it from all . 2 serves as the (base) of the , the foundation of digital computing. In , numbers are represented using only the , with each position representing a . This system aligns with the states (on/off, true/false) of and underpins all modern computer operations. 2 is also notable as a highly composite number, possessing more (two: 1 and 2) than any smaller . Highly composite numbers play an important role in the study of and number theoretic structures. The number 2 appears in many fundamental mathematical concepts, including duality, , and , and its properties influence fields from and to computer science and .

Etymology and representation

Etymology

The word "two" in derives from twā (feminine and neuter) and twegen (masculine), which trace back to twai and ultimately to the root dwo- "two". This root has numerous across , including Latin duo, Ancient Greek dúō, Sanskrit dva, and Germanic forms like German zwei and Dutch twee. These shared forms demonstrate the common ancestry of the numeral in the . In non-Indo-European languages, the word for "two" has independent origins, such as and ithnān (اثنان).

Written representation

The number 2 is most commonly represented in modern usage by the Hindu-Arabic digit 2, which serves as the standard symbol in worldwide (except in certain regional variants described below). This digit evolved from ancient Indian , where early forms of the numeral for two appeared as angular or curved strokes around the ; it subsequently developed through and into more rounded forms adopted in by the , before diverging into the Western European typographic shape familiar today. In Roman numerals, 2 is represented as II, consisting of two vertical strokes derived from the used additively. In the , 2 is written as 二 (a pair of horizontal lines), a form dating back to ancient usage and still standard in everyday writing; a more complex variant 贰 is employed in financial and formal contexts to prevent alteration. In the (a ), 2 is depicted with two dots, typically arranged vertically or horizontally above any bars representing multiples of five. In some Arabic-speaking regions, the ٢ is used instead of the Western 2, reflecting a parallel branch of evolution from the same but preserving a more angular or looped form. Modern typographic and digital representations of 2 vary by and style: common forms include the "closed" version with a looped base (resembling a swan or Z-curve) and the "open" version with a straighter, disconnected base; in , it is formed by five illuminated segments. For its use in specific numeral systems, see .

Mathematics

Elementary properties

2 is the immediately following 1 and preceding 3 in the sequence of . It is the of 1 and the predecessor of 3. As the smallest greater than 1, 2 holds a foundational position in the ordering of . It is also the smallest even , being divisible by 2 with no remainder. 2 is a highly composite number, possessing more than any smaller positive integer (specifically two divisors: 1 and itself). Its prime factorization is simply 2 = 2, reflecting its status as a . It is the only , with all larger even numbers being .

Prime number

2 is the only even . Any even integer greater than 2 is divisible by 2 and greater than 2, hence has a other than 1 and itself, making it . This establishes that no even number larger than 2 can be prime, leaving 2 as the unique . As the smallest and the only even one, 2 plays a foundational role in the Fundamental Theorem of Arithmetic, which states that every greater than 1 has a unique (up to ordering of factors). Every even integer greater than 1 must include 2 as a factor in its unique prime factorization. The uniqueness of 2 as the is central to the Goldbach conjecture, which posits that every even integer greater than 2 can be expressed as the sum of two primes. The base case is 4 = 2 + 2, the only instance using 2 (twice). For larger even integers, any attempt to use 2 would require the other addend to be even and greater than 2, which is ; thus, such representations must use two (if true). are of the form 2p−12^p - 1, where the exponent pp is itself prime. This form, relying on powers of the prime 2, has yielded many of the , with primality testable via efficient methods such as the .

Even number

2 is an even number, as it is divisible by 2 with no and can be expressed as 2=2×12 = 2 \times 1. It is the smallest positive even integer. An even number is an that can be written in the form 2k2k, where kk is any integer. This definition applies to positive, negative, and zero values, with the even integers forming the set …,−4,−2,0,2,4,6,…\dots, -4, -2, 0, 2, 4, 6, \dots. 2 is the only , since any larger even integer is divisible by 2 and thus has additional factors beyond 1 and itself. Parity classifies as even (divisible by 2) or odd (not divisible by 2), with 2 serving as the foundational even number. The arithmetic parity rules follow directly from this classification:
  • The sum of two even numbers is even.
  • The sum of an even number and an odd number is odd.
  • The sum of two odd numbers is even.
  • The product of an even number and any is even.
These properties hold because even numbers contain a factor of 2, which is preserved or introduced in addition and multiplication. In modular arithmetic, even numbers are congruent to 0 modulo 2, while odd numbers are congruent to 1 modulo 2. Thus, 2≡0(mod2)2 \equiv 0 \pmod{2}, and arithmetic operations modulo 2 correspond exactly to parity calculations.

Arithmetic operations

In , the number 2 plays a foundational role in the concepts of doubling and halving, serving as the smallest greater than 1 that meaningfully alters other numbers through multiplication or division. Addition involving 2 corresponds to incrementing by two units. A key is that adding any number to itself yields twice that number: n+n=2nn + n = 2n. This defines the operation of doubling a quantity. Multiplication by 2 produces the same result as doubling, so 2×n=2n2 \times n = 2n. As the smallest non-trivial multiplier (beyond the identity element 1), 2 is the base for the simplest non-identity scaling in . Division by 2 is the inverse process, known as halving. When applied to an even number, division by 2 always yields an , reflecting that even numbers are exactly divisible by 2. In , 2 raised to successive positive integer powers generates the sequence of : 21=22^1 = 2, 22=42^2 = 4, 23=82^3 = 8, 24=162^4 = 16, and so on. Each term doubles the previous one, forming a with ratio 2. These powers of 2 are notable for their , where each corresponds to a single (see ).

Numeral systems

In , a number's representation uses digits whose values are determined by their position, with each position corresponding to a power of the (). The number 2 has a straightforward representation in most . In the (), it is written as the single digit 2, with 2 × 100 = 2. In the (base 2), the only available digits are 0 and 1, so 2 cannot be expressed with a single digit. It is instead written as , representing 1 × + 0 × = 2. In any integer base b > 2, the digit 2 belongs to the (), so 2 is represented simply as 2 in that base, with value . Examples include: The binary case is unique among common bases because it requires two digits to express 2, reflecting the foundational use of as in that system (see ).

In set theory

In set theory, the 2 represents the of any set with exactly two distinct elements, such as {a, b} where a ≠ b. This cardinality is denoted as \mathfrak{2} or simply 2, and it is the smallest greater than 1. In ZFC set theory, are constructed as . The ordinal 0 is defined as the ∅, 1 as {∅}, and 2 as {∅, {∅}}. This construction ensures that 2 is the set whose elements are all ordinals strictly less than itself, and it has exactly two elements, making it the canonical representative for the . The von Neumann definition allows to be built purely from sets without additional , with defined as α + 1 = α ∪ {α}. Thus, 2 arises as the of 1. The Kuratowski definition of the ordered pair, (a, b) = {{a}, {a, b}}, relies on unordered pairs and singletons to encode ordered pairs as sets. The inner sets {a} and {a, b} have 1 and at most 2, respectively, demonstrating how 2 underpins the reduction of ordered structures to unordered sets in foundational set theory. This definition enables relations to be constructed as sets of such ordered pairs.

Other mathematical properties

2 is a highly composite number, meaning that it has more than any smaller positive integer. It is also the smallest , a p for which 2p + 1 is also prime; in this case, 2 × 2 + 1 = 5, which is prime. In the Fibonacci sequence (with the common indexing where F₁ = 1, F₂ = 1), 2 appears as the third term: F₃ = 2. The number 2 plays a fundamental role in various advanced mathematical contexts as the minimal non-trivial dimension, for instance underlying the geometry of the and the representation of as a two-dimensional real vector space.

Science

Astronomy

In astronomy, the number 2 appears prominently in several structural and systemic contexts. , in which two stars are gravitationally bound and orbit a , are widespread and represent a fundamental class of stellar configurations. Such systems are among the most common forms of , playing key roles in studies of , mass transfer, and phenomena such as and gravitational waves. The —an imaginary sphere of infinite radius centered on the observer and used to project the positions of —is divided into two hemispheres: the and the , separated by the . This division facilitates coordinate systems and observational frameworks in positional astronomy. The Earth-Moon system is sometimes described as a (or ) system in certain astronomical contexts. This characterization arises from the Moon's relatively large mass compared to Earth (approximately 1/81 of Earth's mass), which places their common inside Earth but unusually close to the surface compared to typical , distinguishing it from most such systems.

Biology

In biology, the number 2 features prominently in key structures and across various organisms. Most , including animals and plants, are , meaning their contain two complete sets of chromosomes—one inherited from each parent. This diploid state supports during and provides redundancy against deleterious mutations. Many animals display , resulting in paired organs and such as , ears, , , and limbs. This arrangement facilitates efficient and , with the paired structures often mirroring each other across a . reproduce asexually through , in which a parent cell duplicates its genetic material and divides into two genetically identical daughter cells. This rapid, simple process allows to expand quickly under favorable conditions.

Chemistry

In chemistry, the number 2 appears prominently in molecular structures and bonding patterns. Many exist as in their standard states at room temperature, meaning their stable form consists of two identical bonded together. These include (H₂), nitrogen (N₂), oxygen (O₂), fluorine (F₂), chlorine (Cl₂), bromine (Br₂), and iodine (I₂). This diatomic nature arises from the atoms achieving stable through , often involving sharing of two electrons in a or more in multiple bonds, as in the of O₂. Oxygen frequently exhibits a valence of 2 in its compounds, forming two to complete its . A representative example is water (), where one oxygen atom bonds to two atoms. In coordination chemistry, a coordination number of 2 is uncommon but leads to linear geometry around the central metal ion. This occurs primarily in d¹⁰ complexes of group 11 metals (Cu⁺, Ag⁺, Au⁺) and Hg²⁺, such as in [Ag(NH₃)₂]⁺ and linear HgCl₂. Linear structures are also seen in some main-group molecules, such as gaseous BeCl₂, where the central atom bonds to two ligands or atoms.

Physics

In physics, the number 2 plays several fundamental roles, particularly in dimensionality, particle classification, and quantum phenomena. Plane physics, referring to systems confined to , exhibits distinct behaviors compared to . In three spatial dimensions, such as and follow the , where intensity decreases as 1/r21/r^2, a direct consequence of flux spreading over a spherical surface area proportional to r2r^2. In contrast, in two spatial dimensions, the analogous "surface" is a circumference proportional to rr, leading to an inverse-linear law (1/r1/r) for point-source fields. This dimensionality dependence shapes in planar models, such as certain condensed matter systems or theoretical 2+1-dimensional gravity. Spin-1/2 particles constitute a major class of fermions, including , , and , with of ħ/2. These particles obey and the , preventing two identical fermions from occupying the same . The quantum description of spin-1/2 involves a two-dimensional Hilbert space for the spin degree of freedom, corresponding to two (typically spin-up and spin-down along a chosen axis). This two-state nature underpins many quantum mechanical effects. The binary spin states also provide the physical basis for in (see ### Computing). represents a core principle of quantum mechanics, whereby entities such as photons and manifest both particle-like (localized, discrete) and wave-like (extended, continuous) properties, depending on the measurement context. This duality reconciles seemingly contradictory behaviors observed in experiments like the photoelectric effect (particle aspect) and (wave aspect). The concept emphasizes two complementary descriptions of nature, as articulated in the .

Technology

Binary numeral system

The is a with (base) 2, using only the digits 0 and 1. Each position in a corresponds to a , with the rightmost position representing 20=12^0 = 1, the next position to the left representing 21=22^1 = 2, then 22=42^2 = 4, 23=82^3 = 8, and so on. Any positive integer can be uniquely represented as a sum of distinct powers of 2. The themselves have particularly simple representations in : 20=122^0 = 1_2, 21=1022^1 = 10_2, 22=10022^2 = 100_2, 23=100022^3 = 1000_2, and generally 2n=12^n = 1 followed by nn zeros in binary. This pattern arises directly from the being 2, where each successive power shifts the single non-zero digit one place to the left. The modern development of as a complete arithmetic system is attributed to , who formalized it in his 1703 paper Explication de l'arithmétique binaire. In this work, Leibniz demonstrated how all integers and their arithmetic operations could be expressed using only the characters 0 and 1, emphasizing the system's simplicity and universality. presented as a method of calculation that reduces complexity by limiting symbols to two, with positions denoting successive , and he explored its philosophical implications, including analogies to (unity arising from zero). The paper included examples of addition, subtraction, multiplication, and division in binary, showing its viability as a numeral system.

Computing

In digital computing, the number 2 is foundational because computers rely on , representing all data and instructions using only two values: 0 and 1. This stems from the practical limitations and advantages of electronic hardware, where can reliably distinguish between two stable states—typically a high voltage level (1) and a low voltage level (0)—making the most efficient and error-resistant choice for . The fundamental unit of information in computing is the bit (), which can hold either a 0 or a 1. Bits are grouped into larger units for practical use, with the most common being the byte, which consists of 8 bits (equal to 2³). A byte can therefore represent 2⁸ = 256 distinct values, a capacity that historically proved sufficient for like and has made the byte the standard unit for and in modern computers. To represent (both positive and negative), computers predominantly use two's complement representation. In this system, the negative of a number is obtained by inverting all its bits (changing 0s to 1s and 1s to 0s) and then adding 1; this method allows addition and subtraction to be performed using the same hardware circuitry, eliminates the need for separate sign-magnitude handling, provides a unique representation for zero, and simplifies .

Measurement and units

In the measurement of angles using , a full rotation around a circle corresponds to an of 2π2\pi radians. This follows from the definition of the as the angle subtended by an equal in length to the radius of the circle; for a complete of 2πr2\pi r, the angle is therefore 2πr/r=2π2\pi r / r = 2\pi radians. In , () relies on two separate channels—typically designated left and right—to record and reproduce sound with spatial characteristics, creating an illusion of direction and depth when played through two speakers. Two major systems dominate global measurement standards: the (), the with , and the , which remains in use primarily in the United States for everyday applications. Many contexts require dual labeling or between these systems. In digital measurement contexts, (base-2) representation is foundational, as discussed in the Computing subsection.

Culture and symbolism

Religion and mythology

In various religious and mythological traditions, the number 2 is frequently associated with , representing opposing or complementary forces such as good and evil, light and darkness, or unity through pairing. Zoroastrianism prominently features as a core principle, with the benevolent creator god opposed by the destructive spirit (also known as ). This opposition embodies the cosmic struggle between good and evil, with the two forces in complete separation yet interconnection. In the , often conveys meanings of union, division, or . A key example appears in the narrative of , where instructs to bring animals aboard in pairs (male and female) to preserve life through : one pair of each and seven pairs of each . This pairing underscores the theme of duality in reproduction and survival. The number 2 also signifies union in contexts such as , where a man and woman, though two individuals, become one. across cultures include that reflect the significance of 2 as a symbol of or complementarity, though specific examples vary by tradition.

Language and idioms

In , the number 2 frequently appears in and expressions that evoke ideas of duality, similarity, pairing, or efficiency in handling multiple objectives. One such is "two peas in a pod", which refers to two people (or sometimes things) that are extremely similar in appearance, character, or behavior. The expression draws from the observation that peas from the same pod are virtually indistinguishable. The "kill two birds with one stone" means to accomplish two goals with a single action or effort. It is first attested in writing in 1656 in 's The Questions Concerning Liberty, Necessity, and Chance. The phrase "second to none" describes something or someone that is unmatched or superior to all others in quality or excellence. is among the earliest known users of the expression, in his play (circa 1589–1594). Other common English idioms incorporating 2 include "" (both parties are required or responsible for a situation), "two's company, three's a crowd" (two people form an ideal intimate group, and a third is unwelcome), and "" (an evil action is not justified by another evil action). In , several languages feature a dual number as a distinct from and , used specifically to refer to exactly two entities (such as nouns, , or verbs inflected for pairs). This appears in languages including , , and .

Arts and entertainment

In music, the term denotes a or performance for two singers or instrumentalists, in which the participants typically have equal importance and interact closely. The word derives from the Italian , a diminutive of , meaning "two," reflecting the fundamental pairing central to this form. Duets appear across genres, from classical vocal works and to collaborations. In the , two-dimensional (2D) works exist on a flat plane with only length and width, lacking actual depth. This category encompasses , paintings, , photographs, and other media applied to surfaces such as paper, canvas, or . 2D art relies on elements like , , , and to create visual interest within the confines of a single plane. In film and other serialized entertainment, the number 2 frequently appears in titles to indicate a or second part. Many major productions use designations such as "Part 2" or "II" for the continuation of a story, as seen in acclaimed examples including (1974) and (2011). These instances often represent high-impact installments that build on or surpass their predecessors.

Sports and games

The number 2 frequently appears in the rules and structure of sports and games, often symbolizing opposition, partnership, or division of play. Many team sports divide matches into two halves, creating a natural midpoint for rest, strategy, and tactical shifts. In association football (soccer), under and affiliated rules, a standard match consists of two equal halves of 45 minutes each, with a of up to 15 minutes. This two-half format is also used in (two 40-minute halves) and certain formats of basketball, providing clear segments for scoring and momentum changes. In , features two players per team competing against another pair, with specific rules governing serving order, , and partner coordination. The same player serves throughout a , after which service alternates between teams and partners. This format emphasizes teamwork and across the wider . Classic board games often center on two players for direct, head-to-head competition. Chess is played between two opponents on an , with each controlling an army of pieces in a battle of strategy and foresight. Similarly, (also known as draughts) is a two-player strategy game where participants move pieces diagonally to capture opponents by jumping over them. These examples highlight how the number 2 structures competition through paired participants or divided time, fostering balanced and focused play across diverse activities.

Other cultural references

The number 2 is frequently associated with the concept of across philosophical traditions, symbolizing opposition, , and complementary pairs. In , the Pythagoreans regarded the number 2 as embodying duality, the first even number representing opposites and mutual antithesis, often contrasted with the monadic unity of the number 1. This symbolism of duality extends to various philosophical frameworks, including Western mind-body dualism, which posits two fundamentally distinct kinds of substances or properties—mental and physical. In , duality manifests in the complementary yet opposing forces of (see ### Religion and mythology). Such associations highlight the number 2's role in cultural and philosophical thought as a marker of and , distinct from its mathematical properties.

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