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Number

A number is a mathematical object used to count, measure quantities, and label entities, serving as a foundational element in arithmetic and other branches of mathematics.^[1] Numbers are expressed through symbols called numerals, such as digits in positional notation systems like the decimal system, which facilitate calculations including addition, subtraction, multiplication, and division.^[2] The concept of numbers originated in ancient civilizations, with evidence of systematic use dating back over 5,000 years to the Egyptians and Babylonians, who developed methods for arithmetic on whole numbers to support commerce, architecture, and astronomy.^[3]^[4] Initially focused on natural numbers—positive integers starting from 1 used for counting—the number system evolved to incorporate zero as a placeholder and whole number around the 7th century in India, enhancing positional notation.^[4] Further advancements introduced integers (including negatives) by the 3rd century BCE in China and later in India, allowing representation of debts and directions.^[5] Rational numbers, expressible as fractions of integers, emerged to handle divisions, while irrational numbers like √2 were recognized in ancient Greek mathematics around the 5th century BCE for their non-terminating, non-repeating decimal expansions.^[6] The real numbers, comprising rationals and irrationals, form a complete ordered field essential for continuous quantities such as distances on a line.^[7] Complex numbers, involving imaginary units like i where i² = -1, were formalized in the 16th century by mathematicians such as Rafael Bombelli to solve equations like x² + 1 = 0, extending applications to fields like physics and engineering.^[8] In the 19th century, axiomatic foundations like Peano's axioms for natural numbers and Dedekind cuts for reals established numbers rigorously as abstract structures.^[9] In modern mathematics, numbers underpin diverse areas including number theory, which studies properties of integers and primes, and analysis, which relies on real and complex numbers for limits and calculus.^[10] These extensions reflect humanity's progressive abstraction of quantity to model increasingly sophisticated real-world and theoretical problems.^[3]

History

Ancient origins and early numeral systems

The earliest evidence of human engagement with numbers appears in prehistoric tally marks, simple incisions on bones and stones used for counting. One of the oldest known artifacts is the Ishango bone, discovered in the Democratic Republic of Congo and dated to approximately 20,000 BCE, featuring grouped notches that suggest systematic counting or possibly rudimentary arithmetic operations.^[11] These markings represent precursors to formal numerical notation, indicating an innate sense of quantity among early humans without symbolic abstraction.^[12] In ancient Mesopotamia, the Sumerians developed one of the first organized numeral systems around 3500 BCE, using cuneiform script on clay tablets to record quantities for trade, taxation, and administration. This additive system employed wedge-shaped symbols to denote powers of 10 up to 60, combined with a sexagesimal (base-60) structure for smaller units, allowing representation of large numbers through repetition rather than position.^[13] Similarly, in ancient Egypt by about 3000 BCE, hieroglyphic numerals emerged as a decimal (base-10) system, with distinct symbols for 1, 10, 100, and higher powers, repeated additively to form any number; these were inscribed on monuments and papyri for practical purposes like measuring land and building pyramids.^[14] The Babylonians, inheriting and refining Sumerian methods around 2000 BCE, fully embraced the sexagesimal system, which facilitated advanced calculations in astronomy and commerce. This base-60 approach enabled fractional divisions without a zero placeholder, influencing enduring conventions such as dividing the hour into 60 minutes and the minute into 60 seconds, as well as the circle into 360 degrees for angle measurements.^[15] Early Greek mathematicians built on these foundations, shifting toward more theoretical treatments of numbers. In Euclid's Elements (circa 300 BCE), Book VII provides definitions and propositions on arithmetic, including concepts of multiples, proportions, and prime numbers, laying groundwork for axiomatic reasoning about natural numbers that anticipated later formal systems.^[16]

Development of zero and positional notation

The development of zero as a placeholder and the adoption of positional notation marked a pivotal advancement in numeral systems, enabling more efficient arithmetic beyond earlier additive methods. In ancient China, rod numerals emerged around the 2nd century BCE, utilizing bamboo or ivory rods arranged on a counting board to represent digits in a positional framework.^[17] These rods formed a decimal place-value system where position indicated powers of ten—units in the rightmost column, tens to the left, and so on—with empty spaces serving as placeholders for zero, facilitating operations like addition and multiplication through visual patterns and carrying over values.^[18] This system influenced later tools such as the abacus, which by the 14th century CE incorporated beads on wires to maintain positional principles for rapid calculations in commerce and astronomy.^[18] In India, the concept of zero evolved from philosophical notions of emptiness, or shunya, rooted in Buddhist and Hindu traditions that emphasized nothingness as a profound reality, influencing mathematical thought by the 5th century CE.^[19] The earliest recorded use of zero as a placeholder appears in the Bhakshali manuscript from the 3rd or 4th century CE, but it was the 7th-century mathematician Brahmagupta who formalized zero's role in his 628 CE treatise Brahmasphutasiddhanta.^[19] There, Brahmagupta defined zero not merely as an absence but as a number with arithmetic rules, such as a - a = 0 and $0 \div a = 0, integrating it into the decimal positional system alongside digits 1 through 9 to represent quantities compactly.^[19] The synthesis of these Indian innovations into the Hindu-Arabic numeral system occurred in the 9th century through the Persian scholar Al-Khwarizmi, whose work On the Calculation with Hindu Numerals systematically described the place-value method using digits 0–9 for arithmetic and square roots.^[20] This system spread westward, reaching Europe via the Italian mathematician Fibonacci in his 1202 book Liber Abaci, which demonstrated practical applications for merchants, such as currency conversion, and promoted the numerals' superiority over Roman additive systems.^[21] Positional notation's key advantage lies in its scalability and computational efficiency: each digit's value depends on its position relative to the base, allowing concise representation of large numbers and streamlined operations like multiplication without repetitive symbols.^[22] For instance, in base 10, the number 123 denotes

$1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 = 100 + 20 + 3 = 123,

contrasting with additive systems that require multiple instances of symbols for higher values, thus reducing errors and effort in calculations.^[23] This framework revolutionized arithmetic, underpinning advancements in algebra and science across cultures.

Emergence of advanced number concepts

The concept of negative numbers emerged in ancient Chinese mathematics as a practical tool for solving linear equations, particularly in the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), a text compiled during the Han dynasty around 100 BCE. In the chapter on systems of equations, positive quantities were represented by red rods and negative ones by black rods, allowing for the manipulation of debts or deficits in economic and astronomical calculations. This approach marked an early systematic use of negatives, predating their widespread acceptance elsewhere.^[24]^[25] In India, negative numbers gained formal mathematical legitimacy through the work of Brahmagupta in his Brahmasphuṭasiddhānta (628 CE), where he provided explicit rules for arithmetic operations involving positives, negatives, and zero, treating negatives as distinct quantities with properties like "the sum of two negatives is negative." This acceptance facilitated advanced algebra and astronomy, influencing later Islamic mathematics. Brahmagupta's rules, such as the product of two negatives being positive, resolved conceptual ambiguities and integrated negatives into a coherent number system.^[26] Rational numbers, expressible as ratios of integers, appeared in Egyptian mathematics through unit fractions, as documented in the Rhind Papyrus (c. 1650 BCE), which employed sums of distinct unit fractions to represent divisions and proportions in practical problems like resource allocation. The Greeks formalized ratios in Euclid's Elements (c. 300 BCE), particularly in Book V, where magnitudes are compared via proportions that align with modern rational numbers, enabling geometric proofs without explicit fractions. These developments emphasized the utility of ratios in measurement and harmony, laying groundwork for algebraic manipulation.^[27]^[28] The discovery of irrational numbers challenged the Pythagorean belief in all quantities as rational ratios, attributed to Hippasus of Metapontum (c. 450 BCE), who demonstrated that \sqrt{2}—the diagonal of a unit square—cannot be expressed as a fraction p/q with integers p and q. His proof by contradiction assumed \sqrt{2} = p/q in lowest terms, leading to both p and q being even, contradicting the assumption and proving incommensurability. This revelation reportedly caused turmoil among Pythagoreans, highlighting limits in rational-based geometry.^[29] Concepts of infinity arose in Greek philosophy through Zeno of Elea's paradoxes (c. 450 BCE), which argued against motion and plurality by positing infinite divisibility, as in the dichotomy paradox where a path requires infinitely many steps, each halved indefinitely. These thought experiments, aimed at defending Parmenides' monism, forced reflections on continuity and the infinite, influencing later atomic theories. In medieval Europe, Nicole Oresme (c. 1320–1382) advanced infinity discussions in works like De configurationibus qualitatum et motuum, exploring infinitesimals as arbitrarily small quantities to analyze infinite series and uniform acceleration, bridging qualitative philosophy with proto-calculus ideas.^[30]

Modern historical milestones

In the 16th century, Italian mathematician Gerolamo Cardano made a significant advancement in the study of numbers by implicitly employing complex numbers to solve cubic equations in his seminal work Ars Magna, published in 1545. Cardano's solutions for depressed cubics of the form x^3 + px + q = 0 involved expressions like

\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}

, where the discriminant \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 could be negative, leading to square roots of negative quantities that he acknowledged as "sophistic" but necessary for real roots. This marked the first systematic use of what would later be formalized as complex numbers, bridging algebraic solutions with previously dismissed imaginary quantities.^[31]^[32] The formal acceptance and notation of complex numbers advanced in the 18th century by Leonhard Euler, who introduced the symbol i to denote \sqrt{-1}, and further promoted in the 19th century through Carl Friedrich Gauss, who in his 1831 paper "Theoria residuorum biquadraticorum. Commentatio secunda" emphasized the geometric interpretation of complex numbers as points in the plane. Gauss's work on biquadratic residues treated complex integers a + bi as a natural extension of the integers, establishing their arithmetic properties and arguing for their legitimacy in number theory, which helped dispel earlier skepticism and integrated them into mainstream mathematics.^[33]^[34] The rigorous foundation of real numbers was solidified in the late 19th century by Richard Dedekind, who in his 1872 pamphlet Stetigkeit und irrationale Zahlen defined real numbers via "Dedekind cuts," partitions of the rational numbers into two non-empty sets A and B such that all elements of A are less than all of B, with no greatest element in A. This construction ensures the completeness of the reals, capturing irrationals like \sqrt{2} as the cut where A contains all rationals whose square is less than 2, providing an arithmetic-free definition that aligns with the continuum's intuitive density. Complementing this, Georg Cantor developed an equivalent construction around the 1880s using nested intervals and set-theoretic principles, where real numbers are equivalence classes of Cauchy sequences of rationals, further emphasizing the uncountable nature of the reals through his diagonal argument.^[35]^[36]^[37] In the late 19th century, Giuseppe Peano provided axioms for natural numbers in 1889, while Gottlob Frege developed a logical foundation for arithmetic in his 1884 work Begriffsschrift, influencing foundational studies.^[38]^[39] A cornerstone of analytic number theory emerged in 1896 with the independent proofs of the Prime Number Theorem by Jacques Hadamard and Charles Jean de la Vallée Poussin, stating that the number of primes less than or equal to x, denoted \pi(x), is asymptotically \frac{x}{\ln x} as x \to \infty. Hadamard's proof in "Sur la distribution des zéros de la fonction \zeta(s) et ses conséquences arithmétiques" utilized the non-vanishing of the Riemann zeta function on the line \Re(s) = 1, while de la Vallée Poussin's in "Recherches analytiques sur la théorie des nombres premiers" showed an analogous zero-free region, confirming the theorem's prediction of prime distribution density. This result quantified the intuitive scarcity of primes and resolved a conjecture dating to Gauss and Legendre.^[40]^[41] In the 20th century, Kurt Gödel's incompleteness theorems, published in 1931 in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," demonstrated fundamental limits in formal arithmetic systems, showing that any consistent axiomatization of Peano arithmetic contains undecidable statements, such as the Gödel sentence asserting its own unprovability. This demonstrated fundamental limits in formal arithmetic systems, showing undecidable statements and influencing later developments in computability theory, such as Turing's halting problem and the hierarchy of recursive functions.^[42]^[43] Another profound 20th-century milestone was the Atiyah-Singer Index Theorem, announced in 1963 by Michael Atiyah and Isadore Singer, which equates the analytic index of an elliptic differential operator on a compact manifold to a topological index computed via characteristic classes in K-theory. For instance, on a closed oriented 4-manifold, the theorem relates the index of the Dirac operator to the signature and Euler characteristic, forging deep connections between numerical invariants in analysis, algebraic topology, and geometry, with applications to gauge theories and quantum field theory.^[44]

Fundamental classifications

Natural numbers

Natural numbers form the foundational set for counting and enumeration in mathematics, consisting of the positive integers beginning with 1, denoted as the set \{[1](/page/1), 2, [3, \dots](/page/3_Dots)\}.^[45] In some contexts, particularly in set theory and modern axiomatic treatments, the set includes 0, yielding \{0, [1](/page/1), 2, [3, \dots](/page/3_Dots)\}, though the exclusion of 0 aligns with traditional counting applications.^[45] This set serves as the starting point for constructing more complex number systems. The natural numbers are rigorously defined through the Peano axioms, introduced by Giuseppe Peano in 1889.^[46] These axioms establish 0 (or 1 in some formulations) as the base element and introduce a successor function S(n), where S(n) = n + 1, ensuring each natural number has a unique successor that is also a natural number.^[47] The axioms further specify that 0 is not the successor of any natural number, that distinct numbers have distinct successors, and include the induction axiom: if a property holds for 0 and for the successor of every natural number possessing it, then it holds for all natural numbers.^[47] This framework guarantees the existence and uniqueness of the natural numbers as an infinite sequence generated by successive applications of the successor function. Arithmetic operations on natural numbers are defined recursively using the Peano axioms. Addition satisfies:

\begin{align*} n + 0 &= n, \\ n + S(m) &= S(n + m) \end{align*}

for all natural numbers n and m, allowing addition to be built as repeated succession.^[48] Multiplication is similarly recursive:

\begin{align*} n \times 0 &= 0, \\ n \times S(m) &= (n \times m) + n \end{align*}

enabling multiplication as repeated addition.^[48] These definitions ensure the operations are well-defined and align with intuitive arithmetic. The natural numbers admit a total order under the standard less-than relation, where for any two distinct elements m and n, either m < n or n < m, with 0 (or 1) serving as the least element.^[49] This order is discrete rather than dense, meaning that between any two consecutive numbers n and S(n), no other natural number exists, contrasting with dense orders like the rationals where elements lie between any pair.^[50] The well-ordering property ensures every non-empty subset has a least element.^[51] The set of natural numbers is countably infinite, possessing the smallest infinite cardinality \aleph_0, as established by Georg Cantor in his development of set theory.^[52] This cardinality reflects the bijection between the natural numbers and themselves, distinguishing them from uncountable sets like the reals.^[52]

Integers

The integers, denoted by the symbol \mathbb{Z}, consist of the set \{\dots, -2, -1, 0, 1, 2, \dots\}, which extends the natural numbers by including their additive inverses and zero. This set forms a complete additive group, closed under both addition and subtraction, allowing for bidirectional counting and the representation of debts or deficits alongside positive quantities. Unlike the natural numbers, which are limited to non-negative values for counting purposes, \mathbb{Z} provides a symmetric structure around zero that supports the full range of integer arithmetic without fractions.^[53]^[54] Algebraically, \mathbb{Z} constitutes a commutative ring with unity under the standard operations of addition and multiplication, where addition forms an abelian group with identity 0, multiplication is associative and distributive over addition, and 1 serves as the multiplicative identity. A key property enabling unique factorization and divisibility in this ring is the division algorithm: for any a, b \in \mathbb{Z} with b > 0, there exist unique integers q (the quotient) and r (the remainder) such that a = bq + r and $0 \leq r < b. This algorithm underpins many computational processes in number theory, ensuring that every integer can be expressed as a multiple of the divisor plus a non-negative remainder smaller than the divisor.^[55]^[56] The Euclidean algorithm leverages the division algorithm to compute the greatest common divisor (GCD) of two integers a and b (with b \neq 0) through iterative application: repeatedly replace (a, b) with (b, r) where r is the remainder from dividing a by b, until the remainder is zero; the last non-zero remainder is then \gcd(a, b). This process can be executed via successive subtractions or, more efficiently, using the modulo operation, and it terminates due to the non-increasing sequence of remainders. In \mathbb{Z}, the units—elements with multiplicative inverses within the ring—are solely \pm 1, as any other integer lacks an integer inverse. Consequently, two non-zero integers are associates if one is the product of the other by a unit, meaning they differ only by sign (e.g., 6 and -6 are associates).^[57]^[55]^[58]

Rational numbers

A rational number is any number that can be expressed as the ratio of two integers p and q, where q \neq 0, formally denoted as the set \mathbb{Q} = \left\{ \frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z} \setminus \{0\} \right\}.^[59] Two such fractions \frac{p}{q} and \frac{p'}{q'} represent the same rational number if they are equivalent, meaning p q' = p' q.^[60] The set \mathbb{Q} forms a field under the standard operations of addition and multiplication, with every non-zero element having a multiplicative inverse.^[60] Addition of two rational numbers \frac{a}{b} and \frac{c}{d} (with b \neq 0, d \neq 0) is defined as \frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}.^[60] Multiplication is given by \frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}.^[60] The multiplicative inverse of a non-zero rational \frac{a}{b} (with a \neq 0) is \frac{b}{a}, since \frac{a}{b} \times \frac{b}{a} = 1.^[60] These operations satisfy the field axioms, including commutativity, associativity, distributivity, and the existence of additive identities and inverses.^[60] The integers \mathbb{Z} embed naturally into \mathbb{Q} as the subset \left\{ \frac{n}{1} \mid n \in \mathbb{Z} \right\}.^[59] The rational numbers are dense in the real numbers, meaning that between any two distinct real numbers x < y, there exists a rational number r such that x < r < y.^[61] This property follows from the Archimedean axiom of the reals, which ensures the existence of sufficiently large integers to approximate the interval (x, y).^[61] In base-10 representation, every rational number has a decimal expansion that is either terminating (ending in infinite zeros) or eventually repeating.^[60] For example, \frac{1}{2} = 0.5 terminates, while \frac{1}{3} = 0.\overline{3} repeats.^[60] Conversely, any terminating or repeating decimal corresponds to a rational number.^[60] This holds in any integer base greater than or equal to 2.^[60]

Real numbers

The real numbers, denoted \mathbb{R}, form the complete ordered field that extends the rational numbers to include irrational numbers, ensuring a continuous number line without gaps. This structure satisfies all field axioms and order axioms of the rationals, with the key addition of completeness, which guarantees the existence of limits for convergent sequences and suprema for bounded sets. Unlike the rationals, which are countable and dense but incomplete, the reals provide a foundation for calculus and analysis by filling in the "holes" represented by irrationals.^[62] One standard construction of the real numbers begins with Dedekind cuts, introduced by Richard Dedekind in 1872. A Dedekind cut is a partition of the rational numbers \mathbb{Q} into two non-empty subsets A and B such that all elements of A are less than all elements of B, A has no greatest element, and A is downward closed. Each real number corresponds uniquely to such a cut, where rational numbers are represented by cuts with a greatest element in A, and irrationals by those without; arithmetic and order operations are defined set-theoretically on these cuts to form the ordered field \mathbb{R}.^[63] An equivalent construction uses Cauchy sequences of rational numbers, developed by Georg Cantor in 1872. A Cauchy sequence \{q_n\} in \mathbb{Q} satisfies the condition that for every \epsilon > 0, there exists N \in \mathbb{N} such that |q_m - q_n| < \epsilon for all m, n > N. Two such sequences are equivalent if their difference converges to zero, and the real numbers are the equivalence classes of these sequences under this relation, with operations induced pointwise to yield the complete ordered field.^[64] The defining property of the reals is their completeness, axiomatized as follows: every non-empty subset of \mathbb{[R](/page/R)} that is bounded above has a least upper bound (supremum) in \mathbb{[R](/page/R)}. This axiom, which distinguishes \mathbb{[R](/page/R)} from \mathbb{[Q](/page/Q)}, ensures that Cauchy sequences converge and underpins theorems like the intermediate value theorem; it is taken as primitive in axiomatic treatments or proved from the constructions above.^[62] A consequence of completeness is the Archimedean property: for any \epsilon > 0 in \mathbb{R}, there exists n \in \mathbb{N} such that n > 1/\epsilon. To see this, suppose otherwise for contradiction; then \mathbb{N} is bounded above by $1/\epsilon. Let \alpha = \sup \mathbb{N} \leq 1/\epsilon. Then \alpha - 1 is not an upper bound, so there exists k \in \mathbb{N} such that k > \alpha - 1, implying k + 1 > \alpha, which contradicts \alpha being the least upper bound. This property implies that \mathbb{N} is unbounded in \mathbb{R} and that rationals are dense in \mathbb{R}.^[65] The cardinality of \mathbb{R} is the continuum |\mathbb{R}| = 2^{\aleph_0}, which exceeds the countable cardinality \aleph_0 of \mathbb{Q}. Georg Cantor proved the uncountability of \mathbb{R} in 1891 using the diagonal argument: assume for contradiction a bijection f: \mathbb{N} \to (0,1) \subset \mathbb{R}, represented in decimal expansions; construct x = 0.x_1 x_2 \dots where x_i \neq f(i)_i (the i-th digit of f(i)), ensuring x \neq f(n) for all n, hence no such bijection exists. This establishes |\mathbb{R}| > \aleph_0, with equality to $2^{\aleph_0} following from binary representations.^[66]

Complex numbers

Complex numbers extend the real numbers by adjoining an imaginary unit i satisfying i^2 = -1, forming the set \mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}.^[67] Addition and multiplication in \mathbb{C} are defined componentwise for the real and imaginary parts, making \mathbb{C} a field that algebraically closes the reals and enables solutions to polynomial equations lacking real roots, such as x^2 + 1 = 0.^[67] The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root, implying exactly n roots (counting multiplicities) for a degree-n polynomial; this result, first rigorously proved by Carl Friedrich Gauss in his 1799 doctoral dissertation, underscores the completeness of \mathbb{C} for algebraic purposes.^[68] Geometrically, complex numbers are represented in the Argand plane, a two-dimensional coordinate system where the horizontal axis denotes the real part and the vertical axis the imaginary part, interpreting each z = a + bi as a vector from the origin to the point (a, b).^[69] This vector interpretation facilitates operations like addition as vector sums and multiplication as scaling and rotation. In polar form, a complex number z = a + bi (with a \neq 0 or b \neq 0) is expressed as z = r(\cos \theta + i \sin \theta), where r = |z| = \sqrt{a^2 + b^2} is the modulus and \theta = \arg(z) is the argument; De Moivre's theorem then simplifies powers via [r(\cos \theta + i \sin \theta)]^n = r^n (\cos (n\theta) + i \sin (n\theta)) for positive integer n, aiding computations in roots of unity and periodic phenomena.^[70]

Properties of integers

Even and odd numbers

In the context of integers, even and odd numbers represent the fundamental classification based on divisibility by 2, known as parity. An integer n is even if it can be expressed as n = 2k for some integer k, and odd otherwise, equivalently n = 2k + 1 for some integer k.^[71]^[72] This distinction corresponds to the residue class modulo 2, where even numbers satisfy n \equiv 0 \pmod{2} and odd numbers satisfy n \equiv 1 \pmod{2}.^[73] The arithmetic properties of even and odd numbers follow directly from these definitions. The sum of two even integers is even, as is the sum of two odd integers; the sum of an even and an odd integer is odd.^[73] For multiplication, the product of an even integer and any integer is even, while the product of two odd integers is odd.^[73] These rules underpin many combinatorial arguments, such as counting problems in parity-based partitions. One notable implication of parity arises in the Goldbach conjecture, which posits that every even integer greater than 2 can be expressed as the sum of two prime numbers.^[74] Proposed by Christian Goldbach in a 1742 letter to Leonhard Euler, this unsolved conjecture highlights the role of even numbers in additive structures involving primes, though it remains unproven despite extensive verification for large values.^[74] In binary representation, the parity of an integer is determined by its least significant bit: even numbers end with a trailing zero, while odd numbers end with a 1.^[75] This property facilitates efficient computation of parity in algorithms, such as checking divisibility by 2 through bitwise operations.^[75]

Prime numbers

A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself.^[76] This property establishes primes as the multiplicative building blocks of the natural numbers, as articulated in the Fundamental Theorem of Arithmetic, which asserts that every integer greater than 1 can be expressed uniquely as a finite product of primes raised to non-negative integer powers, disregarding the order of factors: n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, where each p_i is prime and each a_i \geq 1.^[77] The theorem, first proved by Carl Friedrich Gauss in his 1801 Disquisitiones Arithmeticae, underpins much of number theory by guaranteeing the irreducibility and uniqueness of prime factorizations.^[78] The infinitude of prime numbers was established by Euclid around 300 BCE in his Elements.^[79] Euclid's proof proceeds by contradiction: assume there are only finitely many primes p_1, p_2, \dots, p_k; form the number N = p_1 p_2 \cdots p_k + 1; then N must be divisible by some prime p, but p cannot equal any p_i since N leaves a remainder of 1 when divided by each p_i, yielding a contradiction and implying infinitely many primes.^[80] This elegant argument highlights the inexhaustible nature of primes within the integers. An efficient method for identifying all primes up to a given limit n is the Sieve of Eratosthenes, devised by the Greek mathematician Eratosthenes around 240 BCE.^[81] The algorithm begins with a list of integers from 2 to n, marks multiples of 2 as composite starting from $2^2, then repeats for the next unmarked number (3) and its multiples from $3^2, continuing up to the square root of n; the unmarked numbers remaining are primes.^[82] With a time complexity of O(n \log \log n), it remains a foundational tool for computational number theory despite its antiquity.^[83] The distribution of primes is intimately connected to the Riemann zeta function, defined for complex s with real part greater than 1 as \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. ^[84] Leonhard Euler demonstrated in 1737 that this sum equals the infinite product over all primes p \zeta(s) = \prod_p \frac{1}{1 - p^{-s}}, known as the Euler product formula, which directly encodes the primes and reveals their role in the analytic structure of the zeta function.^[85] This equivalence provides a bridge between additive and multiplicative number theory, enabling profound insights into prime density through the function's analytic continuation and zeros, as explored by Bernhard Riemann in 1859.^[86] Notably, 2 is the only even prime, with all others being odd.^[87]

Other integer classifications

A perfect number is a positive integer equal to the sum of its proper divisors, that is, the divisors excluding the number itself. For instance, 6 is perfect since its proper divisors are 1, 2, and 3, and $1 + 2 + 3 = 6.^[88] Similarly, 28 is perfect because $1 + 2 + 4 + 7 + 14 = 28.^[88] Euclid proved in Proposition 36 of Book IX of the Elements that if $2^p - 1 is prime for some prime p, then $2^{p-1}(2^p - 1) is a perfect number.^[89] Euler later established the converse: every even perfect number has this form, where $2^p - 1 is a Mersenne prime. As of November 2025, 52 even perfect numbers are known, all corresponding to known Mersenne primes.^[88]^[90] No odd perfect numbers are known, and it remains an open question whether any exist.^[88] Square numbers, also known as perfect squares, are integers of the form n^2 for some integer n \geq 0. Examples include 0, 1, 4, 9, and 16. They arise in various contexts, such as the side lengths of squares and in Diophantine equations. Notably, Pythagorean triples consist of three positive integers a, b, and c satisfying a^2 + b^2 = c^2; the primitive triple (3, 4, 5) is the smallest, as $3^2 + 4^2 = 9 + 16 = 25 = 5^2.^[91] The Fibonacci sequence provides another classification of integers, defined by the recurrence F_n = F_{n-1} + F_{n-2} for n \geq 2, with initial conditions F_0 = 0 and F_1 = 1, yielding the terms 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.^[92] All terms beyond the first two are integers, and the sequence exhibits rich properties, such as Cassini's identity: F_{n+1} F_{n-1} - F_n^2 = (-1)^n, which holds for all n \geq 1.^[93] For example, with n=3, F_4 = 3, F_2 = 1, F_3 = 2, so $3 \cdot 1 - 2^2 = 3 - 4 = -1 = (-1)^3.^[93] Figurate numbers generalize polygonal arrangements of dots or objects, with triangular numbers forming one prominent subclass. The nth triangular number is given by T_n = \frac{n(n+1)}{2}, representing the sum of the first n positive integers; the first few are 1, 3, 6, 10, and 15.^[94] These numbers connect to binomial coefficients, as T_n = \binom{n+1}{2}.^[94] More broadly, figurate numbers include squares (n^2) and pentagonal numbers, but triangular numbers illustrate the concept of building integers through cumulative summation in geometric patterns.^[95]

Advanced subclasses

Algebraic numbers

An algebraic number is a complex number \alpha that is a root of a non-zero polynomial p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0 with rational coefficients a_i \in \mathbb{Q} and leading coefficient a_n \neq 0.^[96] Equivalently, \alpha lies in a finite-degree field extension of \mathbb{Q}.^[97] All rational numbers are algebraic, as they satisfy linear polynomials with rational coefficients, but algebraic numbers extend beyond rationals to include roots of higher-degree polynomials.^[96] The degree of an algebraic number \alpha is the degree of its minimal polynomial, defined as the unique monic irreducible polynomial over \mathbb{Q} of least degree that has \alpha as a root.^[98] This minimal polynomial determines the algebraic structure of \alpha, and the degree measures the "complexity" of adjoining \alpha to \mathbb{Q}. For instance, rational numbers have degree 1, while quadratic irrationals like \sqrt{2} have degree 2.^[98] A subset of algebraic numbers, known as algebraic integers, consists of those \alpha whose minimal polynomial is monic with integer coefficients (i.e., leading coefficient 1 and all other coefficients in \mathbb{Z}).^[99] Algebraic integers form a ring under addition and multiplication, generalizing the ordinary integers.^[100] Adjoining an algebraic number \alpha to \mathbb{Q} generates the field extension \mathbb{Q}(\alpha), which is finite-dimensional as a vector space over \mathbb{Q} with dimension equal to the degree of \alpha.^[101] Elements of \mathbb{Q}(\alpha) are linear combinations of $1, \alpha, \alpha^2, \dots, \alpha^{d-1} with rational coefficients, where d is the degree.^[97] For example, \sqrt{2} satisfies the minimal polynomial x^2 - 2 = 0, so [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2, and the extension consists of numbers a + b\sqrt{2} for a, b \in \mathbb{Q}.^[96] Another example is the golden ratio \phi = \frac{1 + \sqrt{5}}{2}, which satisfies x^2 - x - 1 = 0 and generates \mathbb{Q}(\phi) = \mathbb{Q}(\sqrt{5}) of degree 2.^[102] The set of all algebraic numbers is countable. This follows from the fact that it is a countable union over degrees n \geq 1 of the roots of all monic polynomials of degree n with integer coefficients, where the set of such polynomials is countable (as a countable union of countable sets \mathbb{Z}^{n+1}) and each has finitely many roots.^[103] Algebraic numbers thus form a countable dense subset of the complex numbers.^[97]

Irrational and transcendental numbers

Irrational numbers are the real numbers that cannot be expressed as the ratio of two integers, forming the set \mathbb{R} \setminus \mathbb{Q}.^[104] Unlike rational numbers, whose decimal expansions either terminate or repeat periodically, irrational numbers have non-terminating, non-repeating decimal expansions. A classic example is \sqrt{2}, the positive square root of 2, whose continued fraction expansion is [1; \overline{2}], indicating an infinite, periodic but non-terminating sequence of partial quotients. This infinite non-repeating nature underscores their distinction from rationals. The irrationality of \sqrt{2} can be proved by contradiction. Suppose \sqrt{2} = p/q where p and q are positive integers with \gcd(p, q) = 1 and q \neq 0. Then p^2 = 2q^2, implying p^2 is even, so p must be even (since the square of an odd number is odd). Let p = 2k for some integer k; substituting gives $4k^2 = 2q^2, or q^2 = 2k^2, so q^2 is even and thus q is even. But then both p and q are even, contradicting \gcd(p, q) = 1. Therefore, no such rational p/q exists. Transcendental numbers form a proper subset of the irrationals: they are irrational numbers that are not algebraic, meaning they are not roots of any non-zero polynomial equation with rational coefficients. The existence of transcendental numbers was first demonstrated in 1844 by Joseph Liouville, who constructed explicit examples known as Liouville numbers. These are real numbers that can be approximated exceptionally well by rational numbers—far better than any algebraic irrational—using a series like \sum_{k=1}^\infty 10^{-k!}, which violates the approximation bounds for algebraic numbers. Building on this, Charles Hermite proved in 1873 that e, the base of the natural logarithm, is transcendental by showing that it satisfies no algebraic equation of any degree, using integral representations and properties of exponential functions.^[105] Similarly, Ferdinand von Lindemann established in 1882 that \pi is transcendental, employing Hermite's methods extended to the exponential function evaluated at algebraic points, thereby resolving the question of squaring the circle with straightedge and compass.^[106] Among irrationals, normal numbers represent those whose digits in a given base (typically base 10) are equidistributed, meaning every finite sequence of digits appears with the frequency expected from a random distribution. Almost all real numbers are normal in the sense of Lebesgue measure, though proving normality for specific constants like \pi or e remains open. A concrete example of a proven normal number is the Champernowne constant, formed by concatenating the positive integers: $0.123456789101112131415\dots, which was shown to be normal in base 10 and transcendental.^[107] While algebraic irrationals like \sqrt{2} satisfy polynomial equations, transcendentals such as e and \pi transcend this structure entirely.

Constructible and computable numbers

Constructible numbers are the real numbers that can be obtained from the rational numbers \mathbb{Q} through a finite sequence of operations corresponding to compass and straightedge constructions, starting from points 0 and 1. These operations include drawing lines between existing points, drawing circles centered at existing points with radii to other existing points, and finding intersection points of these lines and circles. Algebraically, the set of constructible numbers forms the smallest field extension of \mathbb{Q} closed under taking square roots of positive elements, built via a tower of quadratic extensions. Thus, for any constructible number \alpha, the degree of the extension [\mathbb{Q}(\alpha):\mathbb{Q}] is a power of 2, by the tower law for field extensions.^[108] A classic example is the construction of a regular pentagon, where the cosine of the central angle $2\pi/5 is constructible, as the minimal polynomial of \cos(2\pi/5) over \mathbb{Q} has degree 2, a power of 2. However, \pi itself is not constructible; Pierre Wantzel proved in 1837 that certain geometric problems, including squaring the circle (which requires constructing \sqrt{\pi}), are impossible with compass and straightedge precisely because they would require field extensions of degrees not a power of 2. All constructible numbers are algebraic, forming a proper subfield of the algebraic numbers.^[109] Computable numbers, introduced by Alan Turing in 1936, are the real numbers \alpha for which there exists a Turing machine that, given positive integer input n, outputs the n-th decimal digit (or binary digit) of \alpha after finitely many steps. This definition captures numbers whose decimal expansions can be generated algorithmically to arbitrary precision. For instance, both e and \pi are computable, as their values can be approximated to any desired accuracy using series expansions: e = \sum_{k=0}^\infty 1/k! and \pi = 4 \sum_{k=0}^\infty (-1)^k / (2k+1). In contrast, Chaitin's constant \Omega, defined as the probability that a randomly generated program halts, is uncomputable.^[110] The uncomputability of \Omega follows from the halting problem, which Turing showed is undecidable: determining whether an arbitrary Turing machine halts on a given input cannot be solved by any algorithm. This implies that the set of computable numbers is countable—since there are countably many Turing machines—while the real numbers are uncountable, so most real numbers are non-computable.^[110]^[111]

Extensions beyond standard numbers

p-adic numbers

In number theory, p-adic numbers provide a non-Archimedean completion of the rational numbers \mathbb{Q} with respect to a metric induced by a prime p, enabling the study of congruences and Diophantine equations through infinite series that converge in this topology.^[112] For each prime p, the field \mathbb{Q}_p consists of formal series \sum_{k=n}^\infty a_k p^k with a_k \in \{0, 1, \dots, p-1\} and n \in \mathbb{Z}, where addition and multiplication are defined with carrying over as in base-p arithmetic.^[112] The rational numbers embed densely into \mathbb{Q}_p via their p-adic expansions. The p-adic valuation v_p: \mathbb{[Q](/page/Q)} \to \mathbb{[Z](/page/Z)} \cup \{\infty\} underpins this structure, defined for a nonzero rational x = a/b in lowest terms as v_p(x) = v_p(a) - v_p(b), where v_p(m) is the highest power of p dividing the integer m (i.e., if m = p^k \cdot \ell with p \nmid \ell, then v_p(m) = k), and v_p(0) = \infty.^[112] This valuation satisfies v_p(xy) = v_p(x) + v_p(y) and v_p(x + y) \geq \min(v_p(x), v_p(y)).^[112] The associated p-adic metric is given by d_p(x, y) = p^{-v_p(x - y)} for x, y \in \mathbb{Q}, with d_p(x, x) = 0, which turns \mathbb{Q} into a metric space where "closeness" means divisibility by high powers of p.^[112] This metric is ultrametric, obeying the strong triangle inequality d_p(x, z) \leq \max(d_p(x, y), d_p(y, z)), a consequence of the non-Archimedean property |x + y|_p \leq \max(|x|_p, |y|_p) where | \cdot |_p = p^{-v_p(\cdot)}.^[112] The completion of \mathbb{Q} under this metric yields \mathbb{Q}_p, a complete metric field containing \mathbb{Q} as a dense subfield.^[112] Hensel's lemma facilitates solving polynomial equations in \mathbb{Q}_p by lifting solutions from modulo p to the full p-adics.^[113] Specifically, for a polynomial f(X) \in \mathbb{Z}[X] and a \in \mathbb{Z} such that f(a) \equiv 0 \pmod{p} and f'(a) \not\equiv 0 \pmod{p}, there exists a unique \alpha \in \mathbb{Z}_p (the p-adic integers, the unit ball in \mathbb{Q}_p) satisfying f(\alpha) = 0 and \alpha \equiv a \pmod{p}.^[113] For example, to solve x^2 \equiv 7 \pmod{3}, note that a = 1 works since $1^2 = 1 \equiv 1 \pmod{3} wait no, 7 mod 3 is 1, yes $1^2 \equiv 1 \pmod{3}, and f'(x) = 2x, f'(1) = 2 \not\equiv 0 \pmod{3}, so a unique square root lifts, such as the series $1 + 3 + 3^2 + 2 \cdot 3^4 + \cdots \in \mathbb{Z}_3.^[113]

Hypercomplex numbers

Hypercomplex numbers generalize the complex numbers to higher dimensions, introducing structures that sacrifice certain algebraic properties like commutativity in exchange for expanded dimensionality. The quaternions, denoted \mathbb{H}, form a foundational example, consisting of elements a + bi + cj + dk where a, b, c, d \in \mathbb{R} and the basis elements satisfy i^2 = j^2 = k^2 = ijk = -1.^[114] Discovered by William Rowan Hamilton on October 16, 1843, while walking along the Royal Canal in Dublin, quaternions extend the two-dimensional complex plane to four dimensions over the reals, enabling the representation of three-dimensional space.^[114] Unlike the complexes, quaternion multiplication is non-commutative—for instance, ij = k but ji = -k—yet it remains associative, making \mathbb{H} a non-commutative division algebra where every non-zero element has a multiplicative inverse given by q^{-1} = \bar{q} / N(q), with conjugate \bar{q} = a - bi - cj - dk and norm N(q) = a^2 + b^2 + c^2 + d^2.^[114] A key application of quaternions lies in three-dimensional rotations, where unit quaternions (those with N(q) = 1) provide an efficient parameterization avoiding singularities like gimbal lock encountered in Euler angles. A rotation by angle \theta around a unit vector \mathbf{u} = (u_x, u_y, u_z) is represented as q = \cos(\theta/2) + \sin(\theta/2)(u_x i + u_y j + u_z k), and composing rotations corresponds to quaternion multiplication.^[115] This formulation is widely used in computer graphics, robotics, and aerospace for smooth interpolation via spherical linear interpolation (SLERP), requiring fewer computational operations than equivalent rotation matrices.^[115] Further extension via the Cayley-Dickson construction doubles the dimension at each step, producing the octonions \mathbb{O} as an eight-dimensional algebra over \mathbb{R} from pairs of quaternions: elements are (a, b) with a, b \in \mathbb{H}, addition componentwise, and multiplication (a, b)(c, d) = (ac - d\bar{b}, cb + da).^[116] The octonions form an alternative algebra, meaning subalgebras generated by any two elements are associative, but overall multiplication is non-associative—for example, (e_1 e_2) e_3 = -e_6 \neq e_1 (e_2 e_3) = e_6 using the standard basis \{e_0 = 1, e_1, \dots, e_7\}.^[117] Despite non-associativity, \mathbb{O} remains a division algebra with no zero divisors and a Euclidean norm N(x) = x \bar{x} satisfying N(xy) = N(x)N(y).^[117] Theorems of Frobenius and Hurwitz classify finite-dimensional division algebras over \mathbb{[R](/page/R)}. Frobenius' theorem (1878) establishes that the only associative ones are \mathbb{[R](/page/R)}, \mathbb{C}, and \mathbb{H}.^[118] Hurwitz' theorem (1898) extends this to normed division algebras, proving that only \mathbb{[R](/page/R)}, \mathbb{C}, \mathbb{H}, and \mathbb{O} admit a bilinear multiplication preserving the norm, i.e., \|xy\| = \|x\| \|y\| for all x, y.^[119] These results underscore the exceptional nature of these structures, limiting higher-dimensional analogs while highlighting the progressive loss of commutativity and associativity.^[119]

Transfinite and nonstandard numbers

Transfinite numbers arise in set theory as a means to extend the concept of infinity beyond the finite, distinguishing between different "sizes" of infinite sets through ordinal and cardinal numbers. Ordinal numbers generalize the ordering of natural numbers to infinite sequences, where \omega denotes the smallest infinite ordinal, representing the order type of the natural numbers, and \omega + 1 is the ordinal obtained by adding a single element after an infinite sequence.^[120] Cardinal numbers, in contrast, measure the size of sets without regard to order; \aleph_0 (aleph-null) is the cardinality of the countable infinite set of natural numbers, while \aleph_1 is the smallest uncountable cardinal, assuming the continuum hypothesis.^[121] These concepts were developed by Georg Cantor in the late 19th century to rigorously handle infinite quantities.^[9] A key result establishing the uncountability of the real numbers is Cantor's diagonal argument, which demonstrates that the power set of the natural numbers has cardinality $2^{\aleph_0}, strictly larger than \aleph_0, as no bijection exists between the reals and the naturals.^[122] This cardinality, known as the continuum, raises the continuum hypothesis (CH), which posits that no cardinal lies between \aleph_0 and $2^{\aleph_0}, or equivalently, $2^{\aleph_0} = \aleph_1.^[123] Kurt Gödel proved in 1940 that CH is consistent with the Zermelo-Fraenkel axioms with choice (ZFC), the standard foundation of set theory, while Paul Cohen showed in 1963 that its negation is also consistent with ZFC, rendering CH independent.^[123] Nonstandard numbers extend the real numbers to include infinitesimals and infinities in a rigorous framework provided by nonstandard analysis, pioneered by Abraham Robinson in the 1960s. In this approach, the nonstandard extension * \mathbb{R} of the reals \mathbb{R} incorporates infinitesimal numbers \varepsilon > 0 that are smaller than every positive standard real but nonzero, and correspondingly infinite numbers such as $1/\varepsilon, which exceed every standard real.^[124] A fundamental feature is the transfer principle, which states that any first-order logical statement true in \mathbb{R} holds in * \mathbb{R} when restricted to standard elements, and vice versa, enabling the transfer of theorems between the two systems. The hyperreal numbers, a specific realization of * \mathbb{R}, form a non-Archimedean ordered field that properly extends \mathbb{R} with both infinitesimals and infinities. Their construction typically employs an ultrafilter on the natural numbers to form the ultrapower \mathbb{R}^\mathbb{N}/\mathcal{U}, where sequences of reals are identified modulo the ultrafilter \mathcal{U}, yielding equivalence classes that include nonstandard elements while preserving field operations and order.^[125] This framework allows nonstandard analysis to reformulate classical results, such as the definition of continuity via infinitesimals, in a manner that aligns with intuitive infinitesimal calculus while remaining logically sound.^[126]

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Odd Number -- from Wolfram MathWorld
... Integers which are not odd are called even. Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence n=1 (mod 2) holds for odd n. The ...
[68]
Evens, Odds, and Primes: A Taste of Number Theory
Definition 1.18. An integer n is even if there exists an integer a such that n = 2a. For example, 98 is even since 98 = 2 · ...
[69]
Goldbach Conjecture -- from Wolfram MathWorld
The conjecture that all odd numbers >=9 are the sum of three odd primes is called the "weak" Goldbach conjecture.
[70]
even number - PlanetMath.org
Mar 22, 2013 · Then the above definition simply that even numbers end with a 0 0 , and odd numbers end with a 1 1 . 0.0.1 Properties. 1. Every integer is ...
[71]
The Prime Glossary: prime number - The Prime Pages
An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and ...
[72]
[PDF] The Fundamental Theorem of Arithmetic
Jun 14, 2008 · The Fundamental Theorem of Arithmetic says that every integer greater than 1 can be factored uniquely into a product of primes. • Euclid's lemma ...
[73]
Prime Numbers
For instance, the Fundamental Theorem of Arithmetic says that every integer greater than 1 can be written uniquely as a product of powers of primes.
[74]
Euclid's Elements, Book IX, Proposition 20 - Clark University
This proposition states that there are more than any finite number of prime numbers, that is to say, there are infinitely many primes.
[75]
[PDF] The infinitude of the primes - Keith Conrad
Euclid's proof of the infinitude of the primes uses the fact that all integers greater than. 1 have a prime factor, so let's discuss that first. Lemma 2.1.
[76]
Sieve of Eratosthenes - The Prime Glossary
Make a list of all the integers less than or equal to n (and greater than one). Strike out the multiples of all primes less than or equal to the square root of ...
[77]
Sieve of Eratosthenes
The Sieve of Eratosthenes is a method for finding all primes up to (and possibly including) a given natural n. This method works well when n is relatively ...
[78]
[PDF] The Genuine Sieve of Eratosthenes - Computer Science
The Sieve of Eratosthenes is a beautiful algorithm that has been cited in introduc- tions to lazy functional programming for more than thirty years (Turner, ...
[79]
[PDF] 16 Riemann's zeta function and the prime number theorem
Oct 31, 2018 · The product in the theorem above ranges over primes p. This is ... From the Euler product ζ(s) = Qp(1 − p−s)−1, we see that for Re(s) ...
[80]
[PDF] How Euler discovered the zeta function
... functions is that, as with the zeta function, they can be expressed as an infinite product over the prime numbers (sometimes known as an. Euler product), namely ...
[81]
[PDF] Notes on the Riemann Zeta Function
Jan 25, 2007 · product over primes converges, provided ... which converges for all σ > 1. We are left to show that the value of the Euler product is actually ζ(s) ...
[82]
Prime Numbers - Department of Mathematics at UTSA
Dec 11, 2021 · Introduction. A prime number (or prime for short) is a natural number that has exactly two divisors: itself and the number 1.
[83]
Perfect Number -- from Wolfram MathWorld
Perfect numbers are positive integers n such that n=s(n), (1) where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), ...Missing: natural | Show results with:natural
[84]
Euclid's Elements, Book IX, Proposition 36 - Clark University
Proposition 36. If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if ...
[85]
Pythagorean Triple -- from Wolfram MathWorld
A Pythagorean triple is a set of positive integers (a, b, c) forming a right triangle with legs a and b, and hypotenuse c. (3,4,5) is the smallest.
[86]
Fibonacci Number -- from Wolfram MathWorld
The Fibonacci numbers are the sequence of numbers {F_n}_(n=1)^infty defined by the linear recurrence equation F_n=F_(n-1)+F_(n-2) with F_1=F_2=1.
[87]
Cassini's Identity -- from Wolfram MathWorld
For F_n the nth Fibonacci number, F_(n-1)F_(n+1)-F_n^2=(-1)^n. This identity was also discovered by Simson (Coxeter and Greitzer 1967, p.Missing: sequence definition<|separator|>
[88]
Triangular Number -- from Wolfram MathWorld
A triangular number is a number obtained by adding all positive integers less than or equal to a given positive integer n, ie, where (n; k) is a binomial ...
[89]
Figurate Number -- from Wolfram MathWorld
If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers illustrated above are called triangular, square, ...
[90]
Algebraic Number -- from Wolfram MathWorld
Any algebraic number is an algebraic period, and if a number is not an algebraic period, then it is a transcendental number (Waldschmidt 2006). Note there is a ...
[91]
algebraic number in nLab
Feb 17, 2025 · 1. Definition. An algebraic number is a root of a non-zero polynomial with integer coefficients (or, equivalently, with rational coefficients).
[92]
Algebraic Number Minimal Polynomial -- from Wolfram MathWorld
The minimal polynomial of an algebraic number zeta is the unique irreducible monic polynomial of smallest degree p(x) with rational coefficients such that ...
[93]
Algebraic Integer -- from Wolfram MathWorld
An algebraic integer is a special case of an algebraic number (for which the leading coefficient a_n need not equal 1).
[94]
algebraic integer in nLab
Dec 20, 2021 · In number theory, the concept of algebraic integer is a generalization of that of integer to more general base-number fields.
[95]
Extension Field -- from Wolfram MathWorld
Extension Field ; c_i in F , of powers of alpha less than ; n . In this case, alpha is called an algebraic number over ; F and F(alpha) is an algebraic extension.
[96]
Golden Ratio -- from Wolfram MathWorld
Golden Ratio ; K_(DeltaADE), = 1/2·r(1+phi)·1=1/2rphi ; K_(DeltaBEF), = 1/2·rphi·phi=1/2rphi^2 ; K_(DeltaCDF), = 1/2(1+phi)·r=1/2rphi^2.
[97]
https://ncatlab.org/nlab/show/algebraic%2Bnumber
[98]
Irrational Number -- from Wolfram MathWorld
Weisstein, E. W. "Books about Irrational Numbers." http://www.ericweisstein.com/encyclopedias/books/IrrationalNumbers.html. Referenced on Wolfram|Alpha.
[99]
[PDF] TRANSCENDENCE OF e - KEITH CONRAD
Such sums are not as mathematically interesting as π or e. • In 1873, Hermite [4] proved e is transcendental. • In 1874, Cantor [1] published his first paper on ...
[100]
[PDF] Pi is Transcendental:
Feb 2, 2006 · Lindemann's proof 5 of the transcendence of pi does not actually rely on the transcendence of e, it uses substantially the same methods as ...
[101]
Champernowne Constant -- from Wolfram MathWorld
Champernowne's constant C=0.12345678910111213... (1) (OEIS A033307) is the number obtained by concatenating the positive integers and interpreting them as ...
[102]
[PDF] 3.2 Constructible Numbers
So once we know that each [F : Q]=2n+1, then we know that the characteristic polynomial of each constructible number divides a power of 2.
[103]
[PDF] Impossible constructions. Wed: Gauss-Wantzel Theorem.
May 4, 2020 · Theorem (Descartes, Wantzel 1837):. 1637 point (a,b) constructible a, be Aggrt. Page 2. Today we will use this this to prove the impossibility ...
[104]
[PDF] ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ...
The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.
[105]
Information-Theoretic Limitations of Formal Systems | Journal of the ...
CHAITIN, G.J. Information-theoretic aspects of Post's construction of a simple set. On the difficulty of generating all binary strings of complexity less ...
[106]
[PDF] AN INTRODUCTION TO THE p-ADIC NUMBERS - UChicago Math
The p-adic numbers are useful because they provide another toolset for solving problems, one which is sometimes easier to work with than the real numbers. They ...Missing: primary | Show results with:primary
[107]
[PDF] HENSEL'S LEMMA 1. Introduction In the p-adic integers ...
Introduction. In the p-adic integers, congruences are approximations: for a and b in Zp, a ≡ b mod pn is the same as |a − b|p ≤ 1/pn.Missing: primary | Show results with:primary
[108]
[PDF] quaternion algebras - keith conrad
The quaternions were the first example of a noncommutative division ring, and the following theorem provides a conceptual role for them in algebra among all ...
[109]
[PDF] Quaternions & Rotation in 3D Space - Purdue Engineering
Any 3D rotation matrix can be represented by a rotation. ( ) around a unit vector ( ). • This rotation can be defined by the following unit quaternion: cos. 2.
[110]
[PDF] Supplement. The Cayley-Dickson Construction and Nonassociative ...
Nov 10, 2022 · The Cayley-Dickson construction creates a 2n-dimensional algebra from an n-dimensional one, starting with R, then C, then H, then O.Missing: non- | Show results with:non-
[111]
[PDF] Eigentheory of Cayley-Dickson algebras - University of Oregon
Cayley-Dickson algebras are non-associative finite-dimensional R-division alge- bras that generalize the real numbers, the complex numbers, the quaternions, and.
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None
### Summary of Frobenius’ Theorem on Real Division Algebras
[113]
[PDF] Hurwitz's Theorem
Feb 19, 2015 · Theorem 1 (Hurwitz; 1898) Suppose there is a bilinear product on Rn with the property that. ||v ◦ w|| = ||v||||w||. Then n = 1,2,4, or 8.Missing: RCHO paper
[114]
Ordinal Number -- from Wolfram MathWorld
From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. In order of increasing size, the ordinal numbers are 0, 1 ...Missing: natural | Show results with:natural
[115]
Aleph-1 -- from Wolfram MathWorld
Aleph-1 is the set theory symbol aleph_1 for the smallest infinite set larger than aleph_0 (Aleph-0), which in turn is equal to the cardinal number of the set ...Missing: null | Show results with:null<|control11|><|separator|>
[116]
Set Theory - Stanford Encyclopedia of Philosophy
Oct 8, 2014 · In ZFC one can develop the Cantorian theory of transfinite (i.e., infinite) ordinal and cardinal numbers.2. The Axioms Of Set Theory · 6. The Set Theory Of The... · 10. Large Cardinals
[117]
Cantor Diagonal Method -- from Wolfram MathWorld
c of the real numbers R , which can be shown to be isomorphic to P(N) , where N is the set of natural numbers, is larger than the cardinality aleph_0 of N ...
[118]
The Continuum Hypothesis - Stanford Encyclopedia of Philosophy
May 22, 2013 · Cantor's famous continuum hypothesis (CH) is the statement that 2ℵ0 = ℵ1. This is a special case of the generalized continuum hypothesis (GCH) ...Missing: diagonal | Show results with:diagonal
[119]
Nonstandard Analysis -- from Wolfram MathWorld
Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of genuine infinitesimals.
[120]
Ultrafilters, nonstandard analysis, and epsilon management
Jun 25, 2007 · Non-standard analysis is not a totally “alien” piece of mathematics, and that it is basically only “one ultrafilter away” from standard analysis.
[121]
[PDF] An introduction to nonstandard analysis - UChicago Math
Aug 14, 2009 · In section 2 we will introduce the notion of an ultrafilter, which will allow us to do a typical ultrapower construction of the hyperreal ...