Fact-checked by Grok 2 weeks ago

Irrational number

In mathematics, an irrational number is a real number that cannot be expressed as a simple fraction p/q, where p and q are integers and q \neq 0.^[1] Unlike rational numbers, the decimal expansion of an irrational number is infinite, non-terminating, and non-repeating.^[1] This distinguishes them from rational numbers, which have either terminating or eventually repeating decimals.^[2] The discovery of irrational numbers is attributed to the ancient Greek Pythagorean school in the 5th century BCE, when Hippasus of Metapontum demonstrated the irrationality of \sqrt{2} through a proof by contradiction involving the Pythagorean theorem applied to an isosceles right triangle.^[3] According to legend, this revelation challenged the Pythagoreans' belief that all quantities could be expressed as ratios of integers, leading to Hippasus's supposed punishment by drowning, though the story may be apocryphal.^[3] Key examples include \sqrt{2} \approx 1.414213562\ldots, the ratio of a circle's circumference to its diameter \pi \approx 3.141592653\ldots, and the base of the natural logarithm e \approx 2.718281828\ldots, all of which were later proven irrational—e by Leonhard Euler in 1737 and \pi by Johann Lambert in 1761.^[1] Irrational numbers form a dense subset of the real numbers, meaning that between any two real numbers, there exists an irrational number, and they are uncountably infinite in cardinality.^[4] Irrational numbers can be algebraic (roots of non-constant polynomials with rational coefficients but not rational themselves, such as square roots of non-perfect squares) or transcendental (not roots of any such polynomial, such as \pi and e).^[1] Their properties underpin essential theorems in analysis, geometry, and number theory, such as the completeness of the real numbers, and they appear ubiquitously in physical constants and measurements, making them indispensable in applied mathematics and science.^[5]

Definition and Fundamentals

Definition

In mathematics, an irrational number is defined as a real number that cannot be expressed as the ratio of two integers p and q, where p and q are integers and q \neq 0.^[1] This contrasts with rational numbers, which can be written in such a fractional form. The term "irrational" derives from the ancient Greek word alogos, meaning "without ratio" or "not measurable by a ratio," a concept introduced by early Greek mathematicians to describe quantities that defied expression through integer proportions.^[6] Irrational numbers, alongside rational numbers, form a complete partition of the set of all real numbers, ensuring the real number line is fully populated without gaps.^[1] A fundamental property distinguishing irrational numbers from rationals lies in their decimal expansions: rational numbers yield decimals that either terminate (e.g., 0.5) or repeat periodically (e.g., 0.333...), whereas irrational numbers produce non-terminating decimals that never repeat.^[1]

Basic Properties

Irrational numbers possess several fundamental properties that distinguish them from rational numbers and underscore their role in the structure of the real numbers. One key property is their density within the real line. The set of irrational numbers is dense in \mathbb{R}, meaning that between any two distinct real numbers a < b, there exists at least one irrational number t such that a < t < b.^[7] This density follows from the density of the rationals in \mathbb{R} and the construction of irrationals, such as by adding a suitable irrational to a rational approximation within the interval.^[8] Regarding cardinality, the set of irrational numbers is uncountable. Since the rational numbers are countable and form a proper subset of the uncountable reals, the irrationals must also be uncountable.^[8] Moreover, the irrationals have the same cardinality as the real numbers, which is $2^{\aleph_0}, the cardinality of the continuum.^[8] This implies that the irrationals constitute the "bulk" of the real numbers in terms of measure and size. The irrational numbers exhibit specific closure properties under addition and multiplication when combined with rationals. The sum of a nonzero rational number and an irrational number is always irrational; for instance, if r \in \mathbb{Q} \setminus \{0\} and \alpha is irrational, then r + \alpha is irrational.^[2] Similarly, the product of a nonzero rational and an irrational is irrational. Exceptions occur when the rational is zero, yielding a rational result (e.g., $0 \cdot \alpha = 0), or in cases like \sqrt{2} + (-\sqrt{2}) = 0, but these do not alter the general rule for mixed rational-irrational operations.^[2] From an algebraic perspective, adjoining an irrational number \alpha to the rationals forms a field extension \mathbb{Q}(\alpha) over \mathbb{Q} of degree greater than 1. For algebraic irrationals, such as \alpha = \sqrt{2}, the minimal polynomial over \mathbb{Q} is x^2 - 2 = 0, which is irreducible and of degree 2, so [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2 > 1.^[9] This extension highlights how irrationals generate larger fields beyond the rationals.

Historical Development

Ancient Greece and India

The discovery of irrational numbers is traditionally attributed to the Pythagorean school in ancient Greece around the 5th century BCE, with Hippasus of Metapontum credited for demonstrating the irrationality of the square root of 2 through a geometric proof involving the diagonal of a unit square. This revelation, showing that the diagonal and side of the square are incommensurable—lacking a common unit of measure—challenged the Pythagorean doctrine that all things could be expressed as ratios of whole numbers, precipitating a philosophical crisis within the sect known as the "Pythagorean crisis." According to late ancient sources, Hippasus was punished, possibly by drowning at sea, for publicizing this discovery, which disrupted the harmony of numbers central to Pythagorean cosmology.^[10] Building on this, Theaetetus of Athens (c. 417–369 BCE) advanced the conceptualization by formalizing the distinction between rational and irrational quantities, generalizing proofs of irrationality for square roots of non-square integers beyond 2, such as √3 through √17 initially explored by Theodorus of Cyrene. His work classified various types of irrational lines, including medials, binomials, and apotomes, providing a systematic framework that emphasized their role in geometry without relying on arithmetic ratios. This formalization laid the groundwork for later geometric treatments, resolving incommensurability as a key issue in understanding continuous magnitudes like lengths and areas.^[11] In Euclid's Elements (c. 300 BCE), Book X systematically addresses incommensurable magnitudes, implying the existence of irrational numbers through definitions and propositions on rational and irrational lines relative to a standard unit, without explicitly naming "irrationality" but demonstrating their necessity in geometric proportions. For instance, the text establishes that certain lines, such as the diagonal of a square, are incommensurable in length and square with the side, integrating Eudoxus's theory of proportions to handle such cases rigorously. Philosophically, Greek mathematicians viewed these irrationals as essential for resolving geometric paradoxes, preserving the integrity of deductive reasoning in proofs involving the unit square's diagonal.^[12] In ancient India, mathematicians from the 5th century CE onward engaged with irrational quantities through practical approximations, particularly for square roots like \sqrt{2}, though without explicit proofs of their irrationality. Aryabhata (476–550 CE) introduced algorithmic methods in the Aryabhatiya for extracting square roots digit-by-digit, enabling accurate rational approximations of irrational values encountered in astronomy and geometry, such as those derived from quadratic solutions. These techniques, iterative and efficient for decimal systems, treated irrationals as limits approachable by rationals, reflecting a computational rather than ontological focus.^[13] Brahmagupta (c. 598–668 CE) further developed approximation methods in the Brahmasphutasiddhanta, using solutions to Pell's equation x^2 - 2y^2 = \pm 1 to generate continued fraction convergents for \sqrt{2}, such as the fraction 577/408, which approximates \sqrt{2} to high precision (error less than 10^{-5}). His iterative algorithm for square roots, akin to modern Newton-Raphson, allowed for refining estimates of incommensurable quantities in algebraic contexts like indeterminate equations, prioritizing utility in calculations over theoretical classification of irrationality. Indian traditions thus integrated these approximations into broader algebraic practices, viewing irrationals as operational tools rather than philosophical anomalies.^[14]^[15]

Medieval Islamic World

In the medieval Islamic world, particularly from the 9th to the 11th centuries, mathematicians at centers like the House of Wisdom in Baghdad advanced the study of irrational numbers through innovative algebraic frameworks and geometric techniques, building a synthesis of earlier traditions. This era marked a shift toward treating irrationals not merely as geometric curiosities but as integral components of algebraic problem-solving, often arising in equations whose solutions defied rational expression. Scholars emphasized practical applications, including astronomy, where irrational ratios appeared in computations for celestial movements and trigonometric functions.^[16] Muhammad ibn Musa al-Khwarizmi, working in the early 9th century, laid foundational work in his treatise Hisab al-jabr w'al-muqabala (c. 820 CE), which systematically classified quadratic equations and provided geometric solutions that frequently yielded irrational roots, such as square roots of non-perfect squares. He employed methods like completing the square to resolve these, visualizing them through diagrams of areas and lengths, thereby integrating irrational quantities into a cohesive algebraic system without relying solely on Greek geometric proofs. Al-Khwarizmi's approach unified rational and irrational numbers as "algebraic objects," departing from the purely geometric constraints of ancient traditions and enabling broader numerical applications.^[17]^[16] Omar Khayyam, in the 11th century, extended this algebraic progress in his Treatise on Demonstration of Problems of Algebra (1070 CE), where he classified various cubic equations and devised geometric constructions using conic sections to find their roots, many of which were irrational. For instance, to solve equations of the form x^3 + a x = b, Khayyam intersected a rectangular hyperbola with a circle, obtaining the root as the abscissa of the intersection point; he acknowledged that such solutions inherently involved irrationals but did not pursue explicit proofs of their irrationality, focusing instead on constructive methods. His work highlighted the limitations of arithmetic for cubics, advocating geometric resolution as a reliable path forward.^[18] This period's contributions reflected a cultural synthesis of Greek geometric rigor—drawn from translated works of Euclid and Ptolemy—with Indian arithmetic innovations, such as decimal notation, fostering advancements in practical fields like astronomy. Islamic astronomers, including Khayyam himself, applied these algebraic tools to compute irrational ratios in sine tables and planetary models, as seen in Khayyam's leadership of the Jalali calendar reform, which required precise handling of non-rational lengths and angles for accurate timekeeping.^[18]^[16]

Modern Period

In the Renaissance, Italian mathematician Gerolamo Cardano advanced the study of equations involving irrational numbers through his work on solving cubic equations in Ars Magna (1545), where solutions often required extracting roots that could not be expressed rationally, highlighting the necessity of irrationals in algebraic contexts. During the 18th century, Leonhard Euler provided the first proof of the irrationality of e, the base of the natural logarithm, using continued fraction expansions in a 1737 manuscript published in 1744, establishing that e cannot be expressed as a ratio of integers. In 1761, Johann Heinrich Lambert proved that \pi is irrational using continued fractions.^[1] The 19th century saw significant milestones in distinguishing irrationals from algebraics. In 1844, Joseph Liouville constructed the first explicit examples of transcendental numbers—irrationals not satisfying any polynomial equation with rational coefficients—such as the Liouville constant \sum_{k=1}^\infty 10^{-k!}, proving their existence via Diophantine approximation bounds.^[19] In 1873, Charles Hermite extended this by proving the transcendence of e, showing it satisfies no algebraic equation of finite degree. Ferdinand von Lindemann built on these ideas in 1882, proving the transcendence of \pi and resolving the ancient question of squaring the circle by demonstrating \pi is not constructible via ruler and compass.^[20] In the 20th century, the Gelfond–Schneider theorem, independently proved by Aleksandr Gelfond and Theodor Schneider in 1934, established that if \alpha is algebraic and nonzero (not 0 or 1), and \beta is irrational algebraic, then \alpha^\beta is transcendental; a key example is $2^{\sqrt{2}}, solving part of Hilbert's seventh problem.^[21] These developments profoundly influenced mathematics' foundations. Georg Cantor's work in the 1870s–1890s, proving the uncountability of the real numbers via diagonalization, underscored the vastness of irrationals as the reals minus the countable rationals, laying groundwork for set theory. Simultaneously, efforts by Karl Weierstrass, Richard Dedekind, and others in the mid-19th century rigorized real analysis by defining irrationals through cuts or sequences, enabling precise treatments of limits, continuity, and calculus on the complete real line.^[22]

Proofs of Irrationality

Classical Proofs

The classical proof that \sqrt{2} is irrational employs the method of contradiction, assuming it can be expressed as a ratio of positive integers in lowest terms. Suppose \sqrt{2} = \frac{p}{q}, where p and q are positive integers with no common factors (i.e., \gcd(p, q) = 1). Squaring both sides yields p^2 = 2q^2. Since the right side is even, p^2 must be even, implying p is even (as the square of an odd integer is odd). Let p = 2r for some positive integer r. Substituting gives (2r)^2 = 2q^2, so $4r^2 = 2q^2, or q^2 = 2r^2. Thus, q^2 is even, so q is even. But if both p and q are even, they share a common factor of 2, contradicting the assumption that \frac{p}{q} is in lowest terms. Therefore, no such integers p and q exist, and \sqrt{2} is irrational.^[23] A geometric interpretation of this result, attributed to the Pythagorean school and formalized by Euclid, demonstrates the incommensurability of the side and diagonal of a square without algebraic notation. Consider a square with side length 1; by the Pythagorean theorem, the diagonal d satisfies d^2 = 1^2 + 1^2 = 2. Euclid shows in Elements Book X, Proposition 9, that no integer mean proportional exists between 1 and 2 (i.e., no integer m such that $1 : m = m : 2), as this would require m^2 = 2, leading to an infinite regress of approximations without exact commensurability. This establishes that the side and diagonal cannot be measured by a common unit, confirming the irrationality of their ratio \sqrt{2}.^[24] This argument generalizes to \sqrt{n} for any positive integer n that is not a perfect square. Assume \sqrt{n} = \frac{p}{q} in lowest terms, so p^2 = n q^2. The prime factorization of p^2 has even exponents for all primes. Thus, the exponents in n q^2 must also be even, implying that the exponents in n must be even (since those in q^2 are already even). But if n is not a perfect square, at least one prime in its factorization has an odd exponent, a contradiction. Hence, \sqrt{n} is irrational.^[25] These classical proofs are limited to quadratic irrationals like square roots, as they rely on the properties of degree-2 equations and the structure of integer factorizations, and do not extend directly to higher-degree or transcendental cases.^[23]

General Methods

In the 19th century, mathematicians developed sophisticated methods to prove the irrationality and transcendence of specific numbers and infinite classes, moving beyond classical contradiction proofs for quadratic irrationals. These approaches often rely on Diophantine approximation, integral representations, and series expansions to establish bounds that lead to contradictions under algebraic assumptions. Hermite's pioneering work marked the beginning of systematic transcendence proofs for fundamental constants like e. Charles Hermite proved in 1873 that e is transcendental by assuming it satisfies an algebraic equation with integer coefficients and deriving a contradiction through exponential integrals. Specifically, suppose e is a root of the polynomial \sum_{k=0}^n a_k y^k = 0 where a_k \in \mathbb{Z} and a_n \neq 0. Hermite considered the function f(t) = t^{p-1} g(t)^p / (p-1)!, where g(t) = \prod_{k=1}^n (t - k) and p is a large prime greater than n and the coefficients' magnitudes. Using integration by parts repeatedly on \int_0^x e^{-t} f(t) \, dt, he expressed linear combinations involving e^k \int_0^k e^{-t} f(t) \, dt for k = 0 to n, weighted by a_k. The left side yields a nonzero integer (due to factorial denominators and prime choices ensuring non-divisibility), while bounds on the right side show it is smaller than 1 in absolute value for sufficiently large p, leading to a contradiction.^[26] Building on Hermite's techniques, the Lindemann-Weierstrass theorem, established by Ferdinand von Lindemann in 1882 and rigorously generalized by Karl Weierstrass in 1885, proves that if \alpha is a nonzero algebraic number, then e^\alpha is transcendental. The theorem states more broadly that if \alpha_1, \dots, \alpha_n are distinct algebraic numbers linearly independent over \mathbb{Q}, then e^{\alpha_1}, \dots, e^{\alpha_n} are algebraically independent over the algebraic numbers. The proof assumes a linear dependence \sum_{i=1}^n \beta_i e^{\alpha_i} = 0 with algebraic \beta_i \not\equiv 0, and uses properties of entire functions and Galois actions on conjugates to construct integrals analogous to Hermite's. These integrals, combined with estimates on their growth and algebraic integer norms, imply that the dependence would force a nonzero algebraic integer to have arbitrarily small magnitude, yielding a contradiction. This result immediately implies the transcendence of \pi, since e^{i\pi} = -1 is algebraic.^[27] In Diophantine approximation, continued fraction theory provides tools to quantify how well irrationals can be approximated by rationals, aiding irrationality proofs for broader classes. Klaus Roth's theorem from 1955 states that for any algebraic irrational \alpha of degree d \geq 2 and any \epsilon > 0, there are only finitely many rationals p/q (with q > 0) satisfying |\alpha - p/q| < 1/q^{2+\epsilon}. The proof employs the Thue-Siegel method, reducing the problem to estimating solutions of certain Diophantine inequalities via auxiliary functions and Schmidt's subspace theorem precursors. This sharpens earlier bounds (like those from continued fractions showing exponent 2 for quadratics) and implies that algebraic irrationals have irrationality measure exactly 2, distinguishing them from transcendentals that may have higher measures. Roth's result has profound implications for infinite classes, ruling out overly good rational approximations for all algebraic irrationals. Joseph Liouville's 1844 construction demonstrated the existence of transcendental numbers by explicitly building numbers with exceptionally good rational approximations, violating bounds for algebraics. Liouville's theorem states that if \alpha is algebraic of degree d, then there exists c > 0 such that |\alpha - p/q| > c / q^d for all integers p, q > 0. To construct a transcendental, consider the Liouville constant

L = \sum_{n=1}^\infty 10^{-n!}.

The partial sum up to m is p_m / q_m with q_m = 10^{m!}, and the remainder satisfies |L - p_m / q_m| < 10^{-(m+1)!}, which is less than $1 / q_m^{m} for large m. For any fixed degree d < m, this exceeds the algebraic bound, so L cannot be algebraic of any degree, hence transcendental. This method generates an uncountable class of such numbers by varying bases and factorials.

Key Examples

Square Roots and nth Roots

Square roots of non-square integers provide some of the earliest and most fundamental examples of irrational numbers. The square root of 2, denoted √2, is approximately 1.4142135623 and arises as the length of the diagonal of a unit square, directly connected to the Pythagorean theorem, which states that in a right triangle with legs of length 1, the hypotenuse is √2.^[28] Similarly, the square root of 3, √3 ≈ 1.73205080757, represents the height of an equilateral triangle with side length 2, again illustrating geometric lengths that cannot be expressed as ratios of integers.^[29] Generalizing to nth roots, for integers n > 1 and k a positive integer that is not a perfect nth power, the nth root of k, denoted ⁿ√k, is irrational. A prominent example is the cube root of 2, ∛2 ≈ 1.25992104989, which satisfies (∛2)^3 = 2 but has no rational solution.^[30] These roots are algebraic irrationals, solutions to polynomial equations like x^n - k = 0, and they appear in contexts requiring precise measurements beyond rational approximations.^[31] Nested radicals offer another construction involving roots, where finite nestings yield irrationals while certain infinite ones converge to rationals. For instance, the infinite nested radical √(2 + √(2 + √(2 + ⋯))) converges to 2, solvable via the equation x = √(2 + x) leading to x^2 = 2 + x and x = 2 (discarding the negative root).^[32] However, finite truncations, such as √(2 + √2) ≈ 1.847759065, are irrational.^[32] In applications, these irrational roots are essential in geometry for computing distances, such as diagonals in polygons via the Pythagorean theorem, and in physics for deriving quantities like the speed of an object in uniform circular motion or the distance traveled under constant acceleration, often involving forms like √(2gh) for gravitational fall.^[33]^[34]

Logarithms and Exponentials

Logarithmic and exponential functions provide prominent examples of transcendental irrational numbers, arising naturally in analysis and geometry. The natural logarithm of 2, denoted \ln 2, is approximately 0.6931471805599453 and is known to be transcendental, hence irrational.^[35] Its irrationality can be established using series expansions, such as the Mercator series \ln 2 = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}, though detailed proofs rely on advanced techniques like continued fractions or integral representations developed in the 19th century.^[35] Similarly, the common logarithm \log_{10} 2 approximates 0.30102999566 and is irrational, as 10 and 2 do not share the same prime factors, preventing \log_{10} 2 = p/q for integers p, q > 0 without leading to a contradiction via unique prime factorization.^[36] More generally, for distinct integers a, b > 1 whose prime factorizations differ, \log_b a is irrational by the same elementary argument.^[36] Gelfond's work in the 1930s extended these results, proving that \log_b a is often transcendental when a and b are algebraic integers greater than 1, contributing to the broader theory of transcendental numbers.^[37] Exponential functions yield further examples, with the base of the natural logarithm, e, defined by the infinite series

e = \sum_{n=0}^{\infty} \frac{1}{n!},

approximating 2.718281828459045.^[38] Euler demonstrated e's irrationality in 1737 using its continued fraction expansion [2; \overline{1, 2, 1, 1, 4, 1, 1, 6, \dots}], showing it cannot terminate or repeat as a rational would.^[38] Powers of e are also irrational; for instance, e^2 is irrational, as proven by Liouville in 1840 through analysis of its series expansion.^[39] Another key transcendental irrational is \pi \approx 3.141592653589793, emerging from the geometry of circles as the ratio of circumference to diameter.^[40] Lindemann established its transcendence in 1882, resolving the ancient problem of squaring the circle by showing \pi satisfies no algebraic equation with rational coefficients.^[40] This proof built on Hermite's earlier transcendence result for e, highlighting the deep connections between exponentials, logarithms, and geometric constants in the landscape of irrational numbers.^[40]

Classification of Irrationals

Algebraic Irrational Numbers

Algebraic irrational numbers are the irrational real numbers that are roots of non-zero polynomials with rational coefficients. Specifically, a real number \alpha is algebraic if there exists a polynomial p(x) = a_n x^n + \cdots + a_0 \in \mathbb{Q} with a_n \neq 0 such that p(\alpha) = 0, and \alpha is irrational if it is not rational (i.e., not of the form p/q with integers p, q and q \neq 0). For example, \sqrt{2} is an algebraic irrational of degree 2, as it satisfies the equation x^2 - 2 = 0.^[41] The minimal polynomial of an algebraic number \alpha is the unique monic irreducible polynomial of lowest degree over \mathbb{[Q](/page/Q)} that has \alpha as a root; it has integer coefficients by Gauss's lemma and its degree equals the degree of \alpha. This polynomial is unique for each algebraic number and divides any other polynomial with rational coefficients that vanishes at \alpha. For instance, the minimal polynomial of \sqrt{2} is x^2 - 2.^[42] Adjoining an algebraic irrational \alpha to the rationals generates a field extension \mathbb{Q}(\alpha) whose degree over \mathbb{Q} equals the degree of the minimal polynomial of \alpha. The golden ratio \phi = (1 + \sqrt{5})/2, an algebraic irrational of degree 2, has minimal polynomial x^2 - x - 1 = 0, so [\mathbb{Q}(\phi) : \mathbb{Q}] = 2.^[42]^[43] The set of all algebraic numbers, which includes the algebraic irrationals, is countable, as it can be enumerated by considering polynomials with rational coefficients ordered by degree and height.^[44]

Transcendental Numbers

Transcendental numbers are irrational numbers that are not algebraic, meaning they are not roots of any non-zero polynomial equation with rational coefficients. In other words, a number \alpha \in \mathbb{R} is transcendental if there do not exist integers a_0, a_1, \dots, a_n with a_n \neq 0 and n \geq 1 such that a_n \alpha^n + a_{n-1} \alpha^{n-1} + \dots + a_1 \alpha + a_0 = 0.^[45] Prominent examples include \pi and e, which transcend the algebraic structure of the rationals.^[40]^[38] The existence of transcendental numbers was first established by Joseph Liouville in 1844 through the construction of explicit examples known as Liouville numbers. These are real numbers that can be approximated by rational numbers to an extraordinarily high degree, violating the bounds for algebraic numbers given by Liouville's approximation theorem. A classic example is Liouville's constant, defined as \sum_{k=1}^\infty 10^{-k!}, which is transcendental because it admits rational approximations p/q satisfying |\alpha - p/q| < 1/q^k for arbitrarily large k.^[46]^[47] Furthermore, Georg Cantor proved in 1874 that transcendental numbers are not only existent but form the vast majority of real numbers. By showing that the set of algebraic numbers is countable (as it is the union over degrees of finite sets of roots of polynomials with integer coefficients) and that the real numbers are uncountable via his diagonal argument, Cantor demonstrated that the transcendentals have the cardinality of the continuum, implying that almost all real numbers in the measure-theoretic sense are transcendental.^[48] Key results in transcendence theory include the Lindemann-Weierstrass theorem, proved by Ferdinand von Lindemann in 1882 and generalized by Karl Weierstrass in 1885. The theorem states that if \alpha_1, \dots, \alpha_n are algebraic numbers that are linearly independent over the rationals, then e^{\alpha_1}, \dots, e^{\alpha_n} are algebraically independent over the rationals. This immediately implies the transcendence of e (taking n=1, \alpha_1=1) and of \pi (since if \pi were algebraic, then e^{i\pi} = -1 would contradict the theorem's algebraic independence).^[49]^[50] Transcendental numbers evade the algebraic closure of the rationals, meaning no finite extension of \mathbb{Q} can contain them, which has profound implications for number theory. In particular, their study intersects with Diophantine approximations, where transcendentals like Liouville numbers achieve approximation qualities beyond those possible for algebraics, influencing theorems on how well irrationals can be approximated by rationals and motivating broader transcendence results.^[51]^[52]

Representations and Expansions

Decimal Expansions

Irrational numbers are distinguished by their decimal expansions, which are infinite and non-repeating. This property arises directly from their definition as real numbers that cannot be expressed as a ratio of integers. For instance, the decimal expansion of \sqrt{2} begins as $1.414213562\dots and continues indefinitely without any periodic pattern.^[53] In contrast, the decimal expansions of rational numbers either terminate after a finite number of digits or eventually repeat a sequence of digits. A fundamental theorem states that a real number is rational if and only if its decimal expansion is either finite (terminating) or eventually repeating.^[54] This dichotomy provides a practical test for rationality: if a decimal expansion shows no repetition after sufficiently many digits, the number is irrational. Irrationality thus manifests in the unending, aperiodic nature of these expansions, preventing any finite or cyclic representation in base 10. The decimal expansions of specific irrationals, such as \pi, are computed using algorithms based on infinite series that converge to the desired precision. A classical example is the Leibniz formula, which expresses \pi/4 = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots, allowing successive partial sums to approximate the digits of \pi.^[55] More efficient modern methods, like the Chudnovsky algorithm, accelerate this process by providing faster convergence through a Ramanujan-type series for $1/\pi, enabling computations to trillions of digits on contemporary hardware.^[56] Many irrational numbers are conjectured to be normal in base 10, meaning that in their infinite decimal expansion, every digit from 0 to 9 appears with equal frequency of $1/10, and every finite sequence of digits appears with the expected frequency based on its length. This equidistribution would imply a statistically random-like digit sequence. However, while empirical evidence from billions of computed digits supports normality for numbers like \pi, the conjecture remains unproven.^[57]

Continued Fractions

A continued fraction is an expression of a real number \alpha as \alpha = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}, where a_0 is an integer and the partial quotients a_i for i \geq 1 are positive integers; this is denoted in abbreviated form as \alpha = [a_0; a_1, a_2, a_3, \dots].^[58] For irrational numbers, the continued fraction expansion is infinite, providing a non-terminating representation that contrasts with the finite expansions of rational numbers.^[58] Quadratic irrational numbers, which are roots of quadratic equations with integer coefficients, have continued fraction expansions that are eventually periodic.^[59] This periodicity arises from the algebraic structure of these numbers, where the sequence of partial quotients repeats after a certain point. For example, the continued fraction for \sqrt{2} is [1; \overline{2}] = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \cdots}}}, with the bar indicating the repeating block of 2. The convergents of a continued fraction, denoted p_n / q_n, are rational approximations obtained by truncating the expansion at the nth partial quotient and evaluating the finite continued fraction. These convergents provide the best rational approximations to the irrational \alpha in the sense that any rational p/q with q \leq q_n satisfies | \alpha - p/q | > | \alpha - p_n / q_n |.^[60] Moreover, the approximation quality is bounded by | \alpha - p_n / q_n | < 1 / q_n^2, ensuring rapid convergence relative to the denominator size.^[58] Continued fractions offer advantages over decimal expansions for computational purposes, particularly in generating high-precision approximations with minimal terms and in solving Diophantine equations.^[61] They are especially useful in addressing Pell's equation x^2 - d y^2 = \pm 1, where d is a positive integer that is not a perfect square; the fundamental solutions to this equation correspond to convergents from the periodic continued fraction expansion of \sqrt{d}.^[61]

Advanced Properties

Irrational Powers

The Gelfond–Schneider theorem provides a fundamental result on the nature of irrational powers, stating that if \alpha is an algebraic number not equal to 0 or 1, and \beta is an irrational algebraic number, then \alpha^\beta is transcendental.^[37] This theorem, proved independently by Aleksandr Gelfond in 1934 and Theodor Schneider in 1935, establishes that such expressions transcend the realm of algebraic numbers. A classic example is $2^{\sqrt{2}}, which is transcendental under the theorem's conditions, as 2 is algebraic (not 0 or 1) and \sqrt{2} is an irrational algebraic number.^[37] The theorem resolves Hilbert's seventh problem, posed in 1900, which asked whether numbers of the form \alpha^\beta (with \alpha algebraic, \neq 0,1, and \beta irrational algebraic) are always transcendental.^[62] While the theorem confirms transcendence in the general case, specific chained expressions reveal exceptions where rationality emerges. For instance, \sqrt{2}^{\sqrt{2}} is transcendental (hence irrational), yet raising it to the power \sqrt{2} yields

(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{\sqrt{2} \cdot \sqrt{2}} = \sqrt{2}^2 = 2,

a rational number.^[62] This example illustrates how irrational bases and exponents can produce rational outcomes through composition, without contradicting the theorem. Challenges arise when the exponent is transcendental rather than algebraic irrational, placing such cases beyond the Gelfond–Schneider theorem's direct scope. A prominent example is $2^e, where e is the base of the natural logarithm (transcendental); even its irrationality remains an unresolved question.^[63] Extensions of these ideas appear in Schanuel's conjecture, proposed in 1970, which posits that for any set of n complex numbers linearly independent over the rationals, the field extension generated by their exponentials and the numbers themselves has transcendence degree at least n. If true, the conjecture would imply the transcendence of expressions like e^e, significantly broadening our understanding of irrational powers.^[64]

Irrationality Measures

The irrationality measure of a real number \alpha, denoted \mu(\alpha), quantifies how well \alpha can be approximated by rational numbers. It is defined as the supremum of the set of real numbers \mu such that the inequality \left| \alpha - \frac{p}{q} \right| < \frac{1}{q^\mu} holds for infinitely many integers p and q > 0 with \gcd(p, q) = 1. For rational \alpha, \mu(\alpha) = 1, as better approximations than linear become impossible beyond finitely many rationals. For irrational \alpha, Dirichlet's approximation theorem guarantees \mu(\alpha) \geq 2, reflecting the existence of infinitely many quadratic approximations.^[65] Liouville numbers represent the extreme case among irrationals, with \mu(\alpha) = \infty. These numbers admit rational approximations of arbitrarily high order, satisfying \left| \alpha - \frac{p}{q} \right| < \frac{1}{q^\mu} for infinitely many p/q and any \mu > 0. Constructed by Joseph Liouville in 1844 using infinite series like \sum_{k=1}^\infty 10^{-k!}, they are transcendental and form a dense G_\delta set of Lebesgue measure zero. Roth's theorem provides a fundamental bound for algebraic irrationals. It asserts that if \alpha is an algebraic irrational number (i.e., a root of a non-zero polynomial with integer coefficients and degree at least 2), then \mu(\alpha) = 2. This means that for any \varepsilon > 0, the inequality \left| \alpha - \frac{p}{q} \right| < \frac{1}{q^{2+\varepsilon}} holds for only finitely many rationals p/q, so approximations are effectively no better than quadratic. Proved by Klaus Roth in 1955, the result earned him the Fields Medal in 1958 and implies that algebraic irrationals are poorly approximable relative to Liouville numbers. Specific constants illustrate these concepts. For the base of the natural logarithm, e, the continued fraction expansion and Padé approximants from its series yield \mu(e) = 2, matching the algebraic case despite e's transcendence.^[65] In contrast, for \pi, transcendence implies \mu(\pi) \geq 2, but upper bounds remain higher; the best known is \mu(\pi) \leq 7.103205334137\dots, established using integral representations and acceleration techniques.^[66]

The Set of Irrational Numbers

Topological and Measure Properties

The set of irrational numbers, \mathbb{R} \setminus \mathbb{Q}, with the subspace topology inherited from the Euclidean topology on \mathbb{R}, exhibits rich topological structure. It is homeomorphic to the Baire space \mathbb{N}^\mathbb{N}, the countable product of copies of the discrete space of natural numbers equipped with the product topology.^[67] This homeomorphism establishes that the irrationals form a Polish space, meaning it is separable and completely metrizable. Furthermore, as a consequence, the space of irrationals is zero-dimensional, possessing a basis of clopen sets. In terms of Lebesgue measure, the irrationals occupy the full measure of the real line. The rational numbers \mathbb{Q} form a countable set and thus have Lebesgue measure zero.^[68] Consequently, for any bounded interval [a, b] \subset \mathbb{R}, the Lebesgue measure of the irrationals in [a, b] equals b - a.^[68] The cardinality of \mathbb{R} \setminus \mathbb{Q} is $2^{\aleph_0}, matching the cardinality of the continuum and the real numbers \mathbb{R}.^[69] Topologically, the irrationals are dense in \mathbb{R}, with an irrational between any two distinct reals, and codense, as their complement \mathbb{Q} is also dense in \mathbb{R}.^[68] Fractal-like subsets within the irrationals include certain Cantor sets of positive Lebesgue measure. For instance, symmetric fat Cantor sets can be constructed entirely from irrationals while maintaining positive measure and being nowhere dense.^[70]

Constructive Perspectives

In constructive mathematics, an irrational number is defined positively as a real number that is bounded away from every rational number by some positive distance, meaning for every rational r, there exists a positive rational \epsilon > 0 such that |x - r| \geq \epsilon.^[71] This requires effective bounds on approximations, ensuring that the irrationality can be verified algorithmically rather than merely assumed through non-constructive existence proofs. Such a definition aligns with the foundational principle that mathematical objects must be explicitly constructible.^[72] Classical proofs of irrationality, such as the reductio ad absurdum demonstration that \sqrt{2} is irrational, are non-constructive because they rely on the law of the excluded middle, asserting that either a number is rational or it is not without providing a method to decide.^[71] Brouwer's intuitionism explicitly rejects this law for statements involving infinite processes or irrationals, as it would imply unattainable omniscience about unending constructions; instead, intuitionists demand that proofs furnish explicit witnesses or algorithms for claims about irrationals.^[73] This rejection highlights a core challenge: many classical irrationals cannot be constructively proven irrational without additional effective procedures to separate them from rationals. Examples of constructively acceptable irrationals include computable ones, which can be approximated to arbitrary precision via algorithms, allowing effective digit computation.^[71] In contrast, non-computable irrationals, such as those arising from classical diagonalization arguments, exist in the full classical continuum but cannot be constructively exhibited or proven to exist, as constructive mathematics avoids non-effective existence proofs.^[71] Bishop's constructive approach formalizes real numbers as equivalence classes of Cauchy sequences of rationals equipped with moduli of convergence, ensuring all operations are effective and computable in principle.^[71] Within this framework, an irrational is a real number whose representing Cauchy sequence does not stabilize to any rational, meaning it remains apart from all rationals by a positive margin throughout the construction; for instance, Bishop provides a fully constructive proof that \sqrt{2} is irrational by exhibiting bounds showing it diverges from all rationals.^[72] This method preserves much of classical analysis while grounding it in verifiable approximations.

Open Questions

Unsolved Problems

One prominent unsolved problem in the theory of irrational numbers concerns the sum \pi + [e](/page/E!). It remains unknown whether this sum is rational or irrational, though it is widely believed to be irrational due to the algebraic independence conjectured for \pi and [e](/page/E!).^[74] A related open question involves [e](/page/E!) + \pi, which is also undetermined as rational or irrational. While Yuri Nesterenko proved in 1996 that \pi and e^{\pi} are algebraically independent—and hence both transcendental—this result does not resolve the nature of their sum.^[75] Apéry's constant, denoted \zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3}, was established as irrational by Roger Apéry in 1979 through a proof involving continued fraction approximations and linear forms. However, its transcendence remains an open problem, with no known algebraic relation over the rationals beyond its irrationality.^[76] Schanuel's conjecture, proposed by Stephen Schanuel in the 1970s, posits that for complex numbers z_1, \dots, z_n linearly independent over \mathbb{Q}, the field extension \mathbb{Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n}) has transcendence degree at least n over \mathbb{Q}. If true, it would imply the algebraic independence of e and \pi, thereby establishing the transcendence of e + \pi and e^{\pi}, as well as resolving various logarithmic relations involving these constants.^[77] It is conjectured that the irrational numbers \pi and e are normal in base 10, meaning that in their decimal expansions, every finite sequence of digits appears with limiting frequency equal to its natural probability, as if the digits were randomly and uniformly distributed from 0 to 9.^[78] This property would imply that the digits behave randomly, with no bias toward particular patterns, but the conjecture remains unproven for both constants despite rigorous statistical analyses. Extensive computations of the first 300 trillion digits of \pi (as of April 2025) and over 35 trillion digits of e (as of 2023) have yielded empirical evidence consistent with normality, showing digit frequencies and block distributions that align closely with expected values under the random model, though such tests cannot constitute a proof.^[79])^[78]^[80] Schanuel's conjecture provides a framework for algebraic independence among transcendental numbers, stating that if z_1, \dots, z_n \in \mathbb{C} are linearly independent over \mathbb{Q}, then the field extension \mathbb{Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n}) has transcendence degree at least n over \mathbb{Q}. A direct consequence is the algebraic independence over \mathbb{Q} of \{\pi, e^\pi\}, implying no nonzero polynomial P \in \mathbb{Q}[X,Y] satisfies P(\pi, e^\pi) = 0.^[77] This would resolve broader questions in transcendental number theory, such as the independence of related constants like e + \pi and e\pi, with implications for the structure of exponential fields and Diophantine approximations involving irrationals.^[77] The ABC conjecture, which bounds the product of distinct prime factors (radical) of integers a, b, c with a + b = c and \gcd(a,b)=1 by asserting c \ll \mathrm{rad}(abc)^{1+\varepsilon} for any \varepsilon > 0, extends to number fields and yields bounds on irrational solutions to Diophantine equations. In particular, for any irrational algebraic number \alpha and \varepsilon > 0, it implies that the equation |\alpha - p/q| < 1/q^2 \cdot (\mathrm{rad}(pq))^\varepsilon has only finitely many rational solutions p/q, refining approximation properties and limiting solutions in superelliptic or radical-involved equations where irrationals arise.^[81] This has potential applications in arithmetic geometry, constraining the growth of solutions in equations mixing rational and irrational terms. The Beal conjecture asserts that there are no positive integers a, b, c, x, y, z > 2 satisfying a^x + b^y = c^z where a, b, c are pairwise coprime, generalizing Fermat's Last Theorem to unequal exponents. While the conjecture focuses on integer exponents, variants exploring irrational exponents, such as whether a^r + b^s = c^t admits solutions for irrational r,s,t > 2 and rational a,b,c, remain open and tie into broader questions of transcendence in exponential Diophantine equations, though no counterexamples or proofs are known in these forms.^[82]

References

[1]
Irrational Number -- from Wolfram MathWorld
Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational. There is no standard notation ...
[2]
[PDF] CHAPTER 9 - Irrational Numbers - Purdue Math
Every infinite decimal represents a specific number, and every real number can be represented as an infinite decimal. Numbers with finite expansions also have ...
[3]
How a Secret Society Discovered Irrational Numbers
Jun 13, 2024 · The ancient scholar Hippasus of Metapontum was punished with death for his discovery of irrational numbers—or at least that's the legend.
[4]
Real Numbers:Irrational - Department of Mathematics at UTSA
Oct 21, 2021 · Irrational numbers are real numbers not rational, cannot be expressed as a ratio of two integers, and their decimal expansions do not terminate ...
[5]
Irrational Numbers - Duke Physics
An irrational number is one that cannot be written as a ratio of two integers e.g. $ a/b$ . It is not immediately obvious that numbers like this exist at ...
[6]
Origin of Irrational Numbers and Their Approximations - MDPI
In particular, for book X, Al-Mahani examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational ...
[7]
[PDF] The Real Numbers
For any two real numbers a < b, there exists an irrational number t such that a<t<b. Proof. Exercise. D. Theorem 1.8 (Existence of. √. 2). There exists a real ...
[8]
[PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
They cover the properties of the real numbers, sequences and series of real numbers ... irrational. In particular, the length of the hypotenuse of a right-angled ...
[9]
[PDF] Lecture 6 - Math 5111 (Algebra 1)
If α is a irrational real number and n is a positive integer, there exist infinitely many rational numbers p q with α. − p q. <. 1 q2 . The point is that this ...
[10]
Pythagoreanism (Stanford Encyclopedia of Philosophy)
Summary of each segment:
[11]
Theaetetus - Biography
### Summary of Theaetetus' Contributions to Irrational Numbers and Formalization of Distinction
[12]
Euclid - Biography
### Summary of Euclid's Elements Book X and Irrational Magnitudes
[13]
[PDF] Ancient Indian Square Roots: An Exercise in Forensic Paleo ...
Our modern system of positional decimal notation with zero, together with efficient algorithms for computation, which were discovered in India some time.
[14]
Ancient Indian Mathematics - Sringeri Vidya Bharati Foundation
In this article, we will see a systematic algebraic method for rational approximation of irrational numbers and how ancient Indian algebraists elegantly handled ...
[15]
Brahmagupta - Biography
### Summary of Brahmagupta's Approximations for Square Roots and Handling of Irrationals
[16]
Arabic mathematics - MacTutor - University of St Andrews
Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Astronomy, time- ...
[17]
Al-Khwarizmi - Biography
### Summary of Al-Khwarizmi's Contributions
[18]
Omar Khayyam - Biography
### Summary of Omar Khayyam's Work
[19]
Joseph Liouville (1809 - 1882) - Biography - MacTutor
Joseph Liouville is best known for his work on transcendental numbers. He constructed an infinite class of such numbers. Thumbnail of Joseph Liouville View ...
[20]
Ferdinand von Lindemann (1852 - 1939) - Biography - MacTutor
He is famed for his proof that π is transcendental ... Using methods similar to those of Hermite, Lindemann established in 1882 that π was also transcendental.
[21]
Aleksandr Osipovich Gelfond - Biography - MacTutor
Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational ...
[22]
Richard Dedekind (1831 - 1916) - Biography - MacTutor
Richard Dedekind's major contribution was a redefinition of irrational numbers in terms of Dedekind cuts. He introduced the notion of an ideal in Ring ...<|separator|>
[23]
[PDF] Extreme Proofs I: The Irrationality of 2
It clearly generalizes to show that no rational number can be the nth root of an integer that is not a perfect nth power: for the exponents of the primes in the ...
[24]
Euclid's Elements, Book VIII, Proposition 8
### Summary of Incommensurability of √2 from Proposition 8
[25]
[PDF] Lemma. A positive integer n is a perfect square - CSUSM
Suppose n ≥ 2 is an integer which is not a perfect square. Then √ n is irrational. multiplying both sides by b2 gives us nb2 = a2. By the above lemma, since n ...
[26]
Sur la fonction exponentielle - EuDML
Hermite, Charles. Sur la fonction exponentielle. Paris: Gauthier-Villars, 1874. <http://eudml.org/doc/203956>. @book{Hermite1874, author = {Hermite, Charles} ...
[27]
Ueber die Zahl (..pi..) - EuDML
Lindemann. "Ueber die Zahl (..pi..)." Mathematische Annalen 20 (1882): 213-225. <http://eudml.org/doc/157031>.Missing: ludolphsche | Show results with:ludolphsche
[28]
Pythagoras's Constant -- from Wolfram MathWorld
Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9,and 16) are also irrational (Wells 1986, p. 34). It is not ...
[29]
Square Root -- from Wolfram MathWorld
The square root of 3 is the irrational number sqrt(3) approx 1.73205081 ... Cube Root, nth Root, Nested Radical, Newton's Iteration, Principal Square Root ...
[30]
root(2,n) is irrational for n≥3 (proof using Fermat's last theorem)
Mar 22, 2013 · Theorem 1. If n≥3 n ≥ 3 , then n√2 2 n is irrational. The below proof can be seen as an example of a pathological proof.Missing: nth | Show results with:nth
[31]
https://mathworld.wolfram.com/NthRoot.html
[32]
Nested Radical -- from Wolfram MathWorld
### Summary of Nested Radicals from Wolfram MathWorld
[33]
Consequences of the Pythagorean Theorem
The Square Root of 2 Is Irrational. The Pythagorean Theorem immediately indicates how to construct the square root of each positive integer. See Figure 1. 1.
[34]
[PDF] Square Root Of 2
complexities among irrational numbers and influences their respective applications. Page 34. Square Root Of 2. 34. Pros and Cons of Using √2 in Calculations.
[35]
Natural Logarithm of 2 -- from Wolfram MathWorld
### Summary of Irrationality, Transcendence, Approximations, and Theorems for ln(2)
[36]
Logarithm -- from Wolfram MathWorld
### Summary on Irrationality of Logarithms
[37]
Gelfond's Theorem -- from Wolfram MathWorld
This provides a partial solution to the seventh of Hilbert's problems. It was proved independently by Gelfond (1934ab) and Schneider (1934ab). This establishes ...
[38]
e -- from Wolfram MathWorld
### Summary of e from Wolfram MathWorld
[39]
Full article: A simple proof e 2 is irrational
Aug 10, 2006 · It is well known that e 2 is irrational: this note presents a simple proof of it. The arguments stay within the realms of a first proof course in mathematical ...
[40]
Pi -- from Wolfram MathWorld
Pi is the ratio of a circle's circumference to its diameter, also known as Archimedes' or Ludolph's constant.Missing: extension | Show results with:extension
[41]
Algebraic Number -- from Wolfram MathWorld
Algebraic numbers are represented in the Wolfram Language as indexed polynomial roots by the symbol Root[f, n], where n is a number from 1 to the degree of the ...
[42]
Algebraic Number Minimal Polynomial -- from Wolfram MathWorld
The minimal polynomial of an algebraic number is the unique irreducible monic polynomial of smallest degree with rational coefficients such that. and whose ...Missing: definition | Show results with:definition
[43]
Golden Ratio -- from Wolfram MathWorld
**Minimal Polynomial of the Golden Ratio:**
[44]
algebraic numbers are countable - PlanetMath.org
Mar 22, 2013 · it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto Z+ ℤ + . References. 1 E. Kamke: ...
[45]
Transcendental Number -- from Wolfram MathWorld
Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. ... "Linear Forms in the ...
[46]
Liouville's Constant -- from Wolfram MathWorld
Liouville (1844) constructed an infinite class of transcendental numbers using continued fractions, but the above number was the first decimal constant to ...
[47]
Liouville's Approximation Theorem -- from Wolfram MathWorld
"Liouville's Theorem and the Construction of Transcendental Numbers." §2.6.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed ...
[48]
Countably Infinite -- from Wolfram MathWorld
Countably infinite sets have cardinal number aleph-0. Examples of countable sets include the integers, algebraic numbers, and rational numbers.
[49]
Lindemann-Weierstrass Theorem -- from Wolfram MathWorld
. The Lindemann-Weierstrass theorem is implied by Schanuel's conjecture (Chow 1999). See also. Algebraically Independent, Hermite-Lindemann Theorem, Schanuel's ...
[50]
proof of Lindemann-Weierstrass theorem and that e and π - π
Mar 22, 2013 · This article provides a proof of the Lindemann-Weierstrass theorem Mathworld Planetmath, using a method similar Planetmath Planetmath to those used by ...Missing: paper | Show results with:paper
[51]
Transcendental numbers and diophantine approximations
There is no online version at this time. Please download the PDF instead. Access the abstract. Serge Lang "Transcendental numbers and diophantine approximations ...Missing: implications | Show results with:implications
[52]
[PDF] bulletin de la smf - Numdam
The theory of transcendental numbers and diophantine approxi- mations has only few results, most of which now appear isolated. It is.
[53]
The existence of the square root of two. - DPMMS
Then by trial and error I discover that 1.42=1.96<2 and 1.52=2.25>2 so the decimal expansion must start with 1.4.
[54]
[PDF] Approximating Numbers Part I: Continued Fractions
Theorem 1 A number is rational if and only if its decimal expansion is either finite or eventually repeating. To prove this theorem, we'll have to be able ...
[55]
Pi - The Gregory-Leibniz Series
The Gregory-Leibniz Series ... expansion of tan − 1 ⁡ x is. tan − 1 ⁡ x = x − x 3 3 + x 5 5 − . . . and the formula is obtained by substituting x = 1 .
[56]
[PDF] A Fast Self-correcting π Algorithm - arXiv
Dec 21, 2019 · The computer program, y-cruncher, implemented the Chudnovsky algorithm has been used to compute π to 31.4 trillion digits [19, 12]. In this ...
[57]
[PDF] The Mountains of Pi
Mar 2, 1992 · ... pi: that every digit has an equal chance of appearing in pi. This is known as the normality conjecture for pi. The normality conjecture says ...
[58]
[PDF] Lecture 4 - Math 4527 (Number Theory 2)
Continued fractions are a method for generating rational approximations of a real number, using the form a0 + 1/a1 + 1/a2 + ... + 1/ak-1 + 1/ak.
[59]
[PDF] 19. Quadratic irrationalities - UCSD Math
Theorem 19.1. ξ is a quadratic irrational if and only if its continued fraction expansion is eventually periodic. Proof. Suppose that the continued fraction ...
[60]
Continued Fractions - Convergence
Theorem: Let p / q be a convergent for the nonnegative real a . Then if p ′ / q ′ is a closer rational approximation, then q ′ > q . ... and a b ′ − a ′ b = − 1 .
[61]
[PDF] Continued Fractions and Pell's Equation
Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell's equation. I.
[62]
Hilbert's seventh problem, and powers of 2 and 3 - Terry Tao
Aug 21, 2011 · I will be discussing another of Hilbert's problems, namely Hilbert's seventh problem, on the transcendence of powers {a^b} of two algebraic numbers {a,b}.
[63]
Have any long-suspected irrational numbers turned out to be rational?
Jul 22, 2010 · Are there examples of numbers that, while their status was unknown, were "assumed" to be irrational, but eventually shown to be rational? nt.Missing: founding | Show results with:founding<|control11|><|separator|>
[64]
Schanuel's Conjecture -- from Wolfram MathWorld
Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that e and pi are ...
[65]
Irrationality Measure -- from Wolfram MathWorld
The irrationality measure is the threshold where a number is no longer approximable by rational numbers, and it's the smallest number where an inequality holds.Missing: extension | Show results with:extension
[66]
The Irrationality Measure of Pi is at most 7.103205334137... - arXiv
Dec 13, 2019 · Abstract:We use a variant of Salikhov's ingenious proof that the irrationality measure of \pi is at most 7.606308\dots to prove that, ...Missing: bound | Show results with:bound
[67]
spaces homeomorphic to Baire space - PlanetMath.org
Mar 22, 2013 · ... the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism f:N→(0,1)∖Q f : ...
[68]
[PDF] Brief Notes on Measure Theory - UC Davis Mathematics
follows from the countable additivity of a measure that the Lebesgue measure of any countable set is zero; for example, the set of rationals has Lebesgue.Missing: citation | Show results with:citation
[69]
[PDF] 6. Cardinality And The Strange Nature Of Infinity
Theorem 6.17: Both \ and the set of irrational numbers are uncountable sets. Proof: We start by showing that \ is an uncountable set. It is better to deal with ...
[70]
Fat, Symmetric, Irrational Cantor Sets - Taylor & Francis Online
Apr 11, 2018 · (1981). Fat, Symmetric, Irrational Cantor Sets. The American Mathematical Monthly: Vol. 88, No. 5, pp. 340-341.<|control11|><|separator|>
[71]
Constructive Mathematics - Stanford Encyclopedia of Philosophy
Nov 18, 1997 · This point is illustrated by a well-worn example, the proposition: There exist irrational numbers $a, b$ such that $a^b$ is rational. A ...Varieties of Constructive... · Constructive Reverse... · Constructive Mathematical...
[72]
Constructive Mathematics | Internet Encyclopedia of Philosophy
Constructive mathematics is positively characterized by the requirement that proof be algorithmic. Loosely speaking, this means that when a (mathematical) ...Motivation & History · Constructive Recursive... · Bishop's Constructive...
[73]
https://plato.stanford.edu/entries/intuitionism/
[74]
[PDF] Some unsolved problems in number theory - OU Math
Is π/e a rational number? It is known that both π and e are irrational, but it is not known whether π is a rational multiple of e. Nor is it known whether π ...
[75]
[PDF] Algebraic and Transcendental Numbers
However, the same question with π and eπ has been answered: Theorem 2.2 (Nesterenko, 1996) The numbers π and eπ are algebraically independent. Concerning ...
[76]
[PDF] Transcendental Number Theory: recent results and open problems.
Why is eπ known to be transcendental while πe is not known to be irrational ? Answer : eπ = (−1)−i. 35 / 104. Page 36 ...
[77]
[PDF] Schanuel's Conjecture
transcendental. • the two numbers e and π are algebraically independent. Schanuel's Conjecture implies that both statements are true ! 44 / 106. Page 45 ...
[78]
[PDF] Normality and the Digits of π - David H Bailey
Nov 8, 2011 · It follows, from basic measure theory, that almost all real numbers are b-normal for any specific base b and even for all bases simultaneously.Missing: unproven | Show results with:unproven
[79]
[PDF] Walking on real numbers - CARMA
Jul 23, 2012 · One key question of some significance is whether (and why) numbers such as π and e are “normal.” A real constant α is b-normal if, given the ...<|control11|><|separator|>
[80]
[PDF] The abc conjecture and some of its consequences - HAL
For any irrational algebraic number α and any positive ε the set of ... • The abc conjecture for a cyclotomic number field K implies Greenberg's conjecture for in ...
[81]
[PDF] arXiv:2103.02431v2 [math.NT] 8 Mar 2021
Mar 8, 2021 · Having established that the slopes of a line through the origin and the lattice point (A,B,C) are irrational when gcd(A,B,C) = 1 translates to ...