|
Hilbert's Seventh Problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen''). Statement of the problem Two specific equivalent questions are asked: #In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental? #Is a^b always transcendental, for algebraic a \not\in \ and irrational algebraic b? Solution The question (in the second form) was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational ''b'' is important, since it is easy to see that a^b is algebraic for algebraic ''a'' and rational ''b''.) From the point of view of generalizations, this is the case : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in ''Archiv der Mathematik und Physik''. and Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, 21, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . The quality of a number being transcendental is called transcendence. Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental. Transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the Golden triangle (mathematics), golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the ''legs'' and the third side is called the base (geometry), ''base'' of the triangle. The other dimensions of the triangle, such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude (mathematics), magnitude or Physical quantity, quantity of these types of geometric figures and in this context an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Gelfond
Alexander Osipovich Gelfond (; 24October 19067November 1968) was a Soviet mathematician. Gelfond's theorem, also known as the Gelfond–Schneider theorem, is named after him. Biography Alexander Gelfond was born in Saint Petersburg, Russian Empire, the son of a professional physician and amateur philosopher Osip Gelfond. He entered Moscow State University in 1924, started his postgraduate studies there in 1927, and obtained his Ph.D. in 1930. His advisors were Aleksandr Khinchin (1894-1959) and Vyacheslav Stepanov (1889-1950). In 1930, he stayed for five months in Germany (in Berlin and Göttingen) where he worked with Edmund Landau, Carl Ludwig Siegel, and David Hilbert. In 1931 he started teaching as a Professor at the Moscow State University and worked there until the last day of his life. Since 1933 he also worked at the Steklov Institute of Mathematics. In 1939, he was elected a Corresponding member of the Academy of Sciences of the Soviet Union for his works in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodor Schneider
__NOTOC__ Theodor Schneider (7 May 1911 – 31 October 1988) was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem, who was born in Frankfurt am Main and died in Freiburg im Breisgau. Schneider studied from 1929 to 1934 in Frankfurt; he solved Hilbert's 7th problem in his PhD thesis, which then came to be known as the Gelfond–Schneider theorem. Later, he became an assistant to Carl Ludwig Siegel in Göttingen, where he stayed until 1953. Then, he became a professor in Erlangen (1953–1959) and finally until his retirement in Freiburg (1959–1976). During his time in Freiburg, he also served as the director of the Mathematical Research Institute of Oberwolfach The Oberwolfach Research Institute for Mathematics () is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and ... from 1959 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond–Schneider Theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement Comments The values of ''a'' and ''b'' are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational). In general, is multivalued, where log stands for the complex natural logarithm. (This is the multivalued inverse of the exponential function exp.) This accounts for the phrase "any value of" in the theorem's statement. An equivalent formulation of the theorem is the following: if ''α'' and ''γ'' are nonzero algebraic numbers, and we take any non-zero logarithm of ''α'', then is either rational or transcendental. This may be expressed as saying that if , are linearly independent over the rationals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Baker (mathematician)
Alan Baker (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Life Alan Baker was born in London on 19 August 1939. He attended Stratford School, Stratford Grammar School, East London, and his academic career started as a student of Harold Davenport, at University College London and later at Trinity College, Cambridge, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, Transcendental number theory, transcendence, linear forms in logarithms, Effective results in number theory, effective methods, Diophantine geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker's Theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be of "extraordinarily great significance". The result, proved by , subsumed many earlier results in transcendental number theory. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. History To simplify notation, let \mathbb be the set of logarithms to the base ''e'' of nonzero algebraic numbers, that is \mathbb = \left \, where \Complex denotes the set of complex numbers and \overline denotes the algebraic numbers (the algebraic closure of the rational numbers \Q). Using this notation, several results in transcenden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
