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Proofs From THE BOOK
''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." Content ''Proofs from THE BOOK'' contains 32 sections (45 in the sixth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, Mathematical analysis, analysis, combinatorics and graph theory. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by . It has gone through six editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian, Spanish and Greek. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Aigner
Martin Aigner (28 February 1942 – 11 October 2023) was an Austrian mathematician and professor at Freie Universität Berlin from 1974 with interests in combinatorial mathematics and graph theory. Biography Martin Aigner was born on 28 February 1942. He received his Ph.D from the University of Vienna. His book ''Proofs from THE BOOK'' (co-written with Günter M. Ziegler) has been translated into 12 languages. Aigner died on 11 October 2023, at the age of 81. Awards Aigner was a recipient of a 1996 Lester R. Ford Award from the Mathematical Association of America for his expository article ''Turán's theorem, Turán's Graph Theorem''. In 2018, Aigner received the Leroy P. Steele Prize for Mathematical Exposition (jointly with Günter M. Ziegler). Selected publications *''Combinatorial Theory'' (1997 reprint: , 1979: ; ) *(with Günter M. Ziegler) ''Proofs from THE BOOK'' ** **''A Course in Enumeration'' 2007, * *Mathematics Everywhere. Martin Aigner (Author, Editor), Ehrhard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Pierre de Fermat, Fermat's theorem on sums of two squares states that an Even and odd numbers, odd prime number, prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all Modular_arithmetic#Congruence, congruent to 1 modular arithmetic, modulo 4, and they can be expressed as sums of two squares in the following ways: :5 = 1^2 + 2^2, \quad 13 = 2^2 + 3^2, \quad 17 = 1^2 + 4^2, \quad 29 = 2^2 + 5^2, \quad 37 = 1^2 + 6^2, \quad 41 = 4^2 + 5^2. On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 (if number squared is even) or 1 (if number squared is odd) modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wetzel's Problem
In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign... Let ''F'' be a family of distinct analytic functions on a given domain with the property that, for each ''x'' in the domain, the functions in ''F'' map ''x'' to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that ''F'' is necessarily itself countable. Paul Erdős in turn learned about the problem at the University of Michigan, likely via Lee Albert Rubel. In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each ''x'' is mapped to a finite set of values, ''F'' is necessarily finite. However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schröder–Bernstein Theorem
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if and , then ; that is, and are equipotent. This is a useful feature in the ordering of cardinal numbers. The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as the Cantor–Bernstein theorem or Cantor–Schröder–Bernstein theorem, after Georg Cantor, who first published it (albeit without proof). Proof The following proof is attributed to Julius König. Assume without loss of generality that ''A'' and ''B'' are disjoint. For any ''a'' in ''A'' or ''b'' in ''B'' we can form a unique two-sided sequence of elements that are alternately in ''A'' and ''B'', by repeatedly applying f and g^ to go from ''A'' to ''B'' and g and f^ to go from ''B'' to ''A'' (where defined; the inverses f^ and g^ are unde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borsuk's Conjecture
The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk. Problem In 1932, Karol Borsuk showed that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally -dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question: The question was answered in the positive in the following cases: * — which is the original result by Karol Borsuk (1932). * — shown by Julian Perkal (1947), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes. * For all for smooth convex fields — shown by Hugo Hadwiger (1946). * For all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Theorem (geometry)
Cauchy's theorem is a theorem in geometry, named after Augustin-Louis Cauchy, Augustin Cauchy. It states that convex polytopes in three dimensions with congruence (geometry), congruent corresponding faces must be congruent to each other. That is, any Net (polyhedron), polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape. This is a fundamental result in rigidity theory (structural), rigidity theory: one consequence of the theorem is that, if one makes a physical model of a convex polyhedron by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn–Erdős Theorem (incidence Geometry)
In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős in 1948, states a lower bound on the number of lines determined by ''n'' points in a projective plane. By duality, this is also a bound on the number of intersection points determined by a configuration of lines. Although the proof given by De Bruijn and Erdős is combinatorial, De Bruijn and Erdős noted in their paper that the analogous ( Euclidean) result is a consequence of the Sylvester–Gallai theorem, by an induction on the number of points. Statement of the theorem Let ''P'' be a configuration of ''n'' points in a projective plane, not all on a line. Let ''t'' be the number of lines determined by ''P''. Then, * ''t'' ≥ ''n'', and * if ''t'' = ''n'', any two lines have exactly one point of ''P'' in common. In this case, ''P'' is either a projective plane or ''P'' is a ''near pencil'', meaning that exactly ''n'' - 1 of the points are colline ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester–Gallai Theorem
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944. A line that contains exactly two of a set of points is known as an ''ordinary line''. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of n points in time O(n\log n). History The Sylvester–Gallai theorem was posed as a problem by . suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Third Problem
The third of Hilbert's problems, Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedron, polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Carl Friedrich Gauss, David Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample. The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Wallace–Bolyai–Gerwien theorem. Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków, and was solved by Ludwik Birkenmajer, Ludwik Antoni B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof That E Is Irrational
The number ''e'' was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that ''e'' is irrational; that is, that it cannot be expressed as the quotient of two integers. Euler's proof Euler wrote the first proof of the fact that ''e'' is irrational in 1737 (but the text was only published seven years later). He computed the representation of ''e'' as a simple continued fraction, which is :e = ; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots, 2n, 1, 1, \ldots Since this continued fraction is infinite and every rational number has a terminating continued fraction, ''e'' is irrational. A short proof of the previous equality is known. Since the simple continued fraction of ''e'' is not periodic, this also proves that ''e'' is not a root of a quadratic polynomial with rational coefficients; in particular, ''e''2 is irrational. Fourier's proof The most well-known proof is Joseph Fourier's pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basel Problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in Russian Academy of Sciences#History, ''The Saint Petersburg Academy of Sciences''. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his Riemann zeta function, zeta function and proved its basic properties. The problem is named after the city of Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. The Basel problem asks for the precise summation of the Multiplicative inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard's Inequality
In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893.Maz'ya & Shaposhnikova It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of ''n'' dimensions marked out by ''n'' vectors ''vi'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors , , ''vi'', , . Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns ''vi'', then :\left, \det(N) \ \le \prod_^n \, v_i\, . If the ''n'' vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. Alternate forms and corollaries A corollary is that if the entries of an ''n'' by ''n'' matrix ''N'' are bounded by ''B'', so , ''Nij'', ≤ ''B'' for all ''i'' and ''j'', then :\left, \det(N) \ \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
