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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, 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 and Spanish. In November 2017 the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Aigner
Martin Aigner (born February 28, 1942 in Linz) is an Austrian mathematician and professor at Freie Universität Berlin since 1974 with interests in combinatorial mathematics and graph theory. 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. He is a recipient of a 1996 Lester R. Ford Award from the Mathematical Association of America for his expository article '' Turán's Graph Theorem''. In 2018, Aigner received the Leroy P. Steele Prize for Mathematical Exposition (jointly with Günter M. Ziegler). Bibliography *''Combinatorial Theory'' (1997 reprint: , 1979: ; ) *(with Günter M. Ziegler) ''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 mathema ...'' ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd 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 congruent to 1 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 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monsky's Theorem
In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the ''American Mathematical Monthly'' in 1965, and was proved by Paul Monsky in 1970. Proof Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows: #Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into ''n'' triangles of equal area then the area of each triangle is 1/''n''. #Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates. #Show that a straight line can contain points of only two colours. #Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours. #Conclude from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree ''n'' polynomial with complex coefficients has, counted with multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept. Additionally, it is not fundamental for modern algeb ... [...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 ... [...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 Cantor–Bernstein theorem, or Cantor–Schröder–Bernstein, after Georg Cantor who first published it 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 understood as partial functi ... [...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 ''n''-dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that ''n'' subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question: : ''Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes \mathbb R^n in'' (''n'' + 1) ''Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?'' This can be translated as: : ''The following question remains open: Can every bounded subset E of the space \mathbb R^n be partitioned into'' (''n'' + 1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Theorem (geometry)
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any 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: 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 edges, then this ensemble of plates and hinges will necessarily form a rigid structure. Stat ... [...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 , 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 collinear. Euclidean proof The theorem is clearly true for ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Third Problem
The third of 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 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 Antoni Birkenmajer with a different method than Dehn. Birk ... [...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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
