|
Almost Integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected. Almost integers relating to the golden ratio and Fibonacci numbers Well-known examples of almost integers are high powers of the golden ratio \phi=\frac\approx 1.618, for example: : \begin \phi^ & =\frac\approx 3571.00028 \\ pt\phi^ & =2889+1292\sqrt5 \approx 5777.999827 \\ pt\phi^ & =\frac\approx 9349.000107 \end The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number. The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: * \operatorname(360)/\operatorname(216) \approx 1242282009792667284144565908481.999999999999999999999999999999195 * \operatorname(361)/\operatorname(216) \approx 2010054515457065378082322433761.0000000000000000000000000000 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Integer In Triangle
In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved). For example: *The set S = \ is almost \mathbb for any k in \mathbb, because only finitely many natural numbers are less than ''k''. *The set of prime numbers is not almost \mathbb, because there are infinitely many natural numbers that are not prime numbers. *The set of transcendental numbers are almost \mathbb, because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable). *The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all real numbers in (0, 1) are members of the complement of the Cantor set. See also *Almost all *Almost surely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Constant
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor. Euler's prime-generating polynomial Euler's prime-generating polynomial n^2 + n + 41, which gives (distinct) primes for ''n'' = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1. Rabinowitz proved that n^2 + n + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schizophrenic Number
A schizophrenic number (also known as mock rational number) is an irrational number that displays certain characteristics of rational numbers. Definition ''The Universal Book of Mathematics'' defines "schizophrenic number" as: The sequence of numbers generated by the recurrence relation ''f''(''n'') = 10 ''f''(''n'' − 1) + ''n'' described above is: :0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, ... . :''f''(49) = 1234567901234567901234567901234567901234567901229 The integer parts of their square roots, :1, 3, 11, 35, 111, 351, 1111, 3513, 11111, 35136, 111111, 351364, 1111111, ... , alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root. Characteristics The ''schizophrenic number'' shown above is the special case of a more general phenomenon that appears in the b-ary expansions of square roots of the solutions of the recurre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Coincidence
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation. For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10: :2^ = 1024 \approx 1000 = 10^3. Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another. Introduction A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a pure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. Weisstein continuously improved the notes and accepted corrections and comments from online readers. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and Encyclopedia, encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the ''CRC Concise Encyclopedia of Mathematics''. He works for Wolfram Research. Education Weisstein holds a Ph.D. in Planetary science, planetary astronomy, which he obtained from the California Institute of Technology's Division of Geological and Planetary Sciences in 1996 as well as an M.S. in planetary astronomy in 1993 also from Caltech. Weisstein graduated cum laude from Cornell University with a B.A. in physics and a minor in astronomy in 1990. During his summers away from Cornell, Weisstein participated in research at the Arecibo Observatory, a radio telescope facility in Puerto Rico operated by Cornell. As a graduate student, Weisstein also participated in research at Goddard Space Flight Center in Greenbelt, MD. During his time at Goddard, Weisstein pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond's Constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is , that is, raised to the power . Like both and , this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that e^\pi = (e^)^ = (-1)^, where is the imaginary unit. Since is algebraic but not rational, is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is , known as the Gelfond–Schneider constant. The related value + is also irrational. Numerical value The decimal expansion of Gelfond's constant begins :e^\pi = ... Construction If one defines and k_ = \frac for , then the sequence (4/k_)^ converges rapidly to . Continued fraction expansion e^ = 23+ \cfrac This is based on the digits for the simple continued fraction: e^ = [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, 108, 2, 2, 1, 3, 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series e = \sum\limits_^ \frac = 1 + \frac + \frac + \frac + \cdots. It is also the unique positive number such that the graph of the function has a slope of 1 at . The (natural) exponential function is the unique function that equals its own derivative and satisfies the equation ; hence one can also define as . The natural logarithm, or logarithm to base , is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the area under the curve between and , in which case is the value of for which this area equals one (see image). There are various other characteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Eisenstein series for the modular group Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: :G_(\tau) = \sum_ \frac. This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Relation to modular invariants The modular invariants and of an elliptic curve are given by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegner Number
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–) Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor. Euler's prime-generating polynomial Euler's prime-generating polynomial n^2 + n + 41, which gives (distinct) primes for ''n'' = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1. Rabinowitz proved that n^2 + n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas series has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio. The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. The first few Lucas numbers are : 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349 .... Defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
