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Gauss Circle Problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name. The problem Consider a circle in \mathbb^2 with center at the origin and radius r\ge 0. Gauss's circle problem asks how many points there are inside this circle of the form (m,n) where m and n are both integers. Since the equation of this circle is given in Cartesian coordinates by x^2+y^2= r^2, the question is equivalently asking how many pairs of integers ''m'' and ''n'' there are such that :m^2+n^2\leq r^2. If the answer for a given r is denoted by N(r) then the following list shows the first few values of N(r) for ''r'' an integer between 0 and 12 followed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Points In Radius-5 Circle
Grid, The Grid, or GRID may refer to: Space partitioning * Regular grid, a tessellation of space with translational symmetry, typically formed from parallelograms or higher-dimensional analogs ** Grid graph, a graph structure with nodes connected in a regular grid ** Square grid, a grid of squares ** Triangular grid, a grid of triangles ** Hexagonal grid, a grid of hexagons ** Unstructured grid, a tessellation of a space by simple shapes such as triangles or tetrahedra in an irregular pattern * Grid reference system, a coordinate system relative to a particular map projection * Grid (spatial index), a discretization of a geometric domain into a set of contiguous cells, used to organize information ** Discrete global grid (DGG), a grid that covers the entire Earth's surface * Grid (graphic design) (or typographic grid), organized lines for guiding graphic design * Grid plan, a city design with streets running at right angles * Grid paper, paper with a regular grid printed on it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Squares Function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different. It is denoted by . Definition The function is defined as :r_k(n) = , \, where , \,\ , denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares. For example, r_2(1) = 4 since 1 = 0^2 + (\pm 1)^2 = (\pm 1)^2 + 0^2 where each sum has two sign combinations, and also r_2(2) = 4 since 2 = (\pm 1)^2 + (\pm 1)^2 with four sign combinations. On the other hand, r_2(3) = 0 because there is no way to represent 3 as a sum of two squares. Formulae ''k'' = 2 The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by :r_2(n) = 4(d_1(n)-d_3(n)) where is the number of divisors o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3Blue1Brown
3Blue1Brown is a math YouTube channel created and run by Grant Sanderson. The channel focuses on teaching Higher Mathematics, higher mathematics from a visual perspective, and on the process of discovery and inquiry-based learning in mathematics, which Sanderson calls "inventing math". Grant Sanderson Early life and education Sanderson graduated from Stanford University in 2015 with a bachelor's degree in mathematics. He worked for Khan Academy from 2015 to 2016 as part of their content fellowship program, producing videos and articles about multivariable calculus, after which he started focusing his full attention on 3Blue1Brown. Career In 2020, Grant Sanderson became one of the creators and lecturers of the Massachusetts Institute of Technology, MIT course ''Introduction to Computational Thinking'', together with Alan Edelman, David Sanders, James Schloss, and Benoit Forget. The course uses the Julia (programming language), Julia programming language and Grant Sanderson's anima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ''ζ''(''s'') is a function whose argument ''s'' may be any complex numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Orchard
In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from to , where and are positive integers. The trees visible from the origin are those at lattice points , where and are coprime, i.e., where the fraction is in reduced form. The name ''Euclid's orchard'' is derived from the Euclidean algorithm. If the orchard is projected relative to the origin onto the plane (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point projects to : \left ( \frac , \frac , \frac \right ). The solution to the Basel problem can be used to show that the proportion of points in the grid that have trees on them is approximately \tfrac and that the error of this approximation goes to zero in the limit as goes to infinity. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monatshefte Für Mathematik
'' Monatshefte für Mathematik'' is a peer-reviewed mathematics journal established in 1890. Among its well-known papers is " Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" by Kurt Gödel, published in 1931. The journal was founded by Gustav von Escherich and Emil Weyr in 1890 as ''Monatshefte für Mathematik und Physik'' and published until 1941. In 1947 it was reestablished by Johann Radon under its current title. It is currently published by Springer in cooperation with the Austrian Mathematical Society. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.58, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.764. External links *''Monatshefte für Mathematik und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Planimeter
A dot planimeter is a device used in planimetrics for estimating the area of a shape, consisting of a transparent sheet containing a square grid of dots. To estimate the area of a shape, the sheet is overlaid on the shape and the dots within the shape are counted. The estimate of area is the number of dots counted multiplied by the area of a single grid square. In some variations, dots that land on or near the boundary of the shape are counted as half of a unit. The dots may also be grouped into larger square groups by lines drawn onto the transparency, allowing groups that are entirely within the shape to be added to the count rather than requiring their dots to be counted one by one. The estimation of area by means of a dot grid has also been called the dot grid method or (particularly when the alignment of the grid with the shape is random) systematic sampling. Perhaps because of its simplicity, it has been repeatedly reinvented. Application In forestry, cartography, and geogr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''p''/''q'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''p''/''q'' and ''α'' may not decrease if ''p''/''q'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component (topology), connected components or branches, that are mirror images of each other and resemble two infinite bow (weapon), bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane (mathematics), plane and a double cone (geometry), cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Besides being a conic section, a hyperbola can arise as the locus (mathematics), locus of points whose difference of distances to two fixed focus (geometry), foci is constant, as a curve for each point of which the rays to two fix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
