Gauss–Seidel method In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. Description The Gauss–Seidel method is an iterative technique for solving a square system of n linear equations with unknown x: It is defined by the iteration where the matrix A is decomposed into a lower triangular component , and a strictly upper triangular component U: In more detail, write out A, x and b in their components: and
Stereographic projection 3D illustration of a stereographic projection from the north pole onto a plane below the sphere Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet or Wulff net. History Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aiguillon. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. Definition This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Properties with E.
Avogadro constant Previous definitions of chemical quantity involved Avogadro's number, a historical term closely related to the Avogadro constant but defined differently: Avogadro's number was initially defined by Jean Baptiste Perrin as the number of atoms in one gram-molecule of hydrogen. It was later redefined as the number of atoms in 12 grams of the isotope carbon-12 and still later generalized to relate amounts of a substance to their molecular weight. For instance, to a first approximation, 1 gram of hydrogen, which has a mass number of 1 (atomic number 1), has 6.022×1023 hydrogen atoms. Similarly, 12 grams of carbon 12, with the mass number of 12 (atomic number 6), has the same number of carbon atoms, 6.022×1023. Avogadro's number is a dimensionless quantity and has the numerical value of the Avogadro constant given in base units. The Avogadro constant is fundamental to understanding both the makeup of molecules and their interactions and combinations. History [dubious ] Measurement
'Periodic table of shapes' to give a new dimension to math Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements. The three-year project should provide a resource that mathematicians, physicists and other scientists can use for calculations and research in a range of areas, including computer vision, number theory, and theoretical physics. The researchers, from Imperial College London and institutions in Australia, Japan and Russia, are aiming to identify all the shapes across three, four and five dimensions that cannot be divided into other shapes. As these building block shapes are revealed, the mathematicians will work out the equations that describe each shape and through this, they expect to develop a better understanding of the shapes' geometric properties and how different shapes are related to one another.
Complex plane Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z. In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. Notational conventions In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this: for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In the Cartesian plane the point (x, y) can also be represented in polar coordinates as where Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Stereographic projections Cutting the plane
Warning! High Dimensions Ahead ← Inductio Ex Machina Machine learning is often framed as finding surfaces (usually planes) that separate or fit points in some high dimensional space. Two and three dimensional diagrams are often used as an aid to intuition when thinking about these higher dimensional spaces. The following result (related to me by Marcus Hutter) shows that our low dimensional intuition can easily lead us astray. Consider the diagram in Figure 1. The larger grey spheres have radius 1 and are each touching exactly two other grey spheres. Pythagoras tells us that and so . Figure 1: The blue circle in the middle touches each of the four identical grey circles. We can move to three dimensions and set up an analogue of the two dimensional situation by arranging eight spheres of radius 1 so they are touching exactly three other grey spheres. As in the case with circles, we can place a blue sphere inside the eight grey spheres so that it touches all of them. Figure 2: The blue sphere touches each of the eight identical grey spheres.
Fundamental theorem of algebra The fundamental theorem of algebra 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 zero imaginary part. History Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”, by which he meant that no coefficient is equal to 0. where α is the square root of 4 + 2√7. A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Proofs when |z| > R. Complex-analytic proofs and let
Dave's short course in trigonometry Table of Contents Who should take this course? Trigonometry for you Your background How to learn trigonometry Applications of trigonometry Astronomy and geography Engineering and physics Mathematics and its applications What is trigonometry? Trigonometry as computational geometry Angle measurement and tables Background on geometry The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords What is a chord? Ptolemy’s sum and difference formulas Ptolemy’s theorem The sum formula for sines The other sum and difference formulas Summary of trigonometric formulas Formulas for arcs and sectors of circles Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities More important identities Less important identities Truly obscure identities About the Java applet.
Euclidean algorithm Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because remainder EA is less than CD. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. In mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). The GCD of two positive integers is the largest integer that divides both of them without leaving a remainder (the GCD of two integers in general is defined in a more subtle way). The main principle is that the GCD does not change if the smaller number is subtracted from the larger number.
Spiral of Theodorus The spiral of Theodorus up to the triangle with a hypotenuse of √17 In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral) is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene. History Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories. Hypotenuse Each of the triangles' hypotenuses hi gives the square root of the corresponding natural number, with h1 = √2. Plato, tutored by Theodorus, questioned why Theodorus stopped at √17. Overlapping Extension Colored extended spiral of Theodorus with 120 triangles Theodorus stopped his spiral at the triangle with a hypotenuse of √17. Growth rate Angle If φn is the angle of the nth triangle (or spiral segment), then: with A triangle or section of spiral
Minkowski space In theoretical physics, Minkowski space is often contrasted with Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension. The isometry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. History In 1905 (published 1906) it was noted by Henri Poincaré that, by taking time to be the imaginary part of the fourth spacetime coordinate √−1 ct, a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension. The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. For further historical information see references Galison (1979), Corry (1997), Walter (1999). Structure The Minkowski inner product Standard basis where