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We use addition on a daily basis-yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series-long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms-the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

Number theory. --- Mathematics --- Number study --- Numbers, Theory of --- Algebra --- Absolute value. --- Addition. --- Analytic continuation. --- Analytic function. --- Automorphic form. --- Axiom. --- Bernoulli number. --- Big O notation. --- Binomial coefficient. --- Binomial theorem. --- Book. --- Calculation. --- Chain rule. --- Coefficient. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Congruence subgroup. --- Conjecture. --- Constant function. --- Constant term. --- Convergent series. --- Coprime integers. --- Counting. --- Cusp form. --- Determinant. --- Diagram (category theory). --- Dirichlet series. --- Division by zero. --- Divisor. --- Elementary proof. --- Elliptic curve. --- Equation. --- Euclidean geometry. --- Existential quantification. --- Exponential function. --- Factorization. --- Fourier series. --- Function composition. --- Fundamental domain. --- Gaussian integer. --- Generating function. --- Geometric series. --- Geometry. --- Group theory. --- Hecke operator. --- Hexagonal number. --- Hyperbolic geometry. --- Integer factorization. --- Integer. --- Line segment. --- Linear combination. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Matrix group. --- Modular form. --- Modular group. --- Natural number. --- Non-Euclidean geometry. --- Parity (mathematics). --- Pentagonal number. --- Periodic function. --- Polynomial. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Pythagorean theorem. --- Quadratic residue. --- Quantity. --- Radius of convergence. --- Rational number. --- Real number. --- Remainder. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Semicircle. --- Series (mathematics). --- Sign (mathematics). --- Square number. --- Square root. --- Subgroup. --- Subset. --- Sum of squares. --- Summation. --- Taylor series. --- Theorem. --- Theory. --- Transfinite number. --- Triangular number. --- Two-dimensional space. --- Unique factorization domain. --- Upper half-plane. --- Variable (mathematics). --- Vector space.

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This one-of-a-kind book presents many of the mathematical concepts, structures, and techniques used in the study of rays, waves, and scattering. Panoramic in scope, it includes discussions of how ocean waves are refracted around islands and underwater ridges, how seismic waves are refracted in the earth's interior, how atmospheric waves are scattered by mountains and ridges, how the scattering of light waves produces the blue sky, and meteorological phenomena such as rainbows and coronas.Rays, Waves, and Scattering is a valuable resource for practitioners, graduate students, and advanced undergraduates in applied mathematics, theoretical physics, and engineering. Bridging the gap between advanced treatments of the subject written for specialists and less mathematical books aimed at beginners, this unique mathematical compendium features problems and exercises throughout that are geared to various levels of sophistication, covering everything from Ptolemy's theorem to Airy integrals (as well as more technical material), and several informative appendixes.Provides a panoramic look at wave motion in many different contextsFeatures problems and exercises throughoutIncludes numerous appendixes, some on topics not often coveredAn ideal reference book for practitionersCan also serve as a supplemental text in classical applied mathematics, particularly wave theory and mathematical methods in physics and engineeringAccessible to anyone with a strong background in ordinary differential equations, partial differential equations, and functions of a complex variable

Mathematical physics. --- Physical mathematics --- Physics --- Mathematics --- Airy approximation. --- Airy functions. --- Airy integral. --- Airy theory. --- Airy wavefront. --- Alexander's dark band. --- Bessel functions. --- Earth. --- Fermat's principle. --- Fresnel integrals. --- Hamilton's principle. --- Hamilton-Jacobi equation. --- Hamilton-Jacobi theory. --- Hamiltonian. --- Hooke's law. --- Kepler's laws of planetary motion. --- Lagrangian. --- Liouville transformation. --- Love waves. --- Navier equations. --- Ptolemy's theorem. --- Rayleigh scattering. --- Schrödinger equation. --- Sir George Biddle Airy. --- Snell's laws. --- Taylor–Goldstein equation. --- WKB(J) approximation. --- Wiechert-Herglotz inverse problem. --- acoustic wave propagation. --- action. --- angle of minimum deviation. --- applied mathematics. --- atmospheric waves. --- billow clouds. --- boundary-value problem. --- buoyancy waves. --- caustics. --- classical mechanics. --- classical wave equation. --- colors. --- complex plane. --- constant phase lines. --- contours. --- corona. --- currents. --- cusp catastrophes. --- deep water waves. --- differential equations. --- diffraction catastrophes. --- diffraction. --- dispersion relations. --- dispersion. --- divergence problem. --- earthquakes. --- eikonal equation. --- elastic solid. --- elastic waves. --- elementary mathematics. --- equations of motion. --- fluid equations. --- fold catastrophes. --- free surface. --- geometric wavefronts. --- geometrical optics. --- glory. --- inhomogeneous medium. --- integrals. --- intensity law. --- internal gravity waves. --- inverse scattering problem. --- islands. --- leading waves. --- lee waves. --- light waves. --- long waves. --- mathematics. --- meteorological optics. --- mountain waves. --- ocean acoustic waveguides. --- ocean acoustics. --- ocean waves. --- one-dimensional waves. --- optics. --- path. --- plane wave incident. --- plane waves. --- polarization. --- potential well. --- rainbow. --- ray equations. --- ray optics. --- ray theory. --- rays. --- reflection. --- refraction. --- ridge. --- scattering. --- seafloor. --- seismic rays. --- seismic tomography. --- seismic waves. --- semicircle theorem. --- shallow water waves. --- ship waves. --- short waves. --- strain. --- stratified fluid. --- stress. --- surface gravity waves. --- surface waves. --- transient waves. --- tsunami propagation. --- tsunamis. --- wave energy. --- wave refraction. --- wave trapping. --- wavefront. --- wavepackets. --- waves. --- wind shear.