 | Table of Contents
See also: Dates of creation of all indexed pages Begin with the answers. Then one day, perhaps, you'll find the final question. "The Chinese Maze Murders" by Robert Hans van Gulik (1910-1967) It is better to know some of the questions than all of the answers. James Grover Thurber (1894-1961) - All metric prefixes: Current SI prefixes, obsolete prefixes, bogus prefixes...
- Prefixes for units of information (multiples of the bit only) & brontobyte hoax.
- Density one. Relative and absolute density precisely defined.
- Acids yielding a mole of H+ per liter are normal (1N) solutions.
- Calories: Thermochemical calorie, gram-calorie (g-cal), IST calorie (and Btu).
- Horsepowers: hp, electric horsepower, metric horsepower, boiler horsepower.
- The standard acceleration of gravity (1G) has been 9.80665 m/s2 since 1901.
Time: - Tiny durations; zeptosecond (zs, 10-21s) & yoctosecond (ys, 10-24s).
- A jiffy is either a light-cm or 10 ms (tempons and chronons are shorter).
- The length of a second. Solar time, ephemeris time, atomic time.
- The length of a day. Solar day, atomic day, sidereal or Galilean day.
- Scientific year = 31557600 SI seconds (» Julian year of 365.25 solar days).
Length: - The International inch (1959) is 999998/1000000 of a US Survey inch.
- The typographer's point is exactly 0.013837" = 0.3514598 mm.
- Leagues: Land league, nautical league.
- Radius of the Earth and circumference at the Equator.
- Extreme units of length. The very large and the very small.
Surface Area: - Acres, furlongs, chains and square inches...
Volume, Capacity: - Capitalization of units. You only have a choice for the liter (or litre ).
- Drops or minims: Winchester, Imperial or metric. Teaspoons and ounces.
- Fluid ounces: American ounces (fl oz) are about 4% larger than British ones.
- Gallons galore: Winchester (US) vs. Imperial gallon (UK), dry gallon, etc.
- US bushel and Winchester units of capacity (dry = bushel, fluid = gallon).
- Kegs and barrels: A keg of beer is half a barrel, but not just any "barrel".
Mass, "Weight": - Tiny units of mass. A hydrogen atom is about 1.66 yg.
- Technical units of mass. The slug and the hyl.
- Customary units of mass which survive in the electronic age.
- The poids de marc system: 18827.15 French grains to the kilogram.
- A talent was the mass of a cubic foot of water.
- Tons: short ton, long ton (displacement ton), metric ton (tonne), assay ton, etc.
- Other tons: Energy (kiloton, toe, tce), cooling power, thrust, speed...
- Scientific notation: Nonzero numbers given as multiples of powers of ten.
- So many "significant" digits imply a result of limited precision.
- Standard deviation gives the precision of a result as a form of uncertainty.
- Engineering notation reduces a number to a multiple of a power of 1000.
- The quadratic formula is numerically inadequate in common cases.
- Devising robust formulas which feature a stable floating-point precision.
- The Beaufort scale is now defined in terms of wind speed.
- The Saffir / Simpson scale for hurricanes.
- The Fujita scale for tornadoes.
- The Richter scale for earthquakes and other sudden energy releases.
- Decibels: A general-purpose logarithmic scale for relative power ratios.
- Apparent and absolute magnitudes of stars.
- The scale of animals according to Galileo Galilei.
- Jumping fleas... compared to jumping athletes...
- Drag coefficient of a sphere as a function of the Reynolds number R. Physical Constants:
- For the utmost in precision, physical constants are derived in a certain order.
- Primary conversion factors between customary systems of units.
6+1 Basic Dimensionful Physical Constants (Proleptic SI) - Speed of Light in a Vacuum (Einstein's Constant): c = 299792458 m/s.
- Magnetic Permeability of the Vacuum: An exact value defining the ampere.
- Planck's constant: The ratio of a photon's energy to its frequency.
- Boltzmann's constant: Relating temperature to energy.
- Avogadro's number: The number of things per mole of stuff.
- Mechanical Equivalent of Light (683 lm/W at 540 THz) defines the candela.
- Newton's constant of gravitation and a futuristic definition of the second.
Fundamental Mathematical Constants: - 0: Zero is the most fundamental and most misunderstood of all numbers.
- 1 and -1: The unit numbers.
- p ("Pi"): The ratio of the circumference of a circle to its diameter.
- Ö2: The diagonal of a square of unit side. Pythagoras' Constant.
- f: The diagonal of a regular pentagon of unit side. The Golden Number.
- Euler's e: The base of the exponential function which equals its own derivative.
- ln(2): The alternating sum of the reciprocals of the integers.
- Euler-Mascheroni Constant g : Limit of [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
- Catalan's Constant G : The alternating sum of the reciprocal odd squares.
- Apéry's Constant z(3) : The sum of the reciprocals of the perfect cubes.
- Imaginary i: If "+1" is a step forward, "+ i" is a step sideways to the left.
Exotic Mathematical Constants: - Delian constant: 21/3 is the solution to the classical duplication of the cube.
- Gauss's constant: The reciprocal of the arithmetic-geometric mean of 1 and Ö2.
- Mertens constant: The limit of [1/2 + 1/3 + 1/5 +...+ 1/p] - ln(ln p)
- Artins's constant is the proportion of long primes in decimal or binary.
- Ramanujan-Soldner constant (m): Positive root of the logarithmic integral.
- The Omega constant: W(1) is the solution of the equation x exp(x) = 1.
- Feigenbaum constant (d) and the related reduction parameter (a).
Some Third-Tier Mathematical Constants: - Brun's Constant: A standard uncertainty (s) means a 99% level of ±3s
- Prévost's Constant: The sum of the reciprocals of the Fibonacci numbers.
- Grossman's Constant: One recurrence converges for only one initial point.
- Ramanujan's Number: exp(p Ö163) is almost an integer.
- Viswanath's constant: Mean growth in random additions and subtractions.
- Always change your first guess if you're always told another choice is bad.
- The Three Prisoner Problem predated Monty Hall and Marilyn by decades.
- Seating N children at a round table in (N-1)! different ways.
- How many Bachet squares? A 1624 puzzle using the 16 court cards.
- Choice Numbers: C(n,p) is the number of ways to choose p items among n.
- C(n+2,3) three-scoop sundaes. Several ways to count them (n flavors).
- C(n+p-1,p) choices of p items among n different types, allowing duplicates.
- How many new intersections of straight lines defined by n random points.
- Face cards. The probability of getting a pair of face cards is less than 5%.
- Homework Central: Aces in 4 piles, bad ICs, airline overbooking.
- Binomial distribution. Defective units in a sample of 200.
- Siblings with the same birthday. What are the odds in a family of 5?
- Covariance: A generic example helps illustrate the concept.
- Variance of a binomial distribution, derived from general principles.
- Standard deviation. Two standard formulas to estimate it.
- The Markov inequality is used to prove the Bienaymé-Chebyshev inequality.
- The Bienaymé-Chebyshev inequality is valid for any probability distribution.
- Inclusion-Exclusion: One approach to the probability of a union of 3 events.
- The "odds in favor" of poker hands: A popular way to express probabilities.
- Probabilities of a straight flush in 7-card stud. Generalizing to "q-card stud".
- Probabilities of a straight flush among 26 cards (or any other number).
- The exact probabilities in 5-card, 6-card, 7-card, 8-card and 9-card stud.
- Rearrangements of CONSTANTINOPLE so no two vowels are adjacent...
- Four-letter words from POSSESSES: Counting with generating functions.
- How many positive integers below 1000000 have their digits add up to 19?
- Polynacci Numbers: Flipping a coin n times without p tails in a row.
- 252 decreasing sequences of 5 digits (2002 nonincreasing ones).
- How many ways are there to make change for a dollar? Closed formulas.
- Partitioniong an amount into the parts minted in a certain currency.
- The number of rectangles in an N by N chessboard-type grid.
- The number of squares in an N by N grid: 0, 1, 5, 14, 30, 55, 91, 140, 204...
- Screaming Circles: How many tries until there's no eye contact?
- Average distance between two random points on a segment, a disk, a cube...
- Average distance between two points on the surface of a sphere.
- Poisson Processes: Random arrivals happening at a constant rate (in Bq).
- Simulating a poisson process is easy with a uniform random number generator.
- Markov Processes: When only the present influences the future...
- The Erlang B Formula assumes callers don't try again after a busy signal.
- Markov-Modulated Poisson Processes may look like Poisson processes.
- The Utility Function: A dollar earned is usually worth less than a dollar lost.
- Saint Petersburg Paradox: What would you pay to play the Petersburg game?
- You can only prove a negative (a lack of counterexamples).
- Stochastic proofs leave only a vanishing uncertainty.
- Heuristic arguments establish the likelihood of a conjecture.
- Too few tools are not enough to prove a given statement.
- Center of an arc determined with straightedge and compass.
- Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
- Special points in a triangle. Euler's line and Euler's circle.
- Elliptic arc: Length of the arc of an ellipse between two points.
- Perimeter of an ellipse. Exact formulas and simple ones.
- Surface area of a spheroid (oblate or prolate ellipsoid of revolution).
- Surface of an ellipse.
- Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
- Volume of an ellipsoid [spheroid].
- Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
- Focal point of a parabola. y = x 2 / 4f (where f is the focal distance).
- Parabolic telescope: The path from infinity to focus is constant.
- Make a cube go through a hole in a smaller cube.
- Octagon: The relation between side and diameter.
- Constructible regular polygons and constructible angles (Gauss).
- Areas of regular polygons of unit side: General formula & special cases.
- For a regular polygon of given perimeter, the more sides the larger the area.
- Curves of constant width: Reuleaux Triangle and generalizations.
- Irregular curves of constant width. With or without any circular arcs.
- Solids of constant width. The three-dimensional case.
- Constant width in higher dimensions.
- Fourth dimension. Difficult to visualize, but easy to consider.
- Volume of a hypersphere and hper-surface area, in any number of dimensions.
- Hexahedra. The cube is not the only polyhedron with 6 faces.
- Descartes-Euler Formula: F-E+V=2 but restrictions apply.
- Metric spaces: The motivation behind more general topological spaces.
- Topological spaces: Singling out abstract "open" subsets defines a topology.
- Closed sets are sets (of a topological space) whose complements are open.
- Subspace F of E: Its open sets are the intersections with F of open sets of E.
- Separation axioms: Flavors of topological spaces, according to Trennung.
- Compactness of a topological space: Every open cover has a finite subcover.
- Complete metric space: All Cauchy sequences are convergent.
- Local compactness: Every point of the set has a compact neighborhood in it.
- General properties of sequences are indicative of topological characteristics.
- Continuous functions are such that the inverse image of any open set is open.
- The product topology makes projections continuous on a cartesian product.
- Connected sets cannot be split by open sets; they need not be path-connected.
- Homeomorphic sets are related by a bicontinuous function (homeomorphism).
- Homotopy: A progessive transformation of a function into another.
- The Fundamental Group: The homotopy classes of all loops through a point.
- Homology and Cohomology. Poincaré duality.
- Descartes-Euler Formula: F-E+V = 2, but restrictions apply.
- Euler Characteristic: c (chi) extended beyond its traditional definition...
- Winding number about point O of a continuous planar curve not containing O.
- Fixed-point theorems by Brouwer, Shauder and Tychonoff.
- Turning number of a planar curve with a well-defined oriented tangent.
- Real projective plane and Boy's surface.
- Hadwiger's additive continuous functions of d-dimensional rigid bodies.
- Eversion of the sphere. An homotopy can turn a sphere inside out.
- Classification of surfaces: "Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
- Braid groups: strands, braids and pure braids.
- Planar angles separate two directions. In an oriented plane, they are signed.
- Solid angles are to spherical patches what planar angles are to circular arcs.
- Circular measures: Angles and solid angles aren't quite dimensionless quantities.
- Formulas for solid angles subtended by patches with simple shapes.
- Curvature of a planar curve: The variation of inclination with distance dj/ds.
- Curvature and torsion of a three-dimensional curve.
- Geodesic and nornal curvatures and torsions of a curve drawn on a surface.
- Lines of curvature and geodesic lines. Lines of extremal curvature or least length.
- Meusnier's theorem: Tangent lines have the same normal curvature.
- Gaussian curvature of a surface and its integral: The Gauss-Bonnet theorem.
- Parallel-transport of a vector around a closed curve. Holonomic angle of a loop.
- Total curvature of a curve. The Fary-Milnor theorem for knotted curves.
- Linearly independent components of the Riemann curvature tensor.
- Cartesian equation of a straight line: passing through two given points.
- Confocal Conics: Ellipses and hyperbolae sharing the same pair of foci.
- Spiral of Archimedes: Paper on a roll, or groove on a vinyl record.
- Catenary: The shape of a thin chain under its own weight.
- Witch of Agnesi. How the versiera (Agnesi's cubic) got a weird name.
- Folium of Descartes.
- Lemniscate of Bernoulli: The shape of the infinity symbol is a quartic curve.
- Along a Cassini oval, the product of the distances to the two foci is constant.
- Limaçons of Pascal: The cardioid (unit epicycloid) is a special case.
- On a Cartesian oval, the weighted average distance to two poles is constant.
- Parallel curves share the same normal, along which their distance is constant.
- Bézier curves are algebraic splines. The cubic type is the most popular.
- Piecewise circular curves: The traditional way to specify curved forms.
- Intrinsic equation [curvature as a function of arc length] may include spikes.
- The quadratrix (or trisectrix) of Hippias can square the circle and trisect angles.
- The parabola is a curve that's constructible with straightedge and compass.
- Mohr-Mascheroni constructions use the compass alone (no straightedge).
- Glossary of terms related to gears.
- Planar curves rolling without slipping while rotating about two fixed points.
- Congruent ellipses roll against each other while revolving around their foci.
- Elliptical gears: A family of gears which include ellipses and sine curves.
- Cycloidal gears : Traditional profiles used by watchmakers.
- Epicycloidal gears : Philippe de la Hire (1640-1718).
- Involute tooth profile provides a constant rotational speed ratio.
- Harmonic Drive: The flexspline has 2 fewer teeth than the circular spline.
- Hexahedra. The cube is not the only polyhedron with 6 faces.
- Fat tetragonal antiwedge: Chiral hexahedron of least area for a given volume.
- Enumeration of polyhedra: Tally of polyhedra with n faces and k edges.
- The 5 Platonic solids: Cartesian coordinates of the vertices.
- The 13 Archimedean solids and their duals (Catalan solids).
- Some special polyhedra may have a traditional (mnemonic) name.
- Polyhedra in certain families are named after one of their prominent polygons.
- Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex.
- Johnson Polyhedra and the associated nomenclature.
- Polytopes are the n-dimensional counterparts of 3-D polyhedra.
- A simplex of touching unit spheres may allow a center sphere to bulge out.
- Regular Antiprism: Height and volume of a regular n-gonal antiprism.
- The Szilassi polyhedron features 7 pairwise adjacent hexagonal faces.
- Wooden buckyball: Cutting 32 blocks to make a truncated icosahedron.
- The bridges of Königsberg: Eulerian graphs and the birth of graph theory.
- Undirected graphs are digraphs with symmetrical adjacency matrix.
- Adjacency matrix of a directed graph (digraph) or of a bipartite graph.
- Silent Circles: An enumeration based on adjacency matrices (Max Alekseyev).
- Silent Prisms: Another version of the screaming game, for short-sighted people.
- Tallying all markings of one edge per node in which no edge is marked twice.
- Line graph: The nodes of L(G) are edges of G (connected iff adjacent in G).
- Transitivity: Vertex-transitive and/or edge-transitive graphs.
- Factorial zero is 1, so is an empty product; an empty sum is 0.
- Anything raised to the power of 0 is equal to 1, including 0 to the power of 0.
- Idiot's Guide to Complex Numbers.
- Using the Golden Ratio (f) to express the 5 [complex] fifth roots of unity.
- "Multivalued" functions are functions defined over a Riemann surface.
- Square roots are inherently ambiguous for negative or complex numbers.
- The difference of two numbers, given their sum and their product.
- Symmetric polynomials of 3 variables: Obtain the value of one from 3 others.
- Geometric progression of 6 terms. Sum is 14, sum of squares is 133.
- Quartic equation involved in the classic "Ladders in an Alley" problem.
- Permutation matrices include the identity matrix and the exchange matrix.
- Operations on matrices are conveniently defined using Dirac's notation.
- Vandermonde matrix: The successive powers of elements in its second row.
- Toeplitz matrix: Constant diagonals.
- Circulant matrix: Cyclic permutations of the first row.
- Wendt's Determinant: The circulant of the binomial coefficients.
- Hankel matrix: Constant skew-diagonals.
- Catberg matrix: Hankel matrix of the reciprocal of Catalan numbers.
- Hadamard matrix: Unit elements and orthogonal columns.
- Sylvester matrix of two polynomials has their resultant for determinant.
- The discriminant of a polynomial is the resultant of itself and its derivative.
- Numerical functions: Polynomial, rational, algebraic, transcendental, special...
- Trigonometric functions: Memorize a simple picture for 3 basic definitions.
- Solving triangles with the law of sines, law of cosines, and law of tangents.
- Spherical trigonometry: Triangles drawn on the surface of a sphere.
- Sum of tangents of two half angles, in terms of sums of sines and cosines.
- The absolute value of the sine of a complex number.
- Exact solutions to transcendental equations.
- All positive rationals (and their square roots) as trigonometric functions of zero!
- The sine function: How to compute it numerically.
- Chebyshev economization saves billions of operations on routine computations.
- The Gamma function: Its definition(s) properties and values.
- Lambert's W function is used to solve practical transcendental equations.
- Pochhammer's symbol: Upper factorial of k increasing factors, starting with x.
- Gauss's hypergeometric function: 2+1 parameters (and one variable).
- Kummer's transformations relate different values of the hypergeometric function.
- Sum of the reciprocal of Catalan numbers, in closed hypergeometric form.
- Derivative: Usually, the slope of a function, but there's a more abstract approach.
- Integration: The Fundamental Theorem of Calculus.
- 0 to 60 mph in 4.59 s, may not always mean 201.96 feet.
- Integration by parts: Reducing an integral to another one.
- Length of a parabolic arc.
- Top height of a curved bridge spanning a mile, if its length is just a foot longer.
- Sagging: A cable which spans 28 m and sags 30 cm is 28.00857 m long.
- The length of the arch of a cycloid is 4 times the diameter of the wheel.
- Integrating the cube root of the tangent function.
- Changing inclination to a particle moving along a parabola.
- Algebraic area of a "figure 8" may be the sum or the difference of its lobes.
- Area surrounded by an oriented planar loop which may intersect itself.
- Linear differential equations of higher order and/or in several variables.
- Theory of Distributions: Convolution products and their usage.
- Laplace Transforms: The Operational Calculus of Oliver Heaviside.
- Integrability of a function and of its absolute value.
- Analytic functions of a linear operator; defining f (D) when D is d/dx...
- Ordinary differential equations. Several examples.
- A singular change of variable is valid over a domain which may not be maximal.
- Vertical fall against fluid resistance (including both viscous and quadratic drag).
- Generalizing the fundamental theorem of calculus.
- The surface of a loop is a vector determining its apparent area in any direction.
- Practical identities of vector calculus
Optimization: Operations Research, Calculus of Variations - Stationary points (or saddlepoints ) are where all partial derivatives vanish.
- Single-variable optimization: Derivative vanishes unless the variable is extreme.
- Extrema of a function of two variables must satisfy a second-order condition.
- Saddlepoints of a multivariate function. One equation to satisfy per variable.
- Lagrange multipliers: Optimizing an objective function under various constraints.
- Minimizing the lateral surface area of a cone of given base and volume.
- Euler-Lagrange equations hold along the path of a stationary integral.
- Noether's theorem: A symmetry of the integrand yields a conserved quantity.
- Isoperimetric Inequality: The largest area enclosed by a loop of unit perimeter.
- Plateau's problem extends the calculus of variations from paths to membranes.
- Embedded minimal surfaces: Plane, catenoid, helicoid, Costa's surface, etc.
- Connecting blue dots to red dots in the plane, without any crossings...
- The shortest way to connect 3 dots can be to join them to a fourth point.
- The Honeycomb Theorem: A conjecture of old, proved by Thomas Hales.
- Counterexamples to Kelvin's conjecture. Tiling space with unit cells of least area.
- Cauchy sequences help define real numbers rigorously.
- Permuting the terms of a series may change its sum arbitrarily.
- Uniform convergence implies properties for the limit of a sequence of functions.
  - Defining integrals: Cauchy, Riemann, Darboux, Lebesgue.
- Cauchy principal value of an integral.
- Fourier series. A simple example.
- Infinite sums may sometimes be evaluated with Fourier Series.
- A double sum is often the product of two sums, which may be Fourier series.
- At a jump, the sum of a Fourier series is the half-sum of its left and right limits.
- Gibbs phenomenon; 9% overshoot of partial Fourier series near a jump.
- Method of Froebenius about a regular singularity of a differential equation.
- Laurent series of a function about one of its poles.
- Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
- Tame complex functions: Holomorphic and meromorphic functions.
- Radius of convergence. The convergence disk of a complex power series.
- Analytic continuation: Power series that coincide whenever their disks overlap.
- Decimated power series are equal to finite sums involving roots of unity.
- Convolution as an inner operation among numerical functions.
- Duality: The product of a bra by a ket is a (complex) scalar.
- A distribution associates a scalar to every test function.
- Schwartz functions are suitable rapidly decreasing test functions.
- Tempered distributions are functionals over Schwartz functions.
- The Fourier Transform associates a tempered distribution to another.
 - Parseval's theorem (1799). The Fourier transform is unitary.
- Noteworthy distributions and their Fourier transforms:
- Dirac's d and the uniform distribution ( f (x) = 1).
- The signum function sign(x) and its transform: i / ps
- The Heaviside step function H(x) = ½ (1+sign(x)) and its transform.
- The square function P(x) = H(x+½)-H(x-½) and sinc ( ps )
- The triangle function L(x) and sinc2 ( ps )
- The normalized Gaussian distribution is its own Fourier transform.
- Poisson summation formula: The unit comb (
 ) is its own Fourier transform. - Far image of a translucent film is the Fourier transform of its optical density.
- The Radon transform, corresponding to lateral tomography, is easily inverted.
- Competing definitions of the Fourier transform. For the record.
- Discrete Fourier Transform, defined as a unitary involution.
- The Barber's Dilemma. Not a paradox if analyzed properly.
- What is infinity? More than a pretty symbol (¥).
- There are more real than rational numbers. Cantor's argument.
- Cantor's ternary set. A vanishing set or reals equipollent to the whole line.
- The axioms of set theory: Fundamental axioms and the Axiom of Choice.
- A set is smaller than its powerset: A simple proof applies to all sets.
- Transfinite cardinals, transfinite ordinals: Two different kinds of infinite numbers.
- The continuum hypothesis: What's between the countable and the continuum?
- Surreal Numbers: These include reals, transfinite ordinals, infinitesimals & more.
- Numbers: From integers to surreals. From reals to quaternions and beyond.
- The number 1 is not prime, as definitions are chosen to make theorems simple.
- Composite numbers are not prime, but the converse need not be true...
- Two prime numbers whose sum is equal to their product.
- Gaussian integers: Factoring into primes on a two-dimensional grid.
- The least common multiple may be obtained without factoring into primes.
- Standard Factorizations: n4 + 4 is never prime for n > 1 because...
- Euclid's algorithm gives the greatest common divisor and Bézout coefficients.
- Bézout's Theorem: The GCD of p and q is of the form u p + v q.
- Greatest Common Divisor (GCD) defined for all commensurable numbers.
- Linear equation in integers can be solved using Bézout's theorem.
- Pythagorean Triples: Right triangles whose sides are coprime integers.
- The number of divisors of an integer.
- Perfect numbers and Mersenne primes.
- Multiperfect and hemiperfect numbers divide twice the sum of their divisors.
- Fast exponentiation by repeated squaring.
- Partition function. How many collections of positive integers add up to 15?
- A Lucas sequence whose oscillations never carry it back to -1.
- A bit sequence with intriguing statistics. Counting squares between cubes.
- Binet's formulas: N-th term of a sequence obeying a second-order recurrence.
- The square of a Fibonacci number is almost the product of its neighbors.
- D'Ocagne's identity relates conjugates products of Fibonacci numbers.
- Catalans's identity generalizes Cassini's Identity (about Fibonacci squares).
- Faulhaber's formula gives the sum of the p-th powers of the first n integers.
- Multiplicative functions: If a and b are coprime, then f(ab) = f(a) f(b).
- Moebius function: Getting N values with only O(N Log(Log N)) additions.
- Dirichlet convolution is especially interesting for multiplicative functions.
- Dirichlet powers of arithmetic functions (especially, of the Moebius function).
- Dirichlet powers of multiplicative functions are given by a superb formula.
- Totally multiplicative functions are the simplest type of multiplicative functions.
- Dirichlet characters are important totally multiplicative functions.
- Euler products and generalized zeta functions.
- Modular Arithmetic may be used to find the last digit(s) of very large numbers.
- Powers of ten expressed as products of two factors without zero digits.
- Divisibility by 7, 13, and 91 (or by B2-B+1 in base B).
- Lucky 7's. Any integer divides a number composed of only 7's and 0's.
- Binary and/or hexadecimal numeration for floating-point numbers as well.
- Extract a square root the old-fashioned way.
- Ternary system: Is base 3 really the best radix for positional numeration?
- A prime number is a positive integer with just two distinct divisors (1 and itself).
- Euclid's proof: There are infinitely many primes.
- Dirichlet's theorem: There are infinitely many primes of the form kN+a.
- Green-Tao theorem: There are arbitrarily long arithmetic progressions of primes.
- The von Mangoldt function is Log p for a power of a prime p, 0 otherwise.
- The Prime Number Theorem: The probability that N is prime is roughly 1/ln(N).
- The average number of factors of a large number N is Log N.
- The average number of distinct prime factors of N is Log Log N.
- The largest known prime: Historical records, old and new.
- The Lucas-Lehmer Test checks the primality of a Mersenne number very fast.
- Formulas giving only primes may not help with new primes.
- Ulam's Lucky Numbers and other sequences generated by sieves.
- Chinese Remainder Theorem: How remainders define an integer (within limits).
- Modular arithmetic: The algebra of congruences, formally introduced by Gauss.
- Fermat's little theorem: For any prime p, ap-1 is 1 modulo p, unless p divides a.
- Euler's totient function: f(n) is the number of integers coprime to n, from 1 to n.
- Fermat-Euler theorem: If a is coprime to n, then a to the f(n) is 1 modulo n.
- Carmichael's reduced totient function (l) : A very special divisor of the totient.
- 91 is a pseudoprime to half of the bases coprime to itself.
- Carmichael Numbers: An absolute pseudoprime n divides (an - a) for any a.
- Chernik's Carmichael numbers: 3 prime factors (6k+1)(12k+1)(18k+1).
- Large Carmichael numbers may be obtained in various ways.
- Conjecture: Any odd number coprime to its totient has a Carmichael multiple.
- Monoids are endowed with an associative operation and a neutral element.
- The inverse of an element comes in two flavors which coincide when both exist.
- Free monoid: All the finite strings (words) in a given alphabet.
- Groups are monoids in which every element is invertible.
- A subgroup is a group contained in another group.
- Generators of a group are not contained in any proper subgroup.
- Lagrange's Theorem: The order of a subgroup divides the order of the group.
- Normal subgroups and their quotients in a group.
- Group homomorphism: The image of a product is the product of the images.
- The symmetric group on a set E consists of all the bijections of E onto itself.
- Inner automorphisms: Inn(G) is isomorphic to the quotient of G by its center.
- The conjugacy class formula uses conjugacy to tally elements of a group.
- Simple groups are groups without nontrivial normal subgroups.
- The derived subgroup of a group is generated by its commutators.
- Direct product of two groups (also called a direct sum for additive groups).
- Groups of small orders and their families: Cyclic groups, dihedral groups, etc.
- Enumeration of "small" groups. How many groups of order n?
- Classification of finite simple groups, by Gorenstein and many others (1982).
- Sporadic groups: Tits Group, 20 relatives of Fischer's Monster, 6 pariahs.
- Classical groups: Their elements depend on parameters from a field.
- The Möbius group consists of homographic transformations of
 È{¥}. - Lorentz transformations may change spatial orientation or time direction.
- Symmetries of the laws of nature: A short primer.
- Rings are sets endowed with addition, subtraction and multiplication.
- Nonzero characteristic: The least p for which all sums of p like terms vanish.
- Ideals within a ring are multiplicatively absorbent additive subgroups.
- Quotient ring, modulo an ideal: The residue classes modulo that ideal.
- Cauchy multiplication is well-defined for "formal power series" over a ring.
- Ring of polynomials whose coefficients are in a given ring.
- Galois rings. Residues of modular polynomials, modulo one of them.
- Vocabulary: We consider skew fields to be noncommutative. Some don't.
- Fields are commutative rings where every nonzero element has a reciprocal.
- Wedderburn's Theorem: Finite division rings are commutative (they're fields).
- Every finite integral domain is a field. A corollary of Wedderburn's theorem.
- Galois fields are the finite fields. Their orders are powers of prime numbers.
- The trivial field has a single element. It's the only field where 0 has a reciprocal.
- The splitting field of PÎF[x] is the smallest extension of F where P fully factors.
- The Nim-Field is algebraically complete. It contains [surreal] infinite ordinals.
- Ternary multiplication compatible with ternary addition (without "carry").
Vector Spaces (over a field) and Modules (over a ring) - Vectors were originally just differences between points in ordinary space...
- Abstract vector spaces: Vectors can be added, subtracted and scaled.
- Modules are vectorial structures over a ring of scalars (instead of a field).
- Banach spaces are complete normed vector spaces.
- Dual space: The set of all [continuous] linear functions with scalar values.
- Tensors: Multilinear functions of vectors and covectors with scalar values.
- An algebra is a vector space with a scalable and distributive internal product.
- Clifford algebras are unital associative algebras endowed with a quadratic form.
- Physical things that are not vectorial because they're not defined intrinsically.
- David Hestenes has proposed geometric algebra as a denotational unification.
- The ring of p-adic integers includes objects with infinitely many radix-p digits.
- Polyadic integers: Greek naming of p-adic integers.
- What if p isn't prime? Dealing with divisors of zero.
- Decadic Integers: The strange realm of 10-adic integers (composite radix).
- The field of p-adic numbers is the quotient field of the ring of p-adic integers.
- Dividing two p-adic numbers looks like "long division", only backwards...
- The p-adic metric can be used to define p-adic numbers analytically.
- The reciprocal of a p-adic number computed by successive approximations.
- Hasse's local-global principle was established for the quadratic case in 1920.
- Integers which double when their digits (in base g) are rotated.
- Pseudoprimes to base a. Poulet numbers are pseudoprimes to base 2.
- Weak pseudoprimes to base a : Composite integers n which divide (an-a).
- Counting the bases to which a composite number is a pseudoprime.
- Strong pseudoprimes to base a are less common than Euler pseudoprimes.
- Rabin-Miller Test: An efficient and trustworthy stochastic primality test.
- The product of 3 primes is a pseudoprime when all pairwise products are.
- Wieferich primes are scarce but there are (probably) infinitely many of them.
- Super-pseudoprimes: All their composite divisors are pseudoprimes.
- Maximal super-pseudoprimes have no super-pseudoprime multiples.
- Jevons Number. Factoring 8616460799 is now an easy task.
- Challenges help tell special-purpose and general-purpose methods apart.
- Special cases of a priori (partial) factorizations may help number theorists.
- Trial division may be used to weed out the small prime factors of a number.
- Ruling out factors can speed up trial divison in special cases.
- Recursively defined sequences (over a finite set) are ultimately periodic.
- Pollard's r (rho) factoring method is based on the properties of such sequences.
- Pollard's p-1 Method finds prime factors p for which p-1 is smooth.
- Williams' p+1 Method is based on the properties of Lucas sequences.
- Lenstra's Elliptic Curve Method is a generalization of Pollard's p-1 approach.
- Dixon's method: Combine small square residues into a solution of x 2 º y 2
- Motivation: On the prime factors of some quadratic forms...
- Quadratic residues: Half of the nonzero residues modulo an odd prime p.
- Euler's criterion: A quadratic residue raised to the power of (p-1)/2 is 1.
- The Legendre symbol (a|p) can be extended to values of p besides odd primes.
- The law of quadratic reciprocity states a simple but surprising fact.
- Gauss' Lemma expresses a Legendre symbol as a product of many signs.
- Eisenstein's Lemma: A variation of Gauss's lemma allowing a simpler proof.
- One of many proofs of the law of quadratic reciprocity.
- Artin's Reciprocity.
- What is a continued fraction? Example: The expansion of p.
- The convergents of a number are its best rational approximations.
- Large partial quotients allow very precise approximations.
- Regular patterns in the continued fractions of some irrational numbers.
- For almost all numbers, partial quotients are ≥ k with probability lg(1+1/k).
- Elementary operations on continued fractions.
- Expanding functions as continued fractions.
- Engel expansion of a positive number. A nondecreasing sequence of integers.
- Pierce expansions of numbers between 0 and 1. Strictly increasing sequences.
- Counterfeit Coin Problem: In 3 weighings, find an odd object among 12, 13, 14.
- General Counterfeit Penny Problem: Find an odd object in the fewest weighings.
- Seven-Eleven: Four prices with a sum and product both equal to 7.11.
- Equating a right angle and an obtuse angle, with a clever false proof.
- Choosing a raise: Trust common sense, beware of fallacious accounting.
- 3 men pay $30 for a $25 hotel room, the bellhop keeps $2... Is $1 missing?
- Chameleons: A situation shown unreachable because of an invariant quantity.
- Sam Loyd's 14-15 puzzle also involves an invariant quantity (and two orbits).
- Einstein's riddle: 5 distinct house colors, nationalities, drinks, smokes and pets.
- Numbering n pages of a book takes this many digits (formula).
- The Ferry Boat Problem (by Sam Loyd): To be or not to be ingenious?
- Hat overboard ! What's the speed of the river?
- All digits once and only once: 48 possible sums (or 22 products).
- Crossing a bridge: 1 or 2 at a time, 4 people (U2), different paces, one flashlight!
- Managing supplies to reach an outpost 6 days away, carrying enough for 4 days.
- Go south, east, north and you're back... not necessarily to the North Pole!
- Icosapolis: Numbering a 5 by 4 grid so adjacent numbers differ by at least 4.
- Unusual mathematical boast for people born in 1806, 1892, or 1980.
- Puzzles for extra credit: From Chinese remainders to the Bookworm Classic.
- Simple geometrical dissection: A proof of the Pythagorean theorem.
- Early bird saves time by walking to meet incoming chauffeur.
- Sharing a meal: A man has 2 loaves, the other has 3, a stranger has 5 coins.
- Fork in the road: Find the way to Heaven by asking only one question.
- Proverbial Numbers: Guess the words which commonly describe many numbers.
- Riddles: The Riddle of the Sphinx and other classics, old and new.
- Flexagons: Hexaflexagons were popularized by Martin Gardner in 1956.
- Polyominoes: The 12 pentominoes and other tiles invented by Sol Golomb.
- Soma: 7 nonconvex solids consisting of 3 or 4 cubes make a larger cube.
- Tessellations by convex pentagons. The contributions of Marjorie Rice.
- Kites and Darts. The aperiodic tilings of Roger Penrose.
- Ambigrams: Calligraphic spellings which change when rotated or flipped.
- The Game of Life. John Conway's endearing cellular automaton (1970).
- Rubik's Cube: Ernõ Rubik (1974) D. Singmaster (1979) M. Gardner (1981).
- 1089: Subtract a 3-digit number and its reverse, then add this to its reverse...
- Mass media mentalism by David Copperfield (1992).
- Grey Elephants in Denmark: "Mental magic" for one-time classroom use.
- The 5-card trick of Fitch Cheney: Tell the fifth card once 4 are known.
- Generalizing the 5-card trick and Devil's Poker...
- Kruskal's Count.
- Paths to God.
- Stacked Deck.
- Magic Age Cards.
- Ternary Cards.
- Magical 21 (or 27).
- Boolean Magic.
- Perfect Faro Shuffles.
- Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
- The Game of Nim: Remove items from one of several rows. Don't play last.
- Grundy numbers are defined for all positions in impartial games.
- Moore's Nim: Remove something from at most (b-1) rows. Play last.
- Normal Kayles: Knocking down one pin, or two adjacent ones, may split a row.
- Grundy's Game: Split a row into two unequal rows. Whoever can't move loses.
- Wythoff's Game: Remove counters either from one heap or equally from both.
- The pigeonhole principle: What must happen with fewer holes than pigeons...
- n+1 of the first 2n integers always include two which are coprime.
- Largest sets of small numbers with at most k pairwise coprime integers.
- Ramsey's Theorem: Monochromatic complete subgraphs of a large graph.
- Infinite alignment among infinitely many lattice points in the plane? Nope.
- Infinite alignment in a lattice sequence with bounded gaps? Almost...
- Large alignments in a lattice sequence with bounded gaps. Yeah!
- Van der Waerden's theorem: Long monochromatic arithmetic progressions.
- Ford circles are nonintersecting circles touching the real line at rational points.
- Farey series: The rationals from 0 to 1, with a bounded denominator.
- The Stern-Brocot tree contains a single occurrence of every positive rational.
- Any positive rational is a unique ratio of two consecutive Stern numbers.
- Pick's formula gives the area of a lattice polygon by counting lattice points. History :
- Earliest mathematics on record. Before Thales was Euphorbe...
- Indian numeration became a positional system with the introduction of zero.
- Roman numerals are awkward for larger numbers.
- The invention of logarithms: John Napier, Bürgi, Briggs, Saint-Vincent, Euler.
- The earliest mechanical calculator(s), by W. Shickard (1623) or Pascal (1642).
- The Fahrenheit Scale: 100°F was meant to be the normal body temperature.
- The revolutionary innovations which brought about new civilizations.
Nomenclature & Etymology : - The origin of the word "algebra", and also that of "algorithm".
- The name of the avoirdupois system: Borrowed from French in a pristine form.
- Long Division: Cultural differences in writing the details of a division process.
- Is a parallelogram a trapezoid? In a mathematical context [only?], yes it is...
- Naming polygons. Greek only please; use hendecagon not "undecagon".
- Chemical nomenclature: Basic sequential names (systematic and/or traditional).
- Fractional Prefixes: hemi (1/2), sesqui (3/2) or weirder hemipenta, hemisesqui...
- Matches, phosphorus, and phosphorus sesquisulphide.
- Zillion. Naming large numbers.
- Zillionplex. Naming huge numbers.
- Abbreviations: Abbreviations of scholarly Latin expressions.
- Typography of long numbers.
- Intervals denoted with square brackets (outward for an excluded extremity).
- Dates in the simplest ISO 8601 form (with customary time stamps or not).
- The names of operands in common numerical operations.
- The word respectively doesn't have the same syntax as "resp."
- The heliocentric Copernican system was known two millenia before Copernicus.
- The assistants of Galileo Galilei and the mythical experiment at the Tower of Pisa.
- Switching calendars: Newton was not born the year Galileo died.
- The Lorenz Gauge is an idea of Ludwig Lorenz (1829-1891) not H.A. Lorentz.
- Special Relativity was first formulated by H. Poincaré (Einstein a close second).
- The Fletcher-Millikan "oil-drop" experiment was not the sole work of Millikan.
- Collected errata about customary physical units.
- Portrait of Legendre: The mathematician was confused with the politician.
- Dubious quotations: Who really said that?
- Obliquity of the ecliptic: An evolving quantity first measured by Eratosthenes.
- Vertical wells at Syene are completely sunlit only once a year, aren't they?
- Eratosthenes sizes up the Earth: 700 stadia per degree of latitude.
- Latitude and longitude: The spherical grid of meridians and parallels.
- Itinerary units: The land league and the nautical league.
- Amber, compass and lightning: Glimpses of electricity and magnetism.
- The "work done" on a point-mass equals the change in its kinetic energy.
- Relativistic work done and the corresponding change in relativistic energy.
- Relativistic thermodynamics: The case of a point endowed with internal heat.
- Spacecraft speeds up upon reentry into the upper atmosphere.
- Lewis Carroll's monkey climbs a rope over a pulley, with a counterweight.
- Two-ball drop can make a light ball bounce up to 9 times the dropping height.
- Normal acceleration is the square of speed divided by the radius of curvature.
- Roller-coasters must rise more than half a radius above any loop-the-loop.
- Conical pendulum: A hanging bob whose trajectory is an horizontal circle.
- Ball in a Bowl: Pure rolling increases the period of oscillation by 18.3%.
- Hooke's Law: Motion of a mass suspended to a spring.
- Speed of an electron estimated with the Bohr model of the atom.
- Thermal expansion coefficients: The cubical coefficient is 3 times the linear one.
- Waves in a solid: P-waves (fastest), S-waves, E-waves (thin rod), SAW...
- Rayleigh Wave: The quintessential surface acoustic wave (SAW).
- Hardest Stuff: Diamond is no longer the hardest material known to science.
- Hardness is an elusive nonelastic property, distinct from stiffness.
- Hot summers, hot equator! The distance to the Sun is not the explanation.
- Kelvin's Thunderstorm: Using falling water drops to generate high voltages.
- The Coriolis effect: A dropped object falls to the east of the plumb line.
- Terminal velocity of an object falling in the air.
- Angular momentum and torque. Spin and orbital angular momentum.
- Rotation vector of a moving rigid body (and/or "frame of reference").
- Angular momentum equals moment of inertia times angular velocity.
- Moments about a point or a plane are convenient mathematical fictions.
- Moment of inertia of a spherical distribution and of an homogeneous ellipsoid.
- Perpendicular Axis Theorem: Axis of rotation perpendicular to a thin plate.
- The Parallel Axis Theorem gives the moment of inertia about an off-center axis.
- Moment of inertia of a thick plate, as obtained from the parallel axis theorem.
- Momenta of homogeneous bodies: List of examples.
- Rigid pendulum moving under its own weight about a fixed horizontal axis.
- Reversible pendulum swings with the same period around two distinct axes.
- All physical theories have a limited range of validity.
- Gravity vs. Electrostatics: Straight comparisons.
- Airy weighs the Earth by timing a pendulum at the bottom of a mine.
- Rigid equilateral triangle formed by three gravitating bodies.
- The five Lagrange points of two gravitating bodies in circular orbit.
- Huygens' Principle is a convenient fiction to describe the propagation of waves.
- Diffraction occurs when when a wave emanates from a bounded source.
- Young's double-slit experiment demonstrates the wavelike nature of light.
- Celerity is the speed with which phase propagates.
- Standing waves feature stationary nodes and antinodes.
- Chladni patterns: The lines formed by nodes in an oscillating plate (or surface).
- Snell's Law (1621) gives the angle of refraction (or says nothing is refracted).
- Birefringence and the discovery of polarization (Erasmus Bartholinus, 1669).
- Brewster's angle is the incidence which yields a 100% polarized reflection.
- Fresnel equations give the reflected or refracted intensities of polarized light.
- Stokes parameters: A standard description of the state of polarization.
- Dispersion relation: pulsatance vs. wave number (or frequency vs. wavelength).
- Group velocity is the traveling speed of a beat phenomenon.
- Rayleigh scattering makes the sky blue and sunsets red.
- Index of refraction of water for light of different colors.
- A spherical drop bounces red light up to 42.34° and violet light up to 40.58°.
- Fermat's principle (least time) for light (c.1655) predates Newton's mechanics.
- Maupertuis principle of least action (1744).
- Virtual Work: A substitute for Newton's laws which factors out constraint forces.
- Phase Space: A phase describes all there is to a classical system.
- Either velocities or momenta can be used (with positions) to specify a phase.
- Relativistic point-mass: Lagrangian, Hamiltonian and momentum in free space.
- Charge in a magnetic field: The canonical momentum isn't the linear momentum.
- Lagrangian mechanics: The Lagrangian is a function of positions and velocities.
- Hamiltonian mechanics: The Hamiltonian depends on positions and momenta.
- Poisson brackets: An abstract synthetic view of analytical mechanics.
- Liouville's theorem: The volume in Hamiltonian phase space doesn't change.
- Noether's Theorem: Conservation laws express the symmetries of physics.
- Field Theory: Lagrangian mechanics on a continuum of values (and derivatives).
- Clarifications by Heaviside & Lorentz: Vector calculus & microscopic view.
- The vexing problem of units is a thing of the past if you stick to SI units.
- The Lorentz force on a test particle defines the local electromagnetic fields.
- Electrostatics: The study of the electric field due to static charges.
- Electric capacity is an electrostatic concept (adequate at low frequencies).
- Electrostatic multipoles: The multipole expansion of an electrostatic field.
- Birth of electromagnetism (1820): Electric currents generate magnetic fields !
- Biot-Savart Law: The static magnetic induction due to steady currents.
- Magnetic scalar potential: A multivalued function whose gradient is induction.
- Magnetic monopoles do not exist : A law stating a fact not yet disproved.
- Ampère's law: The law of static electromagnetism devised by Ampère in 1825.
- Faraday's law: A varying magnetic flux induces an electric circulation.
- Self-induction: The induction received by a circuit from its own magnetic field.
- Ampère-Maxwell law: The dynamic generalization (1861) of Ampère's law.
- Putting it all together: Historical paths to Maxwell's electromagnetism.
- Maxwell's equations unify electricity and magnetism dynamically (1864).
- Continuity equation: Maxwell's equations imply conservation of charge.
- Waves anticipated by Faraday, Maxwell & FitzGerald were observed by Hertz.
- Electromagnetic energy density and the flux of the Poynting vector.
- Planar electromagnetic waves: The simplest type of electromagnetic waves.
- Electromagnetic potentials are postulated to obey the Lorenz gauge.
- Solutions to Maxwell's equations, as retarded or advanced potentials.
- Electrodynamic fields corresponding to retarded potentials.
- The gauge of retarded potentials: is it really the Lorenz gauge?
- Electric and magnetic dipoles: Dipolar solutions of Maxwell's equations.
- Static distributions of magnetic dipoles can be simulated with steady currents.
- Static distributions of electric dipoles can be simulated with static charges.
- Sign reversal in the fields of uniformly distributed magnetic or electric dipoles.
- Fields at the center of uniformly magnetized or polarized spheres (of any size).
- Relativistic dipoles: A moving magnet develops an electric moment.
- Power radiated by an accelerated charge: The Larmor formula (1897).
- Lorentz-Dirac equation for the motion of a point charge is of third order.
- Molecular electric dipole moments were first studied by Peter Debye (1912).
- Force exerted on a dipole by a nonuniform field.
- Torque on a dipole is proportional to its cross-product into the field.
Magnetism, Electromagnetic Properties of Matter - Magnetization and polarization describe densities of dipoles bound to matter.
- Gauge invariance: Many magnetizations and polarizations create the same field.
- Maxwell's equations in matter: Electric displacement D, magnetic strength H.
- Electric susceptibility is the propensity to be polarized by an electric field.
- Electric permittivity and magnetic permeability are related to susceptibilities.
- Paramagnetism: Weak positive susceptibility.
- Diamagnetism: The Lorentz force turns orbital moments against the external B.
- Magnetic levitation: How to skirt the theorem of Samuel Earnshaw (1842).
- Pyrolytic carbon: The most diamagnetic sunstance known, at room temperature.
- Bohr & Van Leeuwen Theorem: Diamagnetism and paramagnetism cancel ?!
- Thermodynamics of dielectric matter: dU = E.dD + ...
- Ferromagnetism: Permanent magnetization without an external magnetic field.
- Antiferromagnetism: When adjacent dipoles tend to oppose each other...
- Ferrimagnetism: With two kinds of dipoles, partial cancellation may occur.
- Magneto-optical effect discovered by Faraday on September 13, 1845.
- Ohm's Law: Current density is proportional to electric field: j = s E.
- Homopolar motor: The first electric motor, by Michael Faraday (1831).
- Faraday's disk can generate huge currents at a low voltage.
- Magic wheels: Two repelling ring magnets mounted on the same axle.
- Beakman's motor. Current switches on and off as the coil spins horizontally.
- Tesla turbine. Stack of spinning disks with outer intake and inner outflow.
- Observers in motion: A simple-minded derivation of the Lorentz Transform.
- Adding up velocities: The combined speed can never be more than c.
- Fizeau's empirical relation between refractive index (n) and Fresnel drag.
- The Harress-Sagnac effect used to measure rotation with fiber optic cable.
- Combining relativistic speeds: Using rapidity, the rule is transparent.
- Relative velocity of two photons: Defined unless both have the same direction.
- Minkowski spacetime: Coordinates of 4-vectors obey the Lorentz transform.
- The Lorentz transform expressed vectorially: A so-called boost of speed V.
- Wave vector: The 4-dimensional gradient of the phase describes propagation.
- Doppler shift: The relativistic effect is not purely radial.
- Kinetic energy: At low speed, the relativistic energy varies like ½ mv 2.
- Photons and other massless particles: Finite energy at speed c.
- The de Broglie celerity (u) is inversely proportional to a particle's speed.
- Compton diffusion: The result of collisions between photons and electrons.
- The Klein-Nishina formula: gives the cross-section in Compton scattering.
- Compton effect is suppressed quantically for visible light and bound electrons.
- Elastic shock: Energy transfer is v.dp. (None is seen from the barycenter.)
- Photon-photon scattering is like an elastic collision of two photons.
- Cherenkov Effect: When the speed of an electron exceeds the celerity of light...
- Constant acceleration over an entire lifetime will take you pretty far...
- The Harress-Sagnac effect seen by an observer rotating with the optical loop.
- Relativistic rigid motion is an equilibrium modified at the speed of sound.
- In the Euclidean plane: Contravariance and covariance.
- In the Lorentzian plane: Contravariance and covariance revisited.
- Tensors of rank n+1 are linear maps that send a vector to a tensor of rank n.
- Signature of the quadratic form defined by a given metric tensor.
- Covariant and contravariant coordinates of tensors of rank n, in 4 dimensions.
- The metric tensor and its inverse. Lowering and raising indices.
- Partial derivatives with respect to contravariant or covariant coordinates.
- Christoffel symbols: Coordinates of the partial derivatives of the basis vectors.
- Covariant derivatives. Absolute differentiation with the nabla operator Ñ.
- Contravariant derivatives: The lesser known flavor of absolute derivatives.
- The antisymmetric part of Christoffels symbols form a fundamental tensor.
- Totally antisymmetric spacetime torsion is described by a vector field.
- Levi-Civita symbols: Antisymmetric with respect to any pair of indices.
- Einstein's equivalence principle implies vanishing spacetime torsion.
- Ricci's theorem: The covariant derivative of the metric tensor vanishes.
- Curvature: The Ricci tensor is obtained by contracting the Riemann tensor.
- The Bianchi identity show that the Einstein tensor is divergence free.
- Stress tensor: Flow of energy density is density of [conserved] momentum.
- Einstein's Field Equations: 16 equations in covariant form (Einstein, 1915).
- Free-falling bodies: Their trajectories are geodesics in curved spacetime.
- The "anomalous" precession of Mercury's perihelion is entirely relativistic.
- The Schwarzschild metric: The earliest exact solution to Einstein's equations.
- What is mass?
- Electromagnetism: Covariant expressions, using tensors.
- Kaluza-Klein theory of electromagnetism involves a fifth dimension.
- Harvard Tower Experiment: The slow clock at the bottom of the tower.
- Shapiro time delay: The effect on radar signals of gravitational time dilation.
String Theory and other "Theories of Everything" - Unification: Consistency is required. Actual high-energy unification is not.
- Kaluza-Klein Theory: Postulating one extra dimension for electromagnetism.
- Gabriele Veneziano: The magic of Euler's beta and gamma functions.
- Leonard Susskind (1940-): The basic idea of a fundamental string.
- Joël Scherk (1946-1979) & John Schwarz: Rediscovering gravity.
- Michael Green & John Schwarz: Hoping for a Theory of Everything.
- String Quintet: Five different consistent string theories!
- M-Theory: Ed Witten's 11-dimensional brainchild, unveiled at String '95.
- The brane world scenarios of Lisa Randall and Burt Ovrut.
- The Magdeburg hemispheres are held together by more than one ton of force.
- The ideal gas laws of Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
- Joule's law: The internal energy of a perfect gas depends only on its temperature.
- The Van der Waals equation and other interesting equations of state.
- Virial equation of state. Virial expansion coefficients. Boyle's temperature.
- Viscosity is the ratio of a shear stress to the shear strain rate it induces.
- Permeability and permeance: Vapor barriers and porous materials.
- Resonant frequencies of air in a box.
- The Earth's atmosphere. Pressure at sea-level and total mass above.
- The first hot-air balloon (Montgolfière) was demonstrated on June 4, 1783.
- Viscosity: The transport of microscopic momentum.
- Brownian motion and Einstein's estimate of molecular sizes.
- Thermal Conductivity: The transport of microscopic energy.
- Diffusivity: The transport of chemical concentration.
- Speed of Sound: Reversible transport of a pressure disturbance in a fluid.
- Complex pulsatance (s) is damping constant (s) plus imaginary pulsatance (iw).
- Complex impedance: Resistance and reactance.
- Quality Factor (Q). Ratio of maximal stored energy to dissipated power.
- Nullators and norators: Strange dipoles for analog electronic design.
- Corner frequency of a simple first-order low-pass filter. -3 dB bandwidth.
- Second-order passive low-pass filter, with inductor and capacitor.
- Sallen key filters: Active filters do not require inductors.
- Lowpass Butterworth filter of order n : The flattest low-frequency response.
- Linkwitz-Riley crossover filters are used in modern active audio crossovers.
- Chebyshev filters: Ripples in either the passband or the stopband.
- Elliptic (Cauer) filters encompass all Butterworth and Chebyshev types.
- Legendre filters maximal roll-off rate with a monotonous frequency response.
- Gegenbauer filters: From Butterworth to Chebyshev, via Legendre.
- Phase response of a filter.
- Bessel-Thomson filters: Phase linearity and group delay.
- Gaussian filters: Focusing on time-domain communication pulses.
- Linear Phase Equiripple: Ripples in group delay to improve on Bessel filters.
- DSL filter allows POTS below 3400 Hz and blocks digital data above 25 kHz.
- Raising the Titanic, with (a lot of) hydrogen.
- Gravitational Subway: From here to anywhere in 42 minutes.
- In a vacuum tube, a drop to the center of the Earth would take 21 minutes.
- The aeolipile: This ancient steam engine demonstrates jet propulsion.
- Edward Somerset of Worcester (1601-1667): Blueprint for a steam fountain.
- Denis Papin (1647-1714): Pressure cooking and the first piston engine.
- Thomas Savery (c.1650-1715): Two pistons and an independent boiler.
- Thomas Newcomen (1663-1729) and John Calley: Atmospheric steam engine.
- Nicolas-Joseph Cugnot (1725-1804): The first automobile (October 1769).
- James Watt (1736-1819): Steam condenser and Watt governor.
- Richard Trevithick (1771-1833) and the first railroad locomotives.
- Sadi Carnot (1796-1832): Carnot's cycle and the theoretical efficiency limit.
- Sir Charles Parsons (1854-1931): The modern steam turbine, born in 1884.
- The elementary concept of temperature. The zeroth law of thermodynamics.
  - Conservation of energy: The first law of thermodynamics.
- Increase of Entropy: The second law of thermodynamics.
- State variables: Extensive and intensive quantities.
- Entropy is missing information, a measure of disorder.
- Nernst Principle (third law): Entropy is zero at zero temperature.
- Thermodynamic potentials can be convenient alternatives to internal energy.
- Latent heat (L) is the heat transferred in a change of phase.
- Calorimetric coefficients, adiabatic coefficient (g) heat capacities, etc.
- Cryogenic coefficients: Lower temperature with an isenthalpic expansion.
- Relativistic considerations: A moving body appears cooler.
- Inertia of energy for an object at nonzero temperature.
- Stefan's Law: A black body radiates as the fourth power of its temperature.
- The "Fourth Law": Is there really an upper bound to temperature?
- Hawking radiation: On the entropy and temperature of a black hole.
- Partition function: The cornerstone of the statistical approach.
- Laplace's Demon: Deducing past and future from a detailed snapshot.
- Maxwell's Demon: Trading information for entropy.
- Shockley's Ideal Diode Equation: Diodes don't violate the Second Law.
- Szilard's engine & Landauer's Principle: The thermodynamic cost of forgetting.
- Lagrange multipliers are associated to the constraints of a maximizing problem.
- Microcanonical equilibrium: All states of an isolated system are equiprobable.
- Canonical equilibrium: In a heat bath, probabilities involve a Boltzmann factor.
- Grand-canonical equilibrium when chemical exchanges are possible.
- Bose-Einstein statistics: One state may be occupied by many particles.
- Fermi-Dirac statistics: One state is occupied by at most one particle.
- Boltzmann statistics: The low-occupancy limit (most states are unoccupied).
- Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.
- Partition function: The cornerstone of the statistical approach.
- Quantum Logic: The surprising way quantum probabilities are obtained.
- Swapping particles either negates the quantum state or leaves it unchanged.
- The Measurement Dilemma: What makes Schrödinger's cat so special?
- Matrix Mechanics: Neither measurements nor matrices can be switched at will.
- Schrödinger's Equation: A nonrelativistic quantum particle in a classical field.
- Noether's Theorem: Conservation laws express the symmetries of physics.
- Kets are Hilbert vectors (their duals are bras) on which observables operate.
- Observables are operators explicitely associated with physical quantities.
- Commutators are the quantities which determine uncertainty relations.
- Uncertainty relations hold whenever the commutator does not vanish.
- Spin is a form of angular momentum without a classical equivalent.
- Pauli matrices: Three 2 by 2 matrices with eigenvalues +1 and -1.
- Quantum Entanglement: The singlet and triplet states of two electrons.
- Bell's inequality is violated for the singlet state of two electron spins.
- Generalizations of Pauli matrices beyond spin ½.
- Density operators are quantum representations of imperfectly known states.
- Elementary particles: Quarks and leptons. Electroweak bosons. Graviton?
- Second Quantization: Particles are modes of a quantized field.
- Bethe-Salpeter Equation: A relativistic equation for bound-state problems.
- Black Powder: An ancient explosive, still used as a propellant (gunpowder).
- Predicting explosive reactions: A useful but oversimplified rule of thumb.
- Thermite generates temperatures hot enough to weld iron.
- Enthalpy of Formation: The tabulated data which gives energy balances.
- Gibbs Function (free energy): Its sign indicates the direction of spontaneity.
- Labile is not quite the same as unstable.
- Inks: India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
- Redox Reactions: Oxidizers are reduced by accepting electrons...
- Gold Chemistry: Aqua regia ("Royal Water") dissolves gold and platinum.
- Who is the "father" of modern chemistry?
- International Unit (IU) is an arbitrarily-defined rating of biological activity.
- Concentration is an amount (either mass or moles) per volume.
- Glycosylated hemoglobin (HbA1c) relates to average blood glucose (bG).
- Kant's Island Universes: The Universe is filled with separate galaxies.
- The Cosmological Principle: The Universe is homogeneous and isotropic.
- The Big Bang: An idea of Georges Lemaître mocked by Fred Hoyle.
- The Cosmic Microwave Background (CMB): Its spectrum and density.
- Cosmic redshift (z): Light emitted in a Universe which was (1+z) times smaller.
- Hubble Law: The relation between redshift and distance for comoving points.
- Omega (W): The ratio of the density of the Universe to the critical density.
- Look-Back Time: The time ellapsed since observed light was emitted.
- Distance: In a cosmological context, there are several flavors to the concept.
- Comoving points are reference points following the expansion of the universe.
- The Anthropic Principle: An obvious explanation which may not be the final one.
- Dark matter & dark energy: Gravity betrays the existence of some dark stuff.
- The Pioneer Effect: The anomalous escape of the Pioneer spaceprobe.
- Nuclear fusion is what powers the stars.
- Brown dwarves fail to ignite fusion. They glow from gravitational contraction.
- The Jeans mass above which a gas at temperature T collapses gravitationally.
- Main sequence: The evolution of a typical star.
- Eta Carinae and hypergiants. The most massive stars possible.
- Betelgeuse and red supergiants.
- Rigel and blue supergiants.
- Planetary nebulae: Aftermaths of stellar explosions.
- White dwarfs: The ultimate fate of our Sun and other small stars.
- Neutron stars: Remnants from the supernova collapse of medium-sized stars.
- Stellar black holes: They form when supermassive stars run out of nuclear fuel.
- Stellar X-ray source: A small accretor in tight orbit around a donor star.
- Astronomical unit: The precise definition of a standard unit of length.
- The solar corona is a very hot region of rarefied gas.
- Solar radiation: The Sun has radiated away about 0.03% of its mass.
- The Titius-Bode Law: A numerical pattern in solar orbits?
- The 4 inner rocky planets: Mercury, Venus, Earth, Mars.
- Earth: This is home.
- The asteroid belt: Planetoids and bolids between Mars and Jupiter.
- The 4 giant gaseous planets: Jupiter, Saturn, Uranus, Neptune.
- The discovery of Neptune: Urbain Le Verrier scooped John Couch Adams.
- Pluto and other Kuiper Belt Objects (KBO).
- Sedna and other planetoids beyond the Kuiper Belt.
- What's a planet? Anything besides the 6 ancient planets, Uranus & Neptune?
- Heliosphere and Heliopause: The domain where solar wind exerts its influence.
- Oort's Cloud is a cometary reservoir at the fringe of the Solar System.
- Easy conversion between Fahrenheit and Celsius scales: F+40 = 1.8 (C+40).
Automotive : - Car speed is proportional to tire diameter and engine rpm, divided by gear ratio.
- Car acceleration. Guessing the curve from standard data.
- "0 to 60 mph" time, obtained from vehicle mass and actual average power.
- Thrust is the power to speed ratio (measuring speed along thrust direction).
- Power of an engine as a function of its size: Rating internal combustion engines.
- Optimal gear ratio to maximize top speed on a flat road (no wind).
Surface Areas : - Heron's Formula (for the area of a triangle) is related to the Law of Cosines.
- Brahmagupta's Formula gives the area of a quadrilateral, inscribed or not.
- Bretschneider's Formula: Area of a quadrilateral of known sides and diagonals.
- The (vector) area of a quadrilateral is half the cross-product of its diagonals.
- Parabolic segment: 2/3 the area of a circumscribed parallelogram or triangle.
Volumes : - Content of a cylindrical tank (horizontal axis), given the height of the liquid in it.
- Volume of a spherical cap, or content of an elliptical vessel, given liquid height.
- Content of a cistern (cylindrical with elliptical ends), as a function of fluid height.
- Volume of a cylinder or prism, possibly with tilted [nonparallel] bases.
- Volume of a conical frustum: Formerly a staple of elementary education...
- Volume of a sphere... obtained by subtracting a cone from a cylinder !
- The volume of a tetrahedron is the determinant of three edges, divided by 6.
- Volume of a wedge of a cone.
Averages : - Splitting a job evenly between two unlike workers.
- Splitting a job unevenly between two unlike workers.
- Alcohol solutions are rated by volume not by mass.
- Mixing solutions to obtain a predetermined intermediate rating.
- Special averages: harmonic (for speeds), geometric (for rates), etc.
- Mean Gregorian month: either 30.436875 days, or 30.458729474253406983...
- The arithmetic-geometric mean is related to a complete elliptic integral.
Geodesy and Astronomy : - Distance to ocean horizon line is proportional to the square root of your altitude.
- Distance between two points on a great circle at the surface of the Earth.
- The figure of the Earth. Geodetic and geocentric latitudes.
- Kepler's Third Law: The relation between orbital period and orbit size.
Below are topics not yet integrated with the rest of this site's navigation.
- Circumference of an ellipse: Introducing exact series and approximate formulas.
- Ramanujan I and Lindner formulas: The journey begins...
- Ramanujan II: An awesome approximation from a mathematical genius (1914).
- Hudson's Formula and other Padé approximations.
- Peano's Formula: The sum of two approximations with cancelling errors.
- The YNOT formula (Maertens, 2000. Tasdelen, 1959).
- Euler's formula is the first step in an exact expansion.
- Naive formula: p ( a + b ) features a -21.5% error for elongated ellipses.
- Cantrell's Formula: A modern attempt with an overall accuracy of 83 ppm.
- From Kepler to Muir. Lower bounds and other approximations.
- Relative error cancellations in symmetrical approximative formulas.
- Complementary convergences of two series yield a nice foolproof algorithm.
- Padé approximants are used in a whole family of approximations...
- Improving Ramanujan II over the whole range of eccentricities.
- The Arctangent Function as a component of several approximate formulas.
- Abed's formula uses a parametric exponent to fine-tune the approximation.
- Zafary's formula.
- Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
- Better accuracy from Cantrell, building on his own previous formula
- Rediscovering a well-known exact expansion due to Euler (1773).
- Exact expressions for the circumference of an ellipse: A summary.
- Surface Area of a Scalene Ellipsoid: The general formula isn't elementary.
- Thomsen's Formula: A simple symmetrical approximation.
- Approximate formulas for the surface area of a scalene ellipsoid.
- Nautical mile: "Average" minute of latitude on an oblate spheroid.
- The Magnetic Field of the Earth.
- Life (1): The mysteries of evolution.
- Life (2): The origins of life on Earth.
- Life (3): Does extraterrestrial life exist? Is there intelligence out there?
- Nemesis: A distant companion to the Sun could explain extinction periodicity.
- Current Challenges to established dogma.
- Unexplained artifacts and sightings.
- The Riemann Hypothesis: {Re(z) > 0 & z(z) = 0} Þ {Re(z) = ½}.
- P = NP ? Can we find in polynomial time whatever we can check that fast?
- Collatz sequences go from n to n/2 (iff n is even) or 3n+1. Do they all lead to 1?
- The Poincaré Conjecture was proven by Grisha Perelman in 2002.
- The only magic hexagon.
- The law of small numbers applied to conversion factors.
- Quadratic formulas yielding long sequences of prime numbers.
- The area under a Gaussian curve involves the square root of p
- Exceptional simple Lie groups.
- Monstrous Moonshine in Number Theory.
- Oldest unsolved mathematical problem: Are there any odd perfect numbers?
- Magnetic Field of the Earth: The south side is near the geographic north pole.
- From the north side, a counterclockwise angle is positive by definition.
- What initiates the wind? Well, primitive answers were not so wrong...
- Why "m" for the slope of a linear function y = m x + b ? [English textbooks]
- The diamond mark on US tape measures corresponds to 8/5 of a foot.
- Naming the largest possible number, in n keystrokes or less (Excel syntax).
- The "odds in favor" of poker hands: A popular way to express probabilities.
- Reverse number sequence(s) on the verso of a book's title page.
- Living species: About 1400 000 have been named, but there are many more.
- Dimes and pennies: The masses of all current US coins.
- Pound of pennies: The dollar equivalent of a pound of pennies is increasing!
- Nickels per gallon: Packing as much as 5252.5523 coins per gallon of space.
- The volume of the Grand Canyon would be 2 cm (3/4") over the entire Earth.
- The Oldest City in the World: Damascus or Jericho?
- USA (States & Territories): Postal and area codes, capitals, statehoods, etc.
- Keyboard and modifier keys. Lesser-used functions require several keystrokes.
- Physical units: A very nice afterthought, with some unfortunate rough edges.
- Real analytical functions may present discontinuity cliffs in the complex realm.
- 68000 Assembly Programming: A primer without the help of an assembler.
- The clock frequency of your calculator: How to measure it with 0.1% accuracy.
- BASIC Programming. TI's built-in interpreted language is convenient but slow.
- Inventing Money: Brass in China, electrum in Lydia, gold and silver staters...
- Prices of Precious Metals: Current market values (Gold, Silver. Pt, Pd, Rh).
- Medieval sysyem: 12 deniers to a sol.
- Ancien Régime French monetary system.
- British coinage before decimalization.
- Exchange rates when the euro was born.
- Worldwide circulation of currencies.
- Counterfeit Coin Problem: In 3 weighings, find an odd object among 12, 13, 14.
- General Counterfeit Penny Problem: Find an odd object in the fewest weighings.
- Explicit tables for detecting one odd marble among 41, in 4 weighings.
- Find-a-birthday: Detect an odd marble among 365, in 6 weighings.
- Error-correcting codes for ternary numeration.
- If the counterfeit is known to be heavier, fewer weighings may be sufficient.
- Fossil calendars: 420 million years ago, a lunar month was only 9 short days.
- Julian Day Number (JDN) Counting days in the simplest of all calendars.
- The Week has not always been a period of seven days.
- Egyptian year of 365 days: Back to the same season after over 1500 years.
- Heliacal rising of Sirius: Sothic dating.
- Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
- The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
- Anno Domini: Counting roughly from the birth of Jesus Christ.
- The Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
- Counting the days between dates, with a simple formula for month numbers.
- Age of the Moon, based on a mean synodic month of 29.530588853 days.
- Easter Sunday is defined as the first Sunday after the Paschal full moon.
- The Muslim Calendar: The Islamic (Hijri) Calendar (AH = Anno Hegirae).
- The Jewish Calendar: An accurate lunisolar calendar, set down by Hillel II.
- Zoroastrian Calendar.
- The Zodiac: Zodiacal signs and constellations. Precession of equinoxes.
- The Iranian Calendar. Solar Hejri [SH] or Anno Persarum [AP].
- The Chinese Calendar.
- The Japanese Calendar.
- Mayan System(s): Haab (365), Tzolkin (260), Round (18980), Long Count.
- Indian Calendar: The Sun goes through a zodiacal sign in a solar month.
- Post-Gregorian Calendars: Painless improvements to the secular calendar.
- Geologic Time Scale: Beyond all calendars.
Roman Numerals (Archaic, Classic and Medieval) - Roman Numeration: The basics and the precise rules (including medieval ones).
- Larger Numbers, like 18034...
- Extending the Roman system.
- IIS (or HS) is for sesterce (originally, 2½ asses, "unus et unus et semis").
- Standard jokes.
- Limericks.
- Proper credit may not always be possible.
- Trick questions can be legitimate ones.
- Ignorance is bliss: Why not read all that mathematical stuff faster ?
- Silly answers to funny questions.
- Why did the chicken cross the road? Scientific and other explanations.
- Humorous or inspirational quotations by famous scientists and others.
- Famous Last Words: Proofs that the guesses of experts are just guesses.
- Famous anecdotes.
- Parodies, hoaxes, and practical jokes.
- Omnia vulnerant, ultima necat: The day of reckoning.
- Funny Units: A millihelen is the amount of beauty that launches one ship.
- Funny Prefixes: A lottagram is many grams; an electron weighs 0.91 lottogram.
- The Lamppost Theory: Look only where there's enough light to find anything.
- Anagrams: Rearranging letters may reveal hidden meanings ;-)
- Mnemonics: Remembering things and/or making fun of them.
- Acronyms: Funny ones and/or alternate interpretations of serious ones.
- Usenet Acronyms: If you can't beat them, join them (and HF, LOL).
- The equality symbol ( = ). The "equal sign" dates back to the 16th century.
- "Lines" among symbols: Vinculum, bar, solidus, virgule, slash, macron, etc.
- The infinity symbol ( ¥ ) introduced in 1655 by John Wallis (1616-1703).
- Transfinite numbers: Mathematical symbols for the multiple faces of infinity.
- Chrevron symbols: Intersection (highest below) or union (lowest above).
- Disjoint union. Square "U" or inverted p symbol.
- Blackboard bold: Doublestruck symbols are often used for sets of numbers.
- The integration sign ( ò ) introduced by Leibniz at the dawn of Calculus.
- The end-of-proof box (or tombstone) is called a halmos symbol (QED).
- Two "del" symbols: ¶ for partial derivatives, and Ñ for Hamilton's nabla.
- The Staff of Aesculapius: Medicine and the 13th zodiacal constellation.
- The Caduceus: Scepter of Hermes, symbol of commerce (not medicine).
- The Tetractys: Mystical Pythagorean symbol, "source of everflowing Nature".
- The Borromean Rings: Three interwoven rings which are pairwise separate.
- The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.
Unabridged Answers (monographs and complements): - Sagan's number: The number of stars, compared to earthly grains of sand.
- The Sand Reckoner: Archimedes fills the cosmos with grains of sand.
- About Zero.
- Wilson's Theorem.
- Counting Polyhedra: A tally of polyhedra with n faces and k edges.
Hall of Fame: - Numericana's list of distinguished Web authors in Science... Links to their sites.
- Giants of Science: Towering characters in the history of Science.
- Two legendary Solvay conferences defined modern physics, in 1911 and 1927.
- Physical Units: A tribute to the late physicist Richard P. Feynman (Nobel 1965).
- The many faces of Nicolas Bourbaki (b. January 14, 1935).
- Lucien Refleu (1920-2005). "Papa" of 600 mathematicians. [ In French ]
- Roger Apéry (1916-1994) and the irrationality of z(3).
- Hergé (1907-1983): Tintin and the Science of Jules Verne (1828-1905).
- Escutcheons of Science (Armorial): Coats of arms of illustrious scientists.
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