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Final Answers
by Gérard P. Michon, Ph.D.

(unless otherwise stated) g.michon@att.net

© 2000 - 2005 Gérard Michon. This entire work (text and illustrations) is copyrighted; only short excerpts may be reproduced from it, according to applicable copyright laws.

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Table of Contents

It is better to know some of the questions than all of the answers.
James Grover Thurber  (1894-1961) 

Measurements and Units

  1. All metric prefixes: Current SI prefixes, obsolete prefixes, bogus prefixes...
  2. Prefixes for units of information. (Multiples of the bit only.)
  3. Density one. Relative and absolute density precisely defined.
  4. Acids yielding a mole of H+ per liter are normal (1N) solutions.
  5. Calories: Thermochemical calorie, IT calorie and gram-calorie (g-cal). Btu.
  6. Horsepower(s): hp, electric horsepower, metric horsepower, boiler horsepower.
  7. The standard acceleration of gravity (1G) has been 9.80665 m/s2 since 1901. 
  8. Tiny durations; zeptosecond (zs, 10-21s) & yoctosecond (ys, 10-24s).
  9. A jiffy is either a light-cm or 10 ms (tempons and chronons are much shorter).
  10. The length of a second. Solar time, ephemeris time, atomic time.
  11. The length of a day. Solar day, atomic day, sidereal or Galilean day.
  12. Scientific year = 31557600 atomic seconds  (» Julian year of  365.25 solar days).
  13. The International inch (1959) is 999998/1000000 of a US Survey inch.
  14. Leagues: Land league, nautical league.
  15. Radius of the Earth and circumference at the Equator.
  16. Extreme units of length. The very large and the very small. 
    Surface Area:
  17. Acres, furlongs, chains and square inches... 
    Volume, Capacity:
  18. Capitalization of units. You only have a choice for the liter (or litre ).
  19. Drops or minims: Winchester, Imperial or metric. Teaspoons and ounces.
  20. Fluid ounces: American ounces (fl oz) are about 4% larger than British ones.
  21. Gallons galore: Winchester gallon (US), Imperial gallon (UK), dry gallon, etc.
  22. US bushel and Winchester basic units of capacity (dry = bushel, fluid = gallon).
  23. Kegs and barrels: A keg of beer is half a barrel, but not just any "barrel". 
    Mass, "Weight":
  24. Tiny units of mass. A hydrogen atom is about 1.66 yg.
  25. Technical units of mass. The slug and the hyl.
  26. A talent was the mass of a cubic foot of water.
  27. Tons: short ton, long ton (displacement ton), metric ton (tonne), assay ton, etc.
    • Other tons: Energy (kiloton, toe, tce), cooling power, thrust, speed...

    Scales and Ratings: Measuring without Units

  28. The Beaufort scale  is now defined in terms of wind speed.
  29. The Saffir / Simpson scale  for hurricanes.
  30. The Fujita scale  for tornadoes.

    Numerical Constants

  31. Primary conversion factors  between customary systems of units.
    6 Basic Dimensionful Physical Constants  (Proleptic SI)
  32. Speed of Light in a Vacuum (Einstein's Constant):   c = 299792458 m/s.
  33. Magnetic Permeability of the Vacuum: An exact value defining the ampere unit.
  34. Planck's constant:  The ratio of a photon's energy to its frequency.
  35. Boltzmann's constant:  Relating temperature to energy.
  36. Avogadro's number:  The number of things per mole of stuff.
  37. Mechanical Equivalent of Light (683 lm/W at 540 THz) defines the candela.
    Fundamental Mathematical Constants: 
  38. 0:  Zero is the most fundamental and most misunderstood of all numbers.
  39. 1 and -1:  The unit numbers.
  40. p ("Pi"): The ratio of the circumference of a circle to its diameter.
  41. Ö2: The diagonal of a square of unit side.  Pythagoras' Constant.
  42. f: The diagonal of a regular pentagon of unit side.  The Golden Number.
  43. Euler's  e:  The base of the exponential function which equals its own derivative.
  44. ln(2):  The alternating sum of the reciprocals of the integers.
  45. Euler-Mascheroni Constant  g :  The limit of   [1 + 1/2 + 1/3 +...+ 1/n] - ln(n).
  46. Catalan's Constant  G :  The alternating sum of the reciprocal odd squares.
  47. Apéry's Constant  z(3) :  The sum of the reciprocals of the perfect cubes.
  48. Imaginary  i:  If "+1" is a step forward, then "+ i" is a step sideways to the left.
    Exotic Mathematical Constants: 
  49. Mertens constant:  How the sum of reciprocal primes (< n) differs from ln(ln n).
  50. Ramanujan-Soldner constant (m):  The positive root of the logarithmic integral.
  51. The Omega constant:  W(1) is the solution of the equation   x exp(x) = 1.
  52. Feigenbaum constant (d) and the related reduction parameter (a).
    Third Tier Mathematical Constants: 
  53. Brun's Constant:  Stated standard uncertainty  (s)  means a 99% level of  ±3s
  54. Prévost's Constant:  The sum of the reciprocals of the Fibonacci numbers.
  55. Grossman's Constant:  The initial point which makes some recurrence converge.
  56. Ramanujan's Number:   exp(p Ö163)   is almost an integer.
  57. Viswanath's constant is the mean growth in random additions and subtractions.

    Counting, Combinatorics, Probability

  58. Always change your first guess if you're always told another choice is bad.
  59. The Three Prisoner Problem predated Monty Hall and Marilyn by decades.
  60. Seating N children at a round table in (N-1)! different ways.
  61. How many Bachet squares?  A 1624 puzzle with the 16 court cards  (AKQJ).
  62. Choice Numbers: C(n,p) is the number of ways to choose p items among n.
  63. C(n+2,3) three-scoop sundaes. Several ways to count them (n flavors).
  64. C(n+p-1,p) choices of p items among n different types, allowing duplicates.
  65. How many new intersections of the straight lines defined by n random points.
  66. Face cards. The probability of getting a pair of face cards is less than 5%.
  67. Homework Central: Aces in 4 piles, bad ICs, airline overbooking.
  68. Binomial distribution. Defective units in a sample of 200.
  69. Siblings with the same birthday. What are the odds in a family of 5?
  70. Variance of a binomial distribution, as obtained quickly from general principles.
  71. Standard deviation. Two standard formulas to estimate it.
  72. Inclusion-Exclusion: One approach to the probability of a union of 3 events.
  73. The "odds in favor" of poker hands: A popular way to express probabilities.
  74. Probabilities of a straight flush in 7-card stud. Generalization to "q-card stud"...
  75. Probabilities of a straight flush among 26 cards... or any other number of cards.
  76. The exact probabilities in 5-card, 6-card, 7-card, 8-card and 9-card stud.
  77. Rearrangements of  CONSTANTINOPLE  so no two vowels are adjacent...
  78. Four-letter words (!) from  POSSESSES:  Counting with generating functions.
  79. How many positive integers below 1000000  have their digits add up to 19?
  80. Polynacci Numbers:  Flipping a coin n times without getting p tails in a row...
  81. 252 decreasing sequences of 5 digits (2002 nonincreasing ones).
  82. How many ways are there to make change for a dollar?  Programs and formulas.
  83. Squares and rectangles in an N by N chessboard-type grid.
  84. Average distance between two random points on a segment, a disk, a cube...
  85. Probability of a Set of Integers. Looking for a "natural" definition.

    Stochastic Processes & Stochastic Models

  86. Poisson Processes: Random arrivals happening at a constant rate (in Bq).
  87. Simulating a poisson process is easy with a uniform random number generator.
  88. Markov Processes: When only the present influences the future...
  89. The Erlang B Formula assumes callers don't try again after a busy signal.
  90. Markov-Modulated Poisson Processes may look like Poisson processes.

    "Utility" and Decision Analysis

  91. The Utility Function: A dollar earned is usually worth less than a dollar lost.
  92. Saint Petersburg Paradox: What would you pay to play the Petersburg game?

    Geometry and Topology   (for Polyhedra page, see below)

  93. Center of an arc determined with straightedge and compass.
  94. Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
  95. Special points in a triangle. Euler's line and Euler's circle.
  96. Elliptic arc: Length of the arc of an ellipse between two points.
  97. Perimeter of an ellipse. Exact formulas and simple ones.
  98. Surface area of an ellipsoid of revolution (oblate or prolate spheroid).
  99. Surface of an ellipse.
  100. Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
  101. Volume of an ellipsoid [spheroid].
  102. Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
  103. Focal point of a parabola. y = x 2 / 4f (where f is the focal distance).
  104. Parabolic telescope: The path from infinity to focus is constant.
  105. Make a cube go through a hole in a smaller cube.
  106. Octagon: The relation between side and diameter.
  107. Constructible regular polygons  and constructible angles (Gauss).
  108. Areas of regular polygons of unit side: General formula & special expressions.
  109. For a regular polygon of given perimeter, the more sides the larger the area.
  110. Curves of constant width: Triangroller (Reuleaux Triangle), Pentagroller, etc.
  111. Irregular curves of constant width. With or without any circular arcs.
  112. Solids of constant width. The three-dimensional case.
  113. Constant width in higher dimensions.
  114. Fourth dimension. Difficult to visualize, but easy to consider.
  115. Volume of a hypersphere in any number of dimensions. Hyper-surface area too!
  116. Hexahedra. The cube is not the only polyhedron with 6 faces.
  117. Descartes-Euler Formula: F-E+V=2 but restrictions apply.

    Planar Curves:

  118. Confocal Conics:  Ellipses and hyperbolae sharing the same pair of  foci.
  119. Spiral of Archimedes:  Paper on a roll, or groove on a vinyl record.
  120. Witch of Agnesi.  How the versiera (Agnesi's cubic) got a weird name.
  121. Folium of Descartes.
  122. Lemniscate of Bernoulli:  The shape of the infinity symbol is a quartic curve.
  123. Along a Cassini oval, the product of the distances to the two foci is constant.
  124. Limaçons of Pascal:  The cardioid  (unit epicycloid) is a special case.
  125. On a Cartesian oval, the weighted average distance to two poles is constant.
  126. Bézier curves  are algebraic splines.  The cubic type is the most popular.
  127. Piecewise circular curves:  The traditional way to specify curved forms.
  128. Intrinsic equation  [curvature as a function of arc length]  may include  spikes.
  129. The quadratrix (or trisectrix) of Hippias can square the circle and trisect angles.
  130. The parabola  is a curve that's  constructible  with straightedge and compass.
  131. Mohr-Mascheroni constructions  use the compass alone  (no straightedge).

    Polyhedra (3D), Polychora (4D), Polytopes (nD)

  132. Hexahedra. The cube is not the only polyhedron with 6 faces.
  133. Enumeration of polyhedra: Tally of polyhedra with n faces and k edges.
  134. The 5 Platonic solids: Cartesian coordinates of the vertices.
  135. Some special polyhedra may have a traditional (mnemonic) name.
  136. Polyhedra in certain families are named after one of their prominent polygons.
  137. Deltahedra have equilateral triangular faces. Only 8 deltahedra are convex.
  138. Naming Polyhedra: Not an easy task...
  139. Polytopes are the n-dimensional counterparts of 3-D polyhedra.
  140. A simplex of touching unit spheres may allow a center sphere to bulge out.
  141. Regular Antiprism:  Height and volume of a regular n-gonal antiprism.


  142. Factorial zero is 1, so is an empty product; an empty sum is 0.
  143. Anything raised to the power of 0 is equal to 1, including 0 to the power of 0.
  144. Idiot's Guide to Complex Numbers.
  145. Using the Golden Ratio (f) to express the 5 [complex] fifth roots of unity.
  146. "Multivalued" functions are functions defined over a Riemann surface.
  147. Square roots are inherently ambiguous for negative or complex numbers.
  148. The difference of two numbers, given their sum and their product.
  149. Symmetric polynomials of 3 variables: Obtain the value of one from 3 others.
  150. Geometric progression of 6 terms. Sum is 14, sum of squares is 133.
  151. Quartic equation involved in the classic "Ladders in an Alley" problem.

    Trigonometry, Elementary Functions, Special Functions

  152. Numerical functions: Polynomial, rational, algebraic, transcendental, special...
  153. Solving triangles using the Law of Sines, Law of Cosines, and Law of Tangents.
  154. Spherical trigonometry: Dealing with triangles drawn on the surface of a sphere.
  155. Sum of tangents of two half angles, in terms of sums of sines and cosines.
  156. The absolute value of the sine of a complex number.
  157. Exact solutions to transcendental equations.
  158. All positive rationals (and their square roots) as trigonometric functions of zero!
  159. The sine function: How to compute it numerically.
  160. Chebyshev economization saves billions of operations on routine computations.
  161. The Gamma function: Its definition(s) properties and values.
  162. Lambert's W function is used to solve practical transcendental equations.


  163. Derivative: Usually, the slope of a function, but there's a more abstract approach.
  164. Integration: The Fundamental Theorem of Calculus.
  165. 0 to 60 mph in 4.59 s, may not always mean 201.96 feet.
  166. Integration by parts:  Reducing an integral to another one.
  167. Length of a parabolic arc.
  168. The length of the arch of a cycloid is 4 times the diameter of the wheel.
  169. Integrating the cube root of the tangent function.
  170. Extrema of a function of 2 variables require a second-order condition.
  171. Changing inclination to a particle moving along a parabola.
  172. Algebraic area of a "figure 8" may be the sum or the difference of its lobes.
  173. Area surrounded by an oriented planar loop  which  may  intersect itself.
  174. Ordinary differential equations. Several examples.
  175. Linear differential equations of higher order and/or in several variables.
  176. Theory of Distributions:  Convolution products and their usage.
  177. Laplace Transforms: The Operational Calculus of Oliver Heaviside.
  178. Integrability of a function and of its absolute value.
  179. Analytic functions of a linear operator; defining  f (D) when D is d/dx...

    Differential Forms  &  Vector Calculus

  180. Generalizing the  fundamental theorem of calculus.
  181. The surface of a loop  is a vector determining its apparent area in any direction.
  182. Practical identities  of vector calculus

    Analysis, Convergence, Series, Complex Analysis

  183. Permuting the terms of a series may change its sum arbitrarily.
  184. Uniform convergence implies properties for the limit of a sequence of functions.
  185. Cauchy sequences help define real numbers rigorously.
  186. Defining integrals: Cauchy, Riemann, Darboux, Lebesgue.
  187. Cauchy principal value of an integral.
  188. Fourier series. A simple example.
  189. Infinite sums may sometimes be evaluated with Fourier Series.
  190. A double sum is often the product of two sums, which may be Fourier series.
  191. At a jump, the sum of a Fourier series is the half-sum of its left and right limits.
  192. Gibbs phenomenon; 9% overshoot of partial Fourier series near a jump.
  193. Method of Froebenius about a regular singularity of a differential equation.
  194. Laurent series of a function about one of its poles.
  195. Cauchy's Residue Theorem is helpful to compute difficult definite integrals.

    Set Theory and Logic

  196. The Barber's Dilemma. Not a paradox if analyzed properly.
  197. What is infinity? More than a pretty symbol (¥).
  198. There are more real than rational numbers. Cantor's argument.
  199. The axioms of set theory: Fundamental axioms and the Axiom of Choice.
  200. A set is smaller than its powerset:  A simple proof applies to all sets.
  201. Transfinite cardinals, transfinite ordinals: Two different kinds of infinite numbers.
  202. Surreal Numbers:  These include reals, transfinite ordinals, infinitesimals & more.
  203. Numbers:  From integers to surreals.  From reals to quaternions and  beyond...

    Number Theory & Numeration

  204. The number 1 is not prime, as definitions are chosen to make theorems simple.
  205. Composite numbers are not prime, but the converse need not be true...
  206. Two prime numbers whose sum is equal to their product.
  207. Gaussian integers:  Factoring into primes on a two-dimensional grid.
  208. The least common multiple may be obtained without factoring into primes.
  209. Modular Arithmetic may be used to find the last digit(s) of very large numbers.
  210. Powers of ten expressed as products of two factors  without zero digits.
  211. Divisibility by 7, 13, and 91 (or by B2-B+1 in base B).
  212. Standard Factorizations:   n4 + 4   is never prime for   n > 1   because...
  213. Linear equation in integers: Use Bezout's theorem and/or Euclid's algorithm.
  214. Lucky 7's. Any integer divides a number composed of only 7's and 0's.
  215. The number of divisors of an integer.
  216. Perfect numbers and Mersenne primes.
  217. Binary and/or hexadecimal numeration for floating-point numbers as well.
  218. Fast exponentiation by repeated squaring.
  219. Partition function. How many collections of positive integers add up to 15?
  220. A Lucas sequence whose oscillations never carry it back to -1.
  221. Faulhaber's formula gives the sum of the p-th powers of the first n integers.
  222. Multiplicative functions form a group under Dirichlet convolution.

    Modular Arithmetic

  223. Chinese Remainder Theorem:  How remainders define an integer (within limits).
  224. Modular arithmetic: The algebra of congruences, formally introduced by Gauss.
  225. Fermat's little theorem:   For any prime p, ap-1 is 1 modulo p, unless p divides a.
  226. Euler's totient functionf(n) is the number of integers less than n coprime to n.
  227. Fermat-Euler theorem:  If  a is coprime to n,  then a to the  f(n)  is 1 modulo n.
  228. Carmichael's reduced totient function (l) : A very special divisor of the totient.
  229. 91 is a pseudoprime to half of the bases coprime to itself.
  230. Carmichael Numbers:  An absolute pseudoprime  n divides  (an - a)  for any a.
  231. Chernik's Carmichael numbers:  3 prime factors   (6k+1)(12k+1)(18k+1).
  232. Large Carmichael numbers may be obtained in various ways.
  233. Conjecture: Any odd number coprime to its totient has a Carmichael multiple.


  234. Pseudoprimes to base aPoulet numbers  are pseudoprimes to base 2.
  235. Weak pseudoprimes to base a :  Composite integers  n  which divide  (an-a).
  236. Strong pseudoprimes to base a  are less common than Euler pseudoprimes.
  237. Counting the bases  to which a composite number is a pseudoprime.
  238. Rabin-Miller Test:  An efficient and trustworthy  stochastic  primality test.
  239. The product of 3 primes  is a pseudoprime when all  pairwise  products are.
  240. Wieferich primes  are scarce but there are (probably) infinitely many of them.
  241. A product of distinct primes  is a pseudoprime when all pairwise products are.

    Factoring into Primes

  242. Trial division  may be used to weed out the small prime factors of a number.
  243. Recursively defined sequences  (over a  finite  set)  are  ultimately periodic.
  244. Pollard's r (rho) factoring method  is based on the properties of such sequences.
  245. Dixon's method:  Combine small square residues into a solution of   x 2 º y 2

    Continued Fractions

  246. What is a continued fraction?  Example:  The expansion of p.
  247. The convergents of a number are its best rational approximations.
  248. Large partial quotients allow very precise approximations.
  249. Regular patterns in the continued fractions of some irrational numbers.
  250. For almost all numbers, partial quotients are ≥ k with probability  lg(1+1/k).
  251. Elementary operations on continued fractions.
  252. Expanding functions as continued fractions.

    Recreational Mathematics

  253. Counterfeit Coin Problem: In 3 weighings, find an odd object among 12, 13, 14.
  254. General Counterfeit Penny Problem: Find an odd object in the fewest weighings.
  255. Seven-Eleven: Four prices with a sum and product both equal to 7.11.
  256. Equating a right angle and an obtuse angle, with a clever false proof.
  257. Choosing a raise: Trust common sense, beware of  fallacious accounting.
  258. 3 men pay $30 for a $25 hotel room, the bellhop keeps $2... Is $1 missing?
  259. Chameleons: A situation shown unreachable because of an invariant quantity.
  260. Sam Loyd's 14-15 puzzle also involves an invariant quantity (and two orbits).
  261. Einstein's riddle: 5 distinct house colors, nationalities, drinks, smokes and pets.
  262. Numbering n pages of a book takes this many digits (formula).
  263. The Ferry Boat Problem (by Sam Loyd): To be or not to be ingenious?
  264. All digits once and only once: 48 possible sums (or 22 products).
  265. Crossing a bridge: 1 or 2 at a time, 4 people (U2), different paces, one flashlight!
  266. Managing supplies to reach an outpost 6 days away, carrying enough for 4 days.
  267. Go south, east, north and you're back... not necessarily to the North Pole!
  268. Icosapolis: Numbering a 5 by 4 grid so adjacent numbers differ by at least 4.
  269. Unusual mathematical boast for people born in 1806, 1892, or 1980.
  270. Puzzles for extra credit: From Chinese remainders to the Bookworm Classic.
  271. Simple geometrical dissection:  A proof of the Pythagorean theorem.
  272. Early bird saves time by walking to meet incoming chauffeur.
  273. Sharing a meal: A man has 2 loaves, the other has 3, a stranger has 5 coins.
  274. Fork in the road: Find the way to Heaven by asking only one question.
  275. Proverbial Numbers: Guess the words which commonly describe many numbers.
  276. Riddles: The Riddle of the Sphinx and other classics, old and new.

    Mathematical "Magic" Tricks

  277. The 5-card trick of Fitch Cheney:  Tell the 5th random card once 4 are shown.
  278. Generalizing the 5-card trick and Devil's Poker...
  279. Grey Elephants in Denmark: "Mental magic" for one-time classroom use...
  280. 1089:  Subtract a 3-digit number and its reverse, then add this to  its  reverse...

    Mathematical Games (Strategies)

  281. Dots and Boxes: The "Boxer's Puzzle" position of Sam Loyd.
  282. The Game of Nim: Remove items from one of several rows. Don't play last.
  283. Grundy numbers are defined for all positions in impartial games.
  284. Moore's Nim: Remove something from at most (b-1) rows. Play last.
  285. Normal Kayles: Knocking down one pin, or two adjacent ones, may split a row.
  286. Grundy's Game: Split a row into two unequal rows. Whoever can't move loses.
  287. Wythoff's Game: Remove counters either from one heap or equally from both.


  288. Extract a square root the old-fashioned way.
  289. Infinite alignment among infinitely many lattice points in the plane?  Nope.
  290. Infinite alignment in a lattice sequence with bounded gaps?  Almost...
  291. Large alignments in a lattice sequence with bounded gaps.  Yeah!
  292. Ford circles are nonintersecting circles touching the real line at rational points.
  293. Farey series:  The rationals from 0 to 1, with a bounded denominator.
  294. The Stern-Brocot tree  contains a single occurrence of every positive rational.
  295. Any positive rational  is a unique ratio of two consecutive Stern numbers.
  296. Pick's formula gives the area of a lattice polygon by counting lattice points.

    History, Nomenclature, Vocabulary, etc.

    History :
  297. Earliest mathematics on record. Before Thales was Euphorbe...
  298. Indian numeration became a positional system with the introduction of zero.
  299. Roman numerals are awkward for larger numbers.
  300. The invention of logarithms: John Napier, Bürgi, Briggs, Saint-Vincent, Euler.
  301. The earliest mechanical calculator(s), by W. Shickard (1623) or Pascal (1642).
  302. The Fahrenheit Scale: 100°F  was meant to be the normal body temperature.
    Nomenclature & Etymology :
  303. The origin of the word "algebra", and also that of "algorithm".
  304. The name of the avoirdupois system:  Borrowed from French in a pristine form.
  305. The names of operands in common numerical operations.
  306. The names of "lines": Vinculum, bar, solidus, virgule, slash.
  307. Long Division: Cultural differences in writing the details of a division process.
  308. Is a parallelogram a trapezoid? In a mathematical context [only?], yes it is...
  309. Naming polygons. Greek only please; use hendecagon not "undecagon".
  310. Chemical nomenclature: Basic sequential names (systematic and/or traditional).
  311. Fractional Prefixes: hemi (1/2), sesqui (3/2) or weirder hemipenta, hemisesqui...
    • Matches, phosphorus, and phosphorus sesquisulphide.
  312. Zillion. Naming large numbers.
  313. Zillionplex. Naming huge numbers.

    Setting the Record Straight

  314. The heliocentric Copernican system was known two millenia before Copernicus.
  315. The assistants of Galileo Galilei and the mythical experiment at the Tower of Pisa.
  316. Switching calendars: Newton was not born the year Galileo died.
  317. The Lorenz Gauge is an idea of Ludwig Lorenz (1829-1891) not H.A. Lorentz.
  318. Special Relativity was first formulated by H. Poincaré (Einstein a close second).
  319. The Fletcher-Millikan "oil-drop" experiment was not the sole work of Millikan.
  320. Collected errata  about customary physical units.


  321. Spacecraft speeds up upon reentry into the upper atmosphere.
  322. Lewis Carroll's monkey climbs a rope over a pulley, with a counterweight.
  323. Hooke's Law: Motion of a mass suspended to a spring.
  324. Speed of an electron estimated with the Bohr model of the atom.
  325. Waves in a solid: P-waves (fastest), S-waves, E-waves (thin rod), SAW...
  326. Rayleigh Wave: The quintessential surface acoustic wave (SAW).
  327. Hardest Stuff:  Diamond is no longer the hardest material known to science.
  328. Hardness  is an elusive  nonelastic  property, distinct from  stiffness.
  329. Hot summers, hot equator! The distance to the Sun is not the explanation.


  330. The vexing problem of units  is a thing of the past if you stick to SI units.
  331. The Lorentz force  on a test particle defines the local electromagnetic fields.
  332. Electrostatics:  The study of the electric field produced by static charges.
  333. Electric capacity  is an electrostatic concept  (adequate at low frequencies).
  334. Faraday's Law:  A varying magnetic flux  induces  an electric circulation.
  335. Electricity and Magnetism:  Historical paths to Maxwell's  electromagnetism.
  336. Maxwell's equations  unify electricity and magnetism dynamically.
  337. Planar electromagnetic waves:  The simplest type of electromagnetic waves.
  338. Electromagnetic energy density  and the flux of the Poynting vector.
  339. Electromagnetic potentials  are postulated to obey the  Lorenz gauge.
  340. Solutions to Maxwell's equations,  as  retarded  or  advanced  potentials.
  341. Electrodynamic fields  corresponding to  retarded  potentials.
  342. Electric and magnetic dipoles:  Dipolar solutions of Maxwell's equations.
  343. Lorentz-Dirac equation  for the motion of a point charge is of  third  order.


  344. Observers in motion:  A simple-minded derivation of the Lorentz Transform.
  345. Adding up velocities:  The combined speed can never be more than c.
  346. Fizeau's empirical relation  between refractive index  (n) and  Fresnel drag.
  347. The Harress-Sagnac effect  used to measure rotation with fiber optic cable.
  348. Combining relativistic speeds:  Using rapidity, the rule is transparent.
  349. Relative velocity of two photons:  Defined unless both have the same direction.
  350. Minkowski spacetime:  Coordinates of 4-vectors obey the Lorentz transform.
  351. Wave vector:  The 4-dimensional gradient of the phase describes propagation.
  352. Doppler shift:  The relativistic effect is not purely radial.
  353. Kinetic energy:  At low speed, the relativistic energy varies like  ½ mv 2.
  354. Photons and other massless particles:  Finite energy at speed  c.
  355. The de Broglie celerity  (u)  is inversely proportional to a particle's speed.
  356. Compton diffusion:  The result of collisions between photons and electrons.
  357. Elastic shock:  Energy transfer is  v.dp.  (None is seen from the barycenter.)
  358. Cherenkov Effect:  When the speed of an electron exceed the celerity of light...

    Physics of Gases and Fluids

  359. The Ideal Gas Law from Boyle, Mariotte, Charles, Gay-Lussac, and Avogadro.
  360. Viscosity is the ratio of a shear stress to the shear strain rate it induces.
  361. Permeability and permeance: Vapor barriers and porous materials.
  362. Resonant frequencies of air in a box.
  363. The Earth's atmosphere. Pressure at sea-level and total mass above.
  364. Raising the Titanic, with (a lot of) hydrogen.

    Steam Engines

  365. The aeolipile:  This ancient steam engine demonstrates jet propulsion.
  366. Edward Somerset of Worcester (1601-1667):  Blueprint for a steam fountain.
  367. Denis Papin (1647-1714):  Pressure cooking and the first piston engine.
  368. Thomas Savery (c.1650-1715):  Two pistons and an independent boiler.
  369. Thomas Newcomen (1663-1729) and John Calley:  Atmospheric steam engine.
  370. Nicolas-Joseph Cugnot (1725-1804):  The first automobile  (October 1769).
  371. James Watt (1736-1819):  Steam condenser and  Watt governor.
  372. Richard Trevithick (1771-1833) and the first railroad locomotives.
  373. Sadi Carnot (1796-1832):  Carnot's cycle and the theoretical  efficiency limit.
  374. Sir Charles Parsons (1854-1931):  The modern steam  turbine, born in 1884.

    Demons of Classical Physics

  375. Laplace's Demon:  Deducing past and future from a detailed snapshot.
  376. Maxwell's Demon:  Trading information for entropy.
  377. Shockley's Ideal Diode Equation:  Diodes don't violate the Second Law.
  378. Szilard's engine & Landauer's Principle: The thermodynamic cost of  forgetting.

    Quantum Mechanics

  379. Quantum Logic:  The surprising way quantum probabilities are obtained.
  380. The Measurement Dilemma:  What makes  Schrödinger's cat  so special?
  381. Matrix Mechanics:  Neither measurements nor matrices can be switched at will.
  382. Schrödinger's Equation:  A nonrelativistic quantum particle in a classical field.
  383. Hamilton's analogy equates the principles of Fermat and Maupertuis.
  384. Noether's Theorem:  Conservation laws express the symmetries of physics.
  385. Kets  are Hilbert vectors (their duals are bras) on which observables operate.
  386. Observables  are operators explicitely associated with physical quantities.
  387. Commutators are the quantities which determine  uncertainty relations.
  388. Density operators  are quantum representations of imperfectly known states.

    Ancient Recipes and Modern Chemistry

  389. Black Powder:  An ancient explosive, still used as a propellant (gunpowder).
  390. Predicting explosive reactions:  A useful but oversimplified rule of thumb.
  391. Enthalpy of Formation:  The tabulated data which gives energy balances.
  392. Inks:  India ink, atramentum, cinnabar (Chinese red HgS), iron gall ink, etc.
  393. Redox Reactions:  Oxidizers are reduced by accepting electrons...
  394. Gold ChemistryAqua regia ("Royal Water") dissolves gold and platinum.
  395. Who is the "father" of modern chemistry?

    Cosmology 101

  396. The Cosmological Principle: The Universe is homogeneous and isotropic.
  397. Cosmic redshift (z):  Light emitted in a Universe which was (1+z) times smaller.
  398. Hubble Law:  The relation between redshift and distance for comoving points.
  399. Omega (W): The ratio of the density of the Universe to the critical density.
  400. Look-Back Time:  The time ellapsed since observed light was emitted.
  401. Distance:  In a cosmological context, there are several flavors to the concept.
  402. Comoving points are reference points following the expansion of the universe.
  403. The Anthropic Principle: An obvious explanation which may not be the final one.
  404. Dark Matter: Its gravitation is there, but what is it?
  405. The Cosmic Microwave Background (CMB): Its spectrum and density.

    The Solar System

  406. Solar radiation:  The Sun has radiated away about 0.03% of its mass.
  407. The Titius-Bode Law: A numerical pattern in solar orbits?
  408. Pluto  and other  Kuiper Belt Objects.

    Practical Formulas

  409. Easy conversion between Fahrenheit and Celsius scales:  F+40  =  1.8 (C+40).
    Automotive :
  410. Car speed is proportional to tire diameter and engine rpm, divided by gear ratio.
  411. Car acceleration. Guessing the curve from standard data.
  412. "0 to 60 mph" time (in seconds), given vehicle mass and actual average power.
  413. Thrust  is the power to speed ratio (measuring speed along thrust direction).
  414. Power of an engine as a function of its size:  Rating internal combustion engines.
  415. Optimal gear ratio  to maximize top speed on a flat road  (no wind).
    Surface Areas :
  416. Heron's Formula (for the area of a triangle) is related to the Law of Cosines.
  417. Brahmagupta's Formula gives the area of a quadrilateral, inscribed or not.
  418. Bretschneider's Formula: Area of a quadrilateral of known sides and diagonals.
  419. Parabolic segment:  2/3 the area of a circumscribed parallelogram or triangle.
    Volumes :
  420. Content of a cylindrical tank (horizontal axis), given the height of the liquid in it.
  421. Volume of a spherical cap, or content of an elliptical vessel, given liquid height.
  422. Content of a cistern (cylindrical with elliptical ends), as a function of fluid height.
  423. Volume of a cylinder or prism, possibly with tilted [nonparallel] bases.
  424. Volume of a conical frustum:  Formerly a staple of elementary education...
  425. Volume of a sphere...  obtained by subtracting a cone from a cylinder !
  426. Volume of a wedge of a cone.
    Averages :
  427. Splitting a job evenly between two unlike workers.
  428. Splitting a job unevenly between two unlike workers.
  429. Alcohol solutions are rated by volume not by mass.
  430. Mixing solutions to obtain a predetermined intermediate rating.
  431. Special averages: harmonic (for speeds), geometric (for rates), etc.
  432. Mean Gregorian month: either 30.436875 days, or 30.458729474253406983...
    Geodesy and Astronomy :
  433. Distance to ocean horizon line is proportional to the square root of your altitude.
  434. Distance between two points on a great circle at the surface of the Earth.
  435. The figure of the Earth. Geodetic and geocentric latitudes.
  436. 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.

    Perimeter of an Ellipse

  437. Circumference of an ellipse: 4 exact series and a dozen approximate formulas!
  438. Ramanujan II:  An awesome approximation from a mathematical genius (1914).
  439. Cantrell's Formula:  A modern attempt with an overall accuracy of 83 ppm.
  440. Padé approximants  are used in a whole family of approximations...
  441. Improving Ramanujan II  over the whole range of eccentricities.
  442. The Arctangent Function as a component of several approximate formulas.
  443. Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
  444. Better accuracy from Cantrell, building on his own previous formula
  445. C.K. Lu  rediscovers a well-known exact expansion due to Euler (1773).
  446. Exact expressions for the circumference of an ellipse:  A summary.

    The Unexplained

  447. The Magnetic Field of the Earth.
  448. Life (1):  The mysteries of evolution.
  449. Life (2):  The origins of life on Earth.
  450. Life (3):  Does extraterrestrial life exist?  Is there intelligence out there?
  451. Nemesis: A distant companion to the Sun could explain extinction periodicity.
  452. Current Challenges to established dogma.
  453. Unexplained artefacts, sightings and other records...
  454. Geologic Time Scale.


  455. Oldest unsolved mathematical problem:  Are there any odd perfect numbers?
  456. Magnetic Field of the Earth: The south side is near the geographic north pole.
  457. What initiates the wind?  Well, primitive answers were not so wrong...
  458. Why "m" for the slope of a linear function  y = m x + b ? [English textbooks]
  459. The diamond mark on US tape measures corresponds to 8/5 of a foot.
  460. Naming the largest possible number, in n keystrokes or less (Excel syntax).
  461. The "odds in favor" of poker hands: A popular way to express probabilities.
  462. Reverse number sequence(s) on the verso of a book's title page.
  463. Living species: About 1400 000 have been named, but there are many more.
  464. Dimes and pennies: The masses of all current US coins.
  465. Pound of pennies: The dollar equivalent of a pound of pennies is increasing!
  466. Nickels per gallon: Packing as much as 5252.5523 coins per gallon of space.
  467. The volume of the Grand Canyon  would be 2 cm (3/4") over the entire Earth.
  468. The Oldest City in the World: Damascus or Jericho?
  469. USA (States & Territories): Postal and area codes, capitals, statehoods, etc.

    Money, Currency, Precious Metals

  470. Inventing Money: Brass in China, electrum in Lydia, gold and silver staters...
  471. Prices of Precious Metals:  Current market values (Gold, Silver. Pt, Pd, Rh).
  472. Exchange rates  on the day the  euro  was born.
  473. Worldwide circulation  of major currencies.

    Calendars & Chronology

  474. Fossil calendars: 420 million years ago, a lunar month was only 9 short days.
  475. Julian Day Number (JDN) Counting days in the simplest of all calendars.
  476. The Week has not always been a period of seven days.
  477. Egyptian year of 365 days: Back to the same season after over 1500 years.
  478. Heliacal rising of Sirius: Sothic dating.
  479. Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
  480. The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
  481. Anno Domini: Counting roughly from the birth of Jesus Christ.
  482. Easter Day is defined as the first Sunday after the Paschal full moon.
  483. The Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
  484. Zoroastrian Calendar.
  485. The Zodiac:  Zodiacal signs and constellations.  Precession of equinoxes.
  486. The Muslim Calendar:  The Islamic (Hijri) Calendar (AH = Anno Hegirae).
  487. The Jewish Calendar:  An accurate lunisolar calendar, set down by Hillel II.
  488. The Chinese Calendar.
  489. The Japanese Calendar.
  490. Mayan System(s)Haab (365), Tzolkin (260), Round (18980), Long Count.
  491. Indian Calendar:  The Sun goes through a zodiacal sign in a solar month.
  492. Post-Gregorian CalendarsPainless  improvements to the secular calendar.


  493. Standard jokes, with due credit where credit is not due.
  494. Silly answers to funny questions.
  495. Why did the chicken cross the road?  Scientific and other explanations.
  496. Humorous or inspirational quotations by famous scientists and others.
  497. Famous Last Words:  Proofs that the guesses of experts are just guesses.
  498. Famous anecdotes.
  499. Parodies, hoaxes, and practical jokes.
  500. Funny Units: A millihelen is the amount of beauty that launches one ship.
  501. Funny Prefixes: A lottagram is many grams; an electron weighs 0.91 lottogram.
  502. Anagrams: Rearranging letters may reveal hidden meanings ;-)
  503. Mnemonics: Remembering things and/or making fun of them.
  504. Acronyms: Funny ones and/or alternate interpretations of serious ones.
  505. Usenet Acronyms: If you can't beat them, join them (and HF, LOL).

    Scientific Symbols and Icons

  506. The equality symbol ( = ).  The "equal sign" dates back to the 16th century.
  507. The infinity symbol ( ¥ ) introduced in 1655 by John Wallis (1616-1703).
  508. Transfinite numbers:  Mathematical symbols for the multiple faces of infinity.
  509. Chrevron symbols:  Intersection (highest below)  or  union (lowest above).
  510. Blackboard boldDoublestruck  symbols are often used for sets of numbers.
  511. The integration sign ( ò ) introduced by Leibniz at the dawn of Calculus.
  512. The end-of-proof box (or tombstone) is called a halmos symbol  (QED).
  513. Two "del" symbols  for partial derivatives, and  Ñ  for Hamilton's nabla.
  514. The Borromean Rings: Three interwoven rings which are pairwise separate.
  515. The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.
    We felt the need to dedicate an entire page to some articles, or groups of articles. Here is the list of our...

    ... Unabridged Answers (monographs and complements):

  516. Surface Area of a General Ellipsoid: Elementary only for ellipsoids of revolution.
  517. Roman numerals: Archaic, classic or medieval (including "large" numbers too).
  518. Counterfeit Coin Problem:  Find an odd coin among n, in k weighings or less.
  519. Physical Units: A tribute to the late physicist Richard P. Feynman (Nobel 1965).
  520. The many faces of Nicolas Bourbaki  (b. January 14, 1935).
  521. About Zero.
  522. Wilson's Theorem.
  523. Counting Polyhedra: Up-to-date tally of polyhedra with n faces and k edges.
  524. Escutcheons of Science (Armorial):  Coats of arms of illustrious scientists.

Note: The above numbering may change, don't use it for reference purposes.

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