
Table of ContentsIt is
better to know some of the questions than all of the answers.
James Grover Thurber
(18941961)
 All metric prefixes:
Current SI prefixes, obsolete prefixes, bogus prefixes...
 Prefixes for units of information.
(Multiples of the bit only.)
 Density one.
Relative and absolute density precisely defined.
 Acids
yielding a mole of H^{+} per liter are
normal (1N) solutions.
 Calories:
Thermochemical calorie, IT calorie and gramcalorie (gcal). Btu.
 Horsepower(s):
hp, electric horsepower, metric horsepower, boiler horsepower.
 The standard acceleration of gravity (1G)
has been 9.80665 m/s^{2} since 1901._{ }
Time:
 Tiny durations;
zeptosecond (zs, 10^{21}s) &
yoctosecond (ys, 10^{24}s).
 A jiffy is either a lightcm
or 10 ms (tempons and chronons are much 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
atomic seconds _{ }(»
Julian year of 365.25 solar days).
Length:
 The International inch
(1959) is 999998/1000000 of a US Survey inch.
 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 gallon (US), Imperial gallon (UK), dry gallon, etc.
 US bushel
and Winchester basic 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.
 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...
 The Beaufort scale
is now defined in terms of wind speed.
 The Saffir / Simpson scale for hurricanes.
 The Fujita scale for tornadoes.
 Primary conversion factors
between customary systems of units.
6 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 unit.
 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.
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.
 EulerMascheroni Constant
g :
The 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, then "+ i" is a step sideways to the left.
Exotic Mathematical Constants:^{ }
 Mertens constant: How the
sum of reciprocal primes (< n) differs from ln(ln n).
 RamanujanSoldner constant
(m): The 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).
Third Tier Mathematical Constants:^{ }
 Brun's Constant:
Stated standard uncertainty (s)
means a 99% level of ±3s
 Prévost's Constant:
The sum of the reciprocals of the Fibonacci numbers.
 Grossman's Constant: The initial
point which makes some recurrence converge.
 Ramanujan's Number:
exp(p Ö163) is almost an integer.
 Viswanath's constant is the 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 (N1)! different ways.
 How many Bachet squares?
A 1624 puzzle with the 16 court cards (AKQJ).
 Choice Numbers:
C(n,p) is the number of ways to choose p items among n.
 C(n+2,3) threescoop sundaes.
Several ways to count them (n flavors).
 C(n+p1,p) choices
of p items among n different types, allowing duplicates.
 How many new intersections
of the 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?
 Variance of a binomial distribution,
as obtained quickly from general principles.
 Standard deviation.
Two standard formulas to estimate it.
 InclusionExclusion:
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 7card stud.
Generalization to "qcard stud"...
 Probabilities of a straight flush
among 26 cards... or any other number of cards.
 The exact probabilities
in 5card, 6card, 7card, 8card and 9card stud.
 Rearrangements of CONSTANTINOPLE
so no two vowels are adjacent...
 Fourletter 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 getting 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? Programs and formulas.
 Squares and rectangles
in an N by N chessboardtype grid.
 Average distance
between two random points on a segment, a disk, a cube...
 Probability of a Set of Integers.
Looking for a "natural" definition.
 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.
 MarkovModulated 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?
 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 an ellipsoid
of revolution (oblate or prolate spheroid).
 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 expressions.
 For a regular polygon of given perimeter,
the more sides the larger the area.
 Curves of constant width:
Triangroller (Reuleaux Triangle), Pentagroller, etc.
 Irregular curves of constant width.
With or without any circular arcs.
 Solids of constant width.
The threedimensional case.
 Constant width in higher dimensions.
 Fourth dimension.
Difficult to visualize, but easy to consider.
 Volume of a hypersphere
in any number of dimensions. Hypersurface area too!
 Hexahedra. The cube is not the only
polyhedron with 6 faces.
 DescartesEuler Formula:
FE+V=2 but restrictions apply.
 Confocal Conics:
Ellipses and hyperbolae sharing the same pair of foci.
 Spiral of Archimedes:
Paper on a roll, or groove on a vinyl record.
 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.
 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.
 MohrMascheroni constructions
use the compass alone (no straightedge).
 Hexahedra. The cube is not the only
polyhedron with 6 faces.
 Enumeration of polyhedra:
Tally of polyhedra with n faces and k edges.
 The 5 Platonic solids:
Cartesian coordinates of the vertices.
 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.
 Naming Polyhedra: Not an easy task...
 Polytopes
are the ndimensional counterparts of 3D polyhedra.
 A simplex of touching unit spheres
may allow a center sphere to bulge out.
 Regular Antiprism:
Height and volume of a regular ngonal antiprism.
 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.
 Numerical functions:
Polynomial, rational, algebraic, transcendental, special...
 Solving triangles
using the Law of Sines, Law of Cosines, and Law of Tangents.
 Spherical trigonometry:
Dealing with 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.
 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.
 The length of the arch of a cycloid
is 4 times the diameter of the wheel.
 Integrating the cube root of the tangent function.
 Extrema of a function of 2 variables
require a secondorder condition.
 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.
 Ordinary differential equations. Several examples.
 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...
 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
 Permuting the terms of a series
may change its sum arbitrarily.
 Uniform convergence
implies properties for the limit of a sequence of functions.
 Cauchy sequences
help define real numbers rigorously.
 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 halfsum 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.
 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.
 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.
 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 twodimensional grid.
 The least common multiple
may be obtained without factoring into primes.
 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 B^{2}B+1 in base B).
 Standard Factorizations: n^{4} + 4
is never prime for
n > 1 because...
 Linear equation in integers:
Use Bezout's theorem and/or Euclid's algorithm.
 Lucky 7's. Any integer divides a number composed
of only 7's and 0's.
 The number of divisors of an integer.
 Perfect numbers and Mersenne primes.
 Binary and/or hexadecimal numeration
for floatingpoint numbers as well.
 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.
 Faulhaber's formula
gives the sum of the pth powers of the first n integers.
 Multiplicative functions
form a group under Dirichlet convolution.
 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, a^{p1} is 1 modulo p,
unless p divides a.
 Euler's totient function:
f(n) is the number of integers less than n coprime to n.
 FermatEuler 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
(a^{n } 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.
 Pseudoprimes to base a.
Poulet numbers are pseudoprimes to base 2.
 Weak pseudoprimes to base a :
Composite integers n which divide
(a^{n}a).
 Strong pseudoprimes to base a
are less common than Euler pseudoprimes.
 Counting the bases to which
a composite number is a pseudoprime.
 RabinMiller 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.
 A product of distinct primes
is a pseudoprime when all pairwise products are.
 Trial division may be used
to weed out the small prime factors of a number.
 Recursively defined sequences (over a
finite set) are ultimately periodic.
 Pollard's r (rho) factoring
method is based on the properties of such sequences.
 Dixon's method: Combine small square residues into
a solution of x^{ 2}
º y^{ 2}
 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.
 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.
 SevenEleven: 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 1415 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?
 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.
 The 5card trick of Fitch Cheney:
Tell the 5^{th} random card once 4 are shown.
 Generalizing the 5card trick
and Devil's Poker...
 Grey Elephants in Denmark:
"Mental magic" for onetime classroom use...
 1089: Subtract a 3digit number and its reverse,
then add this to its reverse...
 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 (b1) 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.
 Extract a square root the oldfashioned way.
 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!
 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 SternBrocot 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, SaintVincent, 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.
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.
 The names of operands
in common numerical operations.
 The names of "lines":
Vinculum, bar, solidus, virgule, slash.
 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.
 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 (18291891) not H.A. Lorentz.
 Special Relativity was first formulated
by H. Poincaré (Einstein a close second).
 The FletcherMillikan "oildrop" experiment
was not the sole work of Millikan.
 Collected errata about customary physical units.
 Spacecraft speeds up upon reentry
into the upper atmosphere.
 Lewis Carroll's monkey
climbs a rope over a pulley, with a counterweight.
 Hooke's Law:
Motion of a mass suspended to a spring.
 Speed of an electron
estimated with the Bohr model of the atom.
 Waves in a solid:
Pwaves (fastest), Swaves, Ewaves (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.
 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 produced by static charges.
 Electric capacity
is an electrostatic concept (adequate at low frequencies).
 Faraday's Law:
A varying magnetic flux induces an electric circulation.
 Electricity and Magnetism:
Historical paths to Maxwell's electromagnetism.
 Maxwell's equations
unify electricity and magnetism dynamically.
 Planar electromagnetic waves:
The simplest type of electromagnetic waves.
 Electromagnetic energy density and
the flux of the Poynting vector.
 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.
 Electric and magnetic dipoles:
Dipolar solutions of Maxwell's equations.
 LorentzDirac equation
for the motion of a point charge is of third order.
 Observers in motion:
A simpleminded 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 HarressSagnac 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 4vectors obey the Lorentz transform.
 Wave vector:
The 4dimensional 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.
 Elastic shock:
Energy transfer is v.dp.
(None is seen from the barycenter.)
 Cherenkov Effect:
When the speed of an electron exceed the celerity of light...
 The Ideal Gas Law from
Boyle, Mariotte, Charles, GayLussac, and Avogadro.
 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 sealevel and total mass above.
 Raising the Titanic, with (a lot of) hydrogen.
 The aeolipile: This
ancient steam engine demonstrates jet propulsion.
 Edward Somerset of Worcester (16011667):
Blueprint for a steam fountain.
 Denis Papin (16471714):
Pressure cooking and the first piston engine.
 Thomas Savery (c.16501715):
Two pistons and an independent boiler.
 Thomas Newcomen (16631729)
and John Calley: Atmospheric steam engine.
 NicolasJoseph Cugnot (17251804):
The first automobile (October 1769).
 James Watt (17361819):
Steam condenser and Watt governor.
 Richard Trevithick (17711833)
and the first railroad locomotives.
 Sadi Carnot (17961832):
Carnot's cycle and the theoretical efficiency limit.
 Sir Charles Parsons (18541931):
The modern steam turbine, born in 1884.
 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.
 Quantum Logic:
The surprising way quantum probabilities are obtained.
 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.
 Hamilton's analogy equates
the principles of Fermat and Maupertuis.
 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.
 Density operators
are quantum representations of imperfectly known states.
 Black Powder:
An ancient explosive, still used as a propellant (gunpowder).
 Predicting explosive reactions:
A useful but oversimplified rule of thumb.
 Enthalpy of Formation:
The tabulated data which gives energy balances.
 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?
 The Cosmological Principle:
The Universe is homogeneous and isotropic.
 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.
 LookBack 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:
Its gravitation is there, but what is it?
 The Cosmic Microwave Background (CMB):
Its spectrum and density.
 Solar radiation:
The Sun has radiated away about 0.03% of its mass.
 The TitiusBode Law:
A numerical pattern in solar orbits?
 Pluto
and other Kuiper Belt Objects.
 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 (in seconds),
given 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.
 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 !
 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...
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:
4 exact series and a dozen approximate formulas!
 Ramanujan II:
An awesome approximation from a mathematical genius (1914).
 Cantrell's Formula:
A modern attempt with an overall accuracy of 83 ppm.
 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.
 Rivera's formula gives the
perimeter of an ellipse with 104 ppm accuracy.
 Better accuracy from
Cantrell, building on his own previous formula
 C.K. Lu
rediscovers a wellknown exact expansion due to Euler (1773).
 Exact expressions for the
circumference of an ellipse: A summary.
 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 artefacts, sightings and other records...
 Geologic Time Scale.
 Oldest unsolved mathematical problem:
Are there any odd perfect numbers?
 Magnetic Field of the Earth:
The south side is near the geographic north pole.
 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.
 Inventing Money: Brass in China, electrum
in Lydia, gold and silver staters...
 Prices of Precious Metals:
Current market values (Gold, Silver. Pt, Pd, Rh).
 Exchange rates
on the day the euro was born.
 Worldwide circulation of major currencies.
 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.
 Easter Day
is defined as the first Sunday after the Paschal full moon.
 The Gregorian Calendar:
Multiples of 100 not divisible by 400 aren't leap years.
 Zoroastrian Calendar.
 The Zodiac:
Zodiacal signs and constellations. Precession of equinoxes.
 The Muslim Calendar:
The Islamic (Hijri) Calendar (AH = Anno Hegirae).
 The Jewish Calendar:
An accurate lunisolar calendar, set down by Hillel II.
 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.
 PostGregorian Calendars:
Painless improvements to the secular calendar.
 Standard jokes,
with due credit where credit is not due.
 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.
 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.
 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.
 The infinity symbol
( ¥ ) introduced in 1655 by John Wallis (16161703).
 Transfinite numbers:
Mathematical symbols for the multiple faces of infinity.
 Chrevron symbols:
Intersection (highest below) or union (lowest above).
 Blackboard bold: Doublestruck
symbols are often used for sets of numbers.
 The integration sign
( ò ) introduced by Leibniz at the dawn of Calculus.
 The endofproof box (or tombstone)
is called a halmos symbol (QED).
 Two "del" symbols:
¶ for partial derivatives, and
Ñ for Hamilton's nabla.
 The Borromean Rings: Three interwoven rings which are
pairwise separate.
 The TaiChi Mandala: The taiji
(YinYang) symbol was Bohr's coatofarms.
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):
 Surface Area of a General Ellipsoid:
Elementary only for ellipsoids of revolution.
 Roman numerals:
Archaic, classic or medieval (including "large" numbers too).
 Counterfeit Coin Problem:
Find an odd coin among n, in k weighings or less.
 Physical Units:
A tribute to the late physicist Richard P. Feynman (Nobel 1965).
 The many faces of Nicolas Bourbaki
(b. January 14, 1935).
 About Zero.
 Wilson's Theorem.
 Counting Polyhedra:
Uptodate tally of polyhedra with n faces and k edges.
 Escutcheons of Science (Armorial):
Coats of arms of illustrious scientists.
Note:
The above numbering may change, don't use it for reference purposes.

Guest Authors:
PublicDomain Texts:
Hall of Fame:


