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Table of ContentsIt 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.)
- 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 gram-calorie (g-cal). Btu.
- Horsepower(s):
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 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.
- Euler-Mascheroni 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).
- Ramanujan-Soldner 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 (N-1)! 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) 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 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.
- 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.
Generalization to "q-card stud"...
- Probabilities of a straight flush
among 26 cards... or any other number of cards.
- 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 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 chessboard-type 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.
- 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?
- 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 three-dimensional case.
- Constant width in higher dimensions.
- Fourth dimension.
Difficult to visualize, but easy to consider.
- Volume of a hypersphere
in any number of dimensions. Hyper-surface area too!
- Hexahedra. The cube is not the only
polyhedron with 6 faces.
- Descartes-Euler Formula:
F-E+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.
- Mohr-Mascheroni 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 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.
- 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 second-order 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 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.
- 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 two-dimensional 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 B2-B+1 in base B).
- Standard Factorizations: n4 + 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 floating-point 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 p-th 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, ap-1 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.
- 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.
- Pseudoprimes to base a.
Poulet numbers are pseudoprimes to base 2.
- Weak pseudoprimes to base a :
Composite integers n which divide
(an-a).
- Strong pseudoprimes to base a
are less common than Euler pseudoprimes.
- Counting the bases to which
a composite number is a pseudoprime.
- 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.
- 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.
- 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?
- 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 5-card trick of Fitch Cheney:
Tell the 5th random card once 4 are shown.
- Generalizing the 5-card trick
and Devil's Poker...
- Grey Elephants in Denmark:
"Mental magic" for one-time classroom use...
- 1089: Subtract a 3-digit 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 (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.
- Extract a square root the old-fashioned 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 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.
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 (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.
- 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:
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.
- 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.
- Lorentz-Dirac equation
for the motion of a point charge is of third order.
- 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.
- 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.
- 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, Gay-Lussac, 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 sea-level 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 (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.
- 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.
- 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:
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 Titius-Bode 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 well-known 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.
- Post-Gregorian 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 (1616-1703).
- 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 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 Borromean Rings: Three interwoven rings which are
pairwise separate.
- 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):
- 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:
Up-to-date tally of polyhedra with n faces and k edges.
- Escutcheons of Science (Armorial):
Coats of arms of illustrious scientists.
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Note:
The above numbering may change, don't use it for reference purposes.
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Guest Authors:
Public-Domain Texts:
Hall of Fame:
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