Benford's law (W3)

grammarbook
Writing Numbers

Rule 1:

Spell out single-digit whole numbers. Use numerals for numbers greater than nine.

Examples:

• I want five copies.
• I want 10 copies.

...

inria
Mathematical Constants

Mathematical Constants
by Steven R. Finch
Dedicated to the memory of Philippe Flajolet

My book Mathematical Constants is now available for online purchase from Cambridge University Press (in the United Kingdom, North America and Australia). It is far more encompassing and detailed than my website ever was. It is also lovingly edited and beautifully produced - many thanks to Cambridge! - please support us in our publishing venture. Thank you. (If you wish, see several very kind reviews. You can also search the book via Amazon and Google by keyword.)

Here are errata and addenda to the book (last updated 8/4/2011), as well sample essays from the book about integer compositions, optimal stopping and Reuleaux triangles. Here also are recent supplementary materials, organized by topic:

Number Theory and Combinatorics

• Bipartite, k-colorable and k-colored graphs (6/5/2003)
• Transitive relations, topologies and partial orders (6/5/2003)
• Series-parallel networks (7/7/2003)
• Two asymptotic series (12/10/2003)
• Multiples and divisors (1/27/2004)
• Discrepancy and uniformity (2/19/2004)
• Unitarism and infinitarism (2/25/2004)
• Erdös' minimum overlap problem (7/2/2004)
• Planar graph growth constants (8/25/2004)
• Tauberian constants (8/30/2004)
• Integer partitions (9/22/2004)
• Class number theory (5/26/2005)
• Quadratic Dirichlet L-series (7/15/2005)
• Elliptic curves over Q (10/10/2005)
• Modular forms on SL2(Z) (12/28/2005)
• Chebyshev's bias (4/26/2006)
• Pattern-avoiding permutations (4/27/2006)
• Cyclic group orders (4/28/2006)
• Dedekind eta products (1/10/2007)
• Series involving arithmetic functions (1/24/2007)
• Riemann zeta moments (3/22/2007)
• Central binomial coefficients (3/28/2007)
• Fractional parts of Bernoulli numbers (4/10/2007)
• Products of consecutive-integer ratios (4/19/2007)
• Prime number theorem (4/27/2007)
• Mertens' formula (5/4/2007)
• Cyclotomic polynomials (3/31/2008)
• Minkowski-Alkauskas constant (5/6/2008)
• Two-colorings of positive integers (5/27/2008)
• Signum equations and extremal coefficients (2/7/2009)
• Monoids of natural numbers (3/17/2009)
• Primitive cusp forms (4/27/2009)
• Cubic and quartic characters (6/5/2009)
• Distribution of error terms (9/9/2009)

Inequalities and Approximation

• Hardy-Littlewood maximal inequalities (10/12/2003)
• Bessel function zeroes (10/23/2003)
• Nash's inequality (11/4/2003)
• Uncertainty inequalities (11/7/2003)
• Airy function zeroes (7/19/2004)
• Projections of minimal norm (11/30/2004)
• Bohr's inequality (12/5/2004)
• Moduli of continuity (12/22/2004)
• Quinn-Rand-Strogatz constant (4/27/2008)
• Tsirelson's constant (8/19/2008)
• Mathieu eigenvalues (10/7/2008)
• Thomas-Fermi model (10/15/2008)
• Prandtl-Blasius flow (11/12/2008)
• Lane-Ritter-Emden constants (11/30/2008)
• Brachistochrone problem (12/17/2008)
• Radiative transfer equations (12/30/2008)

Real and Complex Analysis

• Radii in geometric function theory (1/5/2004)
• Numerical radii of linear operators (4/5/2005)
• Coefficient estimates for univalent functions (12/15/2004)
• Planar harmonic mappings (3/29/2005)
• Constant of interpolation (4/5/2005)
• Dirichlet integral (5/15/2008)
• Dyadic UMD constants (3/24/2009)

Probability and Stochastic Processes

• Hammersley's path process (4/5/2004)
• Moments of sums (4/23/2004)
• Ornstein-Uhlenbeck process (5/15/2004)
• Zero crossings (5/19/2004)
• Variants of Brownian motion (6/12/2004)
• Shapes of binary trees (6/24/2004)
• Expected lifetimes and inradii (2/23/2005)
• Subcritical Galton-Watson trees (3/1/2007)
• Continued fraction transformation (5/23/2007)
• Continued fraction transformation. II (6/15/2007)
• Continued fraction transformation. III (7/1/2007)
• Continued fraction transformation. IV (7/18/2007)
• Lyapunov exponents (11/19/2007)
• Lyapunov exponents. II (11/20/2007)
• Lyapunov exponents. III (11/21/2007)
• Lyapunov exponents. IV (3/18/2008)
• Stars and watermelons (4/26/2008)
• Prophet inequalities (5/9/2008)
• Excursion durations (5/12/2008)
• Gambler's ruin (6/19/2008)
• Self-convolutions (3/2/2009)
• Newcomb-Benford law (8/17/2011)

Geometry and Topology

• Knots, links and tangles (8/8/2003)
• Convex lattice polygons (12/18/2003)
• Volumes of hyperbolic 3-manifolds (9/5/2004)
• Poisson-Voronoi tessellations (3/11/2005)
• Optimal escape paths (5/30/2005)
• Minkowski-Siegel mass constants (1/9/2005)
• Slicing problem (4/5/2005)
• Constant of Theodorus (4/9/2005)
• Nearest-neighbor graphs (4/25/2008)
• Random triangles (1/21/2010)
• Random triangles. II (2/9/2010)
• Random triangles. III (4/30/2010)
• Random triangles. IV (5/14/2010)
• Random triangles. V (12/22/2010)
• Random triangles. VI (1/7/2011)
• Colliding dice probabilities (7/29/2011)

Number (W3)

numberwatch
Number Watch

All about the scares, scams, junk, panics, and flummery cooked up by the media, politicians, bureaucrats, so-called scientists and others who try to confuse you with wrong numbers.

Working to combat Math Hysteria.

It is a pleasure to stand upon the shore, and to see ships tost upon the sea: a pleasure to stand in the window of the castle and to see the battle and the adventures thereof below: but no pleasure is comparable to the standing upon the vantage ground of truth ( a hill not to be commanded and where the air is always clear and serene), and to see the errors, and wanderings, and mists, and tempests, in the vale below.
From Of Truth, Francis Bacon

Contents

• Number of the month
• Book reviews
• Essays
• March of the zealots
• In praise of carbon
• Global warming as religion
• How we know they know they are lying
• Links
• Best of the blogs and the worst
• FAQs
• The complete lists: Cancer and Global Warming
• Recent bookmarked entries
• Contact Information
• Comments and Suggestions
• Sorry, wrong number! The first book of the web site
• The epidemiologists The NEW book of the web site
• Bits and pieces
• Statistical stuff happens (presentation)
• Guest papers
• Home page
• Join The Number Watch Forum
• Correspondence received
• The Laws
• Vocabulary
• And finally

Uni St Andrews
Prime numbers

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.

The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.

A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.

A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.

You can see more about these numbers in the History topics article Perfect numbers.

...

Bentley, Peter J.
The Book of Numbers
The Secret of Numbers and How They Changed the World

Ifrah, Georges
The Universal History of Computing
From the Abacus to the Quantum Computer

From the I Ching to AI, there has been tremendous human brainpower devoted to devising easier means of counting and thinking. Former math teacher Georges Ifrah has devoted his life to tracking down traces of our past calculating tools and reporting on them with charm and verve. The Universal History of Computing: From the Abacus to Quantum Computing gives a grand title to a grand subject, and Ifrah makes good on his promise of universality by leaping far back in time and spanning all the inhabited continents.

If his scope is vast, his stories and details are still engrossing. Readers will hang on the stories of 19th-century inventors converging on multiplication machines and other, more general "engines", and better understand the roots of biological and quantum computation. Ifrah has great respect for our ancestors and their work, and he transmits this feeling to his readers with humour and humility. His timelines, diagrams and concordance help readers unfamiliar with foreign concepts of numbers and computation to keep up with his narrative.

By the end, his slight bias against strong "artificial intelligence" shows, but he is careful to acknowledge the future's unforeseeable nature and suggest that we keep our minds open. How can we resist?

Rob Lightner

Nahin, Paul J. (Autor)
An Imaginary Tale (PSL Edition): The Story of 'I' [the Square Root of Minus One]

At the very beginning of his book on "i", the square root of minus one, Paul Nahin warns his readers: "An Imaginary Tale has a very strong historical component to it, but that does not mean it is a mathematical lightweight. But don't read too much into that either. It is *not* a scholarly tome meant to be read only by some mythical, elite group.... Large chunks of this book can, in fact, be read and understood by a high school senior who has paid attention to his or her teachers in the standard fare of pre-college courses. Still, it will be most accessible to the million or so who each year complete a college course in freshman calculus.... But when I need to do an integral, let me assure you I have not fallen to my knees in dumbstruck horror. And neither should you."

Nahin is a professor of electrical engineering at the University of New Hampshire; he has also written a number of science fiction short stories. His style is far more lively and humane than a mathematics textbook while covering much of the same ground. Readers will end up with a good sense for the mathematics of "i" and for its applications in physics and engineering.
Mary Ellen Curtin

Rogerson, Barnaby - DK
Rogerson’s Book of Numbers
Die Kultur der Zahlen
Von 1001 Nacht bis zu den 7 Weltwundern

London 2014, 11,5 x 19 cm, 288 S., zahlr. farb. und s/w-Abb., geb.

Die Kulturen der Menschheit stecken voller Zahlenmystik, dies gilt auch für die modernen Gesellschaften weiterhin. Dieses Buch beschäftigt sich mit den Zahlenordnungen von Tugenden, spirituellen Attributen, Göttern, Teufeln, heiligen Städten, Mächten, Kalendern, Helden, Heiligen, Ikonen und kulturellen Symbolen. Das Werk versammelt erstaunliche Informationen über die vielen Rollen, die Zahlen in der Folklore und in der Populärkultur, in der Musik und der Dichtung, im Aberglauben und in vielen Religionen und Kulturen dieser Welt spielen. Die Geschichten hinter den Zahlen öffnen sich von der Million bis zur Null: von der »Number of the Beast« (666) bis zu den 7 Todsünden, von den 12 Tierkreiszeichen bis zu den 4 Farben eines Kartenspiels, von Genghis Khans Stadt der 108 Türme bis zu Dantes Göttlicher Komödie mit ihren vielfältigen Zahlenbezügen, von den Millionen Inkarnationen des Buddha bis zu den Ursprüngen unserer angeblich unglückbringenden Zahl 13, die in China die Zahl 14 ist. Herausragend! (Text engl.)