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🔗 Mathematics

A sequence of six 9's occurs in the decimal representation of the number pi (π), starting at the 762nd decimal place. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the digits of π up to that point, recite them and end with "nine nine nine nine nine nine and so on", which seems to suggest that π is rational. The earliest known mention of this idea occurs in Douglas Hofstadter's 1985 book Metamagical Themas, where Hofstadter states

I myself once learned 380 digits of π, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9's, and then impishly say, "and so on!"

This sequence of six nines is sometimes called the "Feynman point", after physicist Richard Feynman, who allegedly stated this same idea in a lecture. It is not clear when, or even if, Feynman made such a statement, however; it is not mentioned in published biographies or in his autobiographies, and is unknown to his biographer, James Gleick.

🔗 Mathematics 🔗 Statistics

In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.

🔗 Mathematics

In number theory, a weird number is a natural number that is abundant but not semiperfect.

In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

🔗 Computer science 🔗 Mathematics

CORDIC (for COordinate Rotation DIgital Computer), also known as Volder's algorithm, or: Digit-by-digit method Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and FPGAs), as the only operations it requires are additions, subtractions, bitshift and lookup tables. As such, they all belong to the class of shift-and-add algorithms. In computer science, CORDIC is often used to implement floating-point arithmetic when the target platform lacks hardware multiply for cost or space reasons.

🔗 History 🔗 Mathematics 🔗 Books 🔗 Greece 🔗 History of Science

The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the Ostomachion and the Method of Mechanical Theorems) and the only surviving original Greek edition of his work On Floating Bodies. The first version of the compilation is believed to have been produced by Isidorus of Miletus, the architect of the geometrically complex Hagia Sophia cathedral in Constantinople, sometime around AD 530. The copy found in the palimpsest was created from this original, also in Constantinople, during the Macedonian Renaissance (c. AD 950), a time when mathematics in the capital was being revived by the former Greek Orthodox bishop of Thessaloniki Leo the Geometer, a cousin of the Patriarch.

Following the sack of Constantinople by Western crusaders in 1204, the manuscript was taken to an isolated Greek monastery in Palestine, possibly to protect it from occupying crusaders, who often equated Greek script with heresy against their Latin church and either burned or looted many such texts (including at least two other copies of Archimedes). The complex manuscript was not appreciated at this remote monastery and was soon overwritten (1229) with a religious text. In 1899, nine hundred years after it was written, the manuscript was still in the possession of the Greek church, and back in Istanbul, where it was catalogued by the Greek scholar Papadopoulos-Kerameus, attracting the attention of Johan Heiberg. Heiberg visited the church library and was allowed to make detailed photographs in 1906. Most of the original text was still visible, and Heiberg published it in 1915. In 1922 the manuscript went missing in the midst of the evacuation of the Greek Orthodox library in Istanbul, during a tumultuous period following the World War I. Concealed for over 70 years by a Western businessman, forged pictures were painted on top of some text to increase resale value. Unable to sell the book privately, in 1998 the businessman's daughter risked a public auction in New York contested by the Greek church; the U.S. court ruled for the auction, and the manuscript was purchased by an anonymous buyer (rumored to be Jeff Bezos). The texts under the forged pictures, and previously unreadable texts, were revealed by analyzing images produced by ultraviolet, infrared, visible and raking light, and X-ray.

All images and transcriptions are now freely available on the web at the Archimedes Digital Palimpsest under the Creative Commons License CC BY.

🔗 Mathematics

Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter. Example notation using the halo system can be seen below.

Various proposals have been made to adopt a linear system where all the notation would be on a single line without the use of superscripts or subscripts. Such a method would be useful for computing where representation of the halo system can be extremely difficult. However, a standard linear system has yet to emerge.

🔗 Biography 🔗 Mathematics 🔗 Astronomy 🔗 Japan 🔗 Japan/Science and technology 🔗 Japan/Biography

Seki Takakazu (関 孝和, 1642 – December 5, 1708), also known as Seki Kōwa (関 孝和), was a Japanese mathematician and author of the Edo period.

Seki laid foundations for the subsequent development of Japanese mathematics known as wasan; and he has been described as "Japan's Newton".

He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. A contemporary of German polymath mathematician and philosopher Gottfried Leibniz and British mathematician Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo period.

While it is not clear how much of the achievements of wasan are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe. For example, he is credited with the discovery of Bernoulli numbers. The resultant and determinant (the first in 1683, the complete version no later than 1710) are attributed to him. These achievements are astonishing, considering that Japanese mathematics before his appearance was at such a primitive stage. For example, comprehensive introduction of 13th century Chinese algebra was made as late as 1671, by Kazuyuki Sawaguchi.

🔗 United States 🔗 Mathematics 🔗 Education 🔗 United States/Massachusetts 🔗 Higher Education

Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard University, founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Honors Abstract Algebra (Math 55a) and Honors Real and Complex Analysis (Math 55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra.

🔗 Mathematics

In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist.

🔗 Biography 🔗 Mathematics 🔗 Physics 🔗 Biography/science and academia 🔗 Robotics 🔗 United Kingdom 🔗 Physics/Biographies 🔗 Christianity

Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-born American theoretical physicist and mathematician known for his work in quantum electrodynamics, solid-state physics, astronomy and nuclear engineering. He was professor emeritus in the Institute for Advanced Study in Princeton, a member of the Board of Visitors of Ralston College and a member of the Board of Sponsors of the Bulletin of the Atomic Scientists.

Dyson originated several concepts that bear his name, such as Dyson's transform, a fundamental technique in additive number theory, which he developed as part of his proof of Mann's theorem; the Dyson tree, a hypothetical genetically-engineered plant capable of growing in a comet; the Dyson series, a perturbative series where each term is represented by Feynman diagrams; the Dyson sphere, a thought experiment that attempts to explain how a space-faring civilization would meet its energy requirements with a hypothetical megastructure that completely encompasses a star and captures a large percentage of its power output; and Dyson's eternal intelligence, a means by which an immortal society of intelligent beings in an open universe could escape the prospect of the heat death of the universe by extending subjective time to infinity while expending only a finite amount of energy.

Dyson believed global warming is caused merely by increased carbon dioxide but that some of the effects of this are favourable and not taken into account by climate scientists, such as increased agricultural yield. He was skeptical about the simulation models used to predict climate change, arguing that political efforts to reduce causes of climate change distract from other global problems that should take priority. He also signed the World Climate Declaration that there "is no Climate Emergency".