Benito Mussolini

The Mathematics Portal

Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of spacetime (blue lines) due to the Sun's mass.

General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.

By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion. (Full article...)
$Image 2Logic studies valid forms of inference like modus ponens.Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } (and) or → {\displaystyle \to } (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (Full article...)$
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Logic studies valid forms of inference like modus ponens.

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like $\land$ (and) or $\to$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (Full article...)
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Rejewski, c. 1932

Marian Adam Rejewski (Polish: [ˈmarjan rɛˈjɛfskʲi] ; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine.

Over the next nearly seven years, Rejewski and fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski, working at the Polish General Staff's Cipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included the cryptologic card catalog and the cryptologic bomb. (Full article...)
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The weighing pans of this balance scale contain zero objects, divided into two equal groups.

In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer multiple of 2, specifically 0 × 2. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if $y$ is even then $y + x$ has the same parity as $x$ —indeed, $0 + x$ and $x$ always have the same parity.

Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all. (Full article...)
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The manipulations of the Rubik's Cube form the Rubik's Cube group.

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. (Full article...)
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In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. (Full article...)
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The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group $S̃ 3$

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers ( $..., - 2, - 1, 0, 1, 2, ...$ ) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in combinatorics and representation theory.

A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)
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Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He shared the 1965 Nobel Prize in Physics with Julian Schwinger and Shin'ichirō Tomonaga "for their fundamental work in quantum electrodynamics (QED), with deep-ploughing consequences for the physics of elementary particles".
He is also known for his work in the path integral formulation of quantum mechanics, the theory of the physics of the superfluidity of supercooled liquid helium, and the parton model. Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and remains widely used.

He assisted in the development of the atomic bomb during World War II and became known to the wider public in the 1980s as a member of the Rogers Commission, the panel that investigated the Space Shuttle Challenger disaster. Along with his work in theoretical physics, Feynman has been credited with having pioneered the field of quantum computing and introducing the concept of nanotechnology. He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
$Image 9Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).It is an example of an algorithm, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)$
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Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
It is an example of an algorithm, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, $21$ is the GCD of $252$ and $105$ (as $252 = 21 \times 12$ and $105 = 21 \times 5)$ , and the same number $21$ is also the GCD of $105$ and $252 - 105 = 147$ . Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, $21 = 5 \times 105 + (-2) \times 252$ ). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)
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Josiah Willard Gibbs (/ɡɪbz/; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.

In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics". (Full article...)
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Noether c. 1900–1910

Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental in mathematical physics. Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, David Hilbert and Felix Klein invited her to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent. (Full article...)
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Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.

In Euclidean plane geometry, the problem of Apollonius (also called Apollonius's problem or the Apollonian problem) is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, 'Tangencies'); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).

In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution uses methods not limited to straightedge and compass constructions. François Viète found a straightedge and compass solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius's method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius's problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN. (Full article...)
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Damage from Hurricane Katrina in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriate reserves, and design appropriate reinsurance and capital management strategies.

An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline is actuarial science.

While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in 17th-century studies of probability and annuities. Actuaries in the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems; actuaries use this knowledge to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks does not exceed the expected cost of those risks actualized. The steps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (Full article...)
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A statue depicting Zhang Heng at his tomb in Nanyang, Henan province, China

Zhang Heng (Chinese: 張衡; AD 78 – 139), courtesy name Pingzi, formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Eastern Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer, cartographer, ethnographer, artist, poet, philosopher, politician, and literary scholar.

Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in present-day Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...)
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Plots of logarithm functions, with three commonly used bases. The special points $log b b = 1$ are indicated by dotted lines, and all curves intersect in $log b 1 = 0$ .

In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of $1000$ to base $10$ is $3$ , because $1000$ is $10$ to the $3$ rd power: $1000 = 10 3 = 10 \times 10 \times 10$ . More generally, if $x = b y$ , then $y$ is the logarithm of $x$ to base $b$ , written $log b x = y$ , so $log 10 1000 = 3$ . As a single-variable function, the logarithm to base $b$ is the inverse of exponentiation with base $b$ .

The logarithm base $10$ is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number $e \approx 2.718$ as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base $2$ and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written $log x$ . (Full article...)

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Sieve of Eratosthenes

Credit: M.qrius

The sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified maximum value. It works by identifying the prime numbers in increasing order while removing from consideration composite numbers that are multiples of each prime. This animation shows the process of finding all primes no greater than 120. The algorithm begins by identifying 2 as the first prime number and then crossing out every multiple of 2 up to 120. The next available number, 3, is the next prime number, so then every multiple of 3 is crossed out. (In this version of the algorithm, 6 is not crossed out again since it was just identified as a multiple of 2. The same optimization is used for all subsequent steps of the process: given a prime

p

, only multiples no less than

p 2

are considered for crossing out, since any lower multiples must already have been identified as multiples of smaller primes. Larger multiples that just happen to already be crossed out—like 12 when considering multiples of 3—are crossed out again, because checking for such duplicates would impose an unnecessary speed penalty on any real-world implementation of the algorithm.) The next remaining number, 5, is the next prime, so its multiples get crossed out (starting with 25); and so on. The process continues until no more composite numbers could possibly be left in the list (i.e., when the square of the next prime exceeds the specified maximum). The remaining numbers (here starting with 11) are all prime. Note that this procedure is easily extended to find primes in any given arithmetic progression. One of several prime number sieves, this ancient algorithm was attributed to the Greek mathematician Eratosthenes (d. c. 194 BCE) by Nicomachus in his first-century (CE) work Introduction to Arithmetic. Other more modern sieves include the sieve of Sundaram (1934) and the sieve of Atkin (2003). The main benefit of sieve methods is the avoidance of costly primality tests (or, conversely, divisibility tests). Their main drawback is their restriction to specific ranges of numbers, which makes this type of method inappropriate for applications requiring very large prime numbers, such as public-key cryptography.

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Image 1
The integer number line, a set of infinitely many points with integer distances. According to the Erdős–Anning theorem, any such set lies on a line.

The Erdős–Anning theorem states that, whenever an infinite number of points in the plane all have integer distances, the points lie on a straight line. The same result holds in higher dimensional Euclidean spaces.
The theorem cannot be strengthened to give a finite bound on the number of points: there exist arbitrarily large finite sets of points that are not on a line and have integer distances.

The theorem is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. Erdős later supplied a simpler proof, which can also be used to check whether a point set forms an Erdős–Diophantine graph, an inextensible system of integer points with integer distances. The Erdős–Anning theorem inspired the Erdős–Ulam problem on the existence of dense point sets with rational distances. Some later publications credit the same result to Gustave Choquet and Germain Kreweras. It was independently rediscovered, with an application to the design of phased arrays, by B. Gleijeses in 1981. (Full article...)
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A Hasse diagram of divisibility relationships among the regular numbers up to 400. The vertical scale is logarithmic.

Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 60² = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular.

These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.
- In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their prime factors. This is a specific case of the more general $k$ -smooth numbers, the numbers that have no prime factor greater than $k$ .* In the study of Babylonian mathematics, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the sexagesimal (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
- In music theory, regular numbers occur in the ratios of tones in five-limit just intonation. In connection with music theory and related theories of architecture, these numbers have been called the harmonic whole numbers.
- In computer science, regular numbers are often called Hamming numbers, after Richard Hamming, who proposed the problem of finding computer algorithms for generating these numbers in ascending order. This problem has been used as a test case for functional programming.
(Full article...)
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Vedic Mathematics is a book written by Indian Shankaracharya Bharati Krishna Tirtha and first published in 1965. It contains a list of mathematical techniques which were falsely claimed to contain advanced mathematical knowledge. The book was posthumously published under its deceptive title by editor V. S. Agrawala, who noted in the foreword that the claim of Vedic origin, made by the original author and implied by the title, was unsupported.

Neither Krishna Tirtha nor Agrawala were able to produce sources, and scholars unanimously note it to be a compendium of methods for increasing the speed of elementary mathematical calculations sharing no overlap with historical mathematical developments during the Vedic period. Nonetheless, there has been a proliferation of publications in this area and multiple attempts to integrate the subject into mainstream education at the state level by right-wing Hindu nationalist governments. (Full article...)
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Hugo Steinhaus (1968)

Hugo Dyonizy Steinhaus (English: /ˈhjuːɡoʊ ˈstaɪnˌhaʊs/ HYOO-goh STYNE-howss, Polish: [ˈxuɡɔ ˈʂtajnxaws]; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.

Author of around 170 scientific articles and books, Steinhaus has left his legacy and contribution in many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founders of game theory and probability theory, which led to later development of more comprehensive approaches by other scholars. (Full article...)
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Advanced Placement (AP) Statistics (also known as AP Stats) is a college-level high school statistics course offered in the United States through the College Board's Advanced Placement program. This course is equivalent to a one semester, non-calculus-based introductory college statistics course and is normally offered to sophomores, juniors and seniors in high school.

One of the College Board's more recent additions, the AP Statistics exam was first administered in May 1997 to supplement the AP program's math offerings, which had previously consisted of only AP Calculus AB and BC. In the United States, enrollment in AP Statistics classes has increased at a higher rate than in any other AP class. (Full article...)
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In computational complexity theory, Yao's principle (also called Yao's minimax principle or Yao's lemma) relates the performance of randomized algorithms to deterministic (non-random) algorithms. It states that, for certain classes of algorithms, and certain measures of the performance of the algorithms, the following two quantities are equal:
- The optimal performance that can be obtained by a deterministic algorithm on a random input (its average-case complexity), for a probability distribution on inputs chosen to be as hard as possible and for an algorithm chosen to work as well as possible against that distribution
- The optimal performance that can be obtained by a random algorithm on a deterministic input (its expected complexity), for an algorithm chosen to have the best performance on its worst case inputs, and the worst case input to the algorithm
Yao's principle is often used to prove limitations on the performance of randomized algorithms, by finding a probability distribution on inputs that is difficult for deterministic algorithms, and inferring that randomized algorithms have the same limitation on their worst case performance.

This principle is named after Andrew Yao, who first proposed it in a 1977 paper. It is closely related to the minimax theorem in the theory of zero-sum games, and to the duality theory of linear programs. (Full article...)
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Roman copy of a Greek bust of Aristotle

Aristotle (Ancient Greek: Ἀριστοτέλης, romanized: Aristotélēs; 384–322 BC) was an ancient Greek philosopher and polymath. His writings span the natural sciences, philosophy, linguistics, economics, politics, psychology, and the arts. As the founder of the Peripatetic school of philosophy in the Lyceum in Athens, he began the wider Aristotelian tradition that followed, which set the groundwork for the development of modern science.

Little is known about Aristotle's life. He was born in the city of Stagira in Chalkidiki (northern Greece) during the Classical period. His father, Nicomachus, died when Aristotle was a child, and he was brought up by a guardian. At around eighteen years old, he joined Plato's Academy in Athens and remained there until the age of thirty seven (c. 347 BC). Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, tutored his son Alexander the Great beginning in 343 BC. He established a library in the Lyceum, which helped him to produce many of his hundreds of books on papyrus scrolls. (Full article...)
$Image 8A set of 20 points in a 10 × 10 grid, with no three points in a lineThe no-three-in-line problem in discrete geometry asks how many points can be placed in the n × n {\displaystyle n\times n} grid so that no three points lie on a straight line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".At most 2 n {\displaystyle 2n} points can be placed, because 2 n + 1 {\displaystyle 2n+1} points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with 2 n {\displaystyle 2n} points for every n {\displaystyle n} up to 70 {\displaystyle 70} , it is conjectured that fewer than 2 n {\displaystyle 2n} points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than 1.5 n {\displaystyle 1.5n} points, not 2 n {\displaystyle 2n} . (Full article...)$
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A set of 20 points in a 10 × 10 grid, with no three points in a line

The no-three-in-line problem in discrete geometry asks how many points can be placed in the $n\times n$ grid so that no three points lie on a straight line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".

At most $2n$ points can be placed, because $2n+1$ points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with $2n$ points for every $n$ up to $70$ , it is conjectured that fewer than $2n$ points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than $1.5n$ points, not $2n$ . (Full article...)
$Image 9The 13 possible strict weak orderings on a set of three elements {a, b, c}In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of n {\displaystyle n} elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race.The ordered Bell numbers were studied in the 19th century by Arthur Cayley and William Allen Whitworth. They are named after Eric Temple Bell, who wrote about the Bell numbers, which count the partitions of a set; the ordered Bell numbers count partitions that have been equipped with a total order. Their alternative name, the Fubini numbers, comes from a connection to Guido Fubini and Fubini's theorem on equivalent forms of multiple integrals. Because weak orderings have many names, ordered Bell numbers may also be called by those names, for instance as the numbers of preferential arrangements or the numbers of asymmetric generalized weak orders. (Full article...)$
Image 9
The 13 possible strict weak orderings on a set of three elements ${a, b, c$ }

In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of $n$ elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race.

The ordered Bell numbers were studied in the 19th century by Arthur Cayley and William Allen Whitworth. They are named after Eric Temple Bell, who wrote about the Bell numbers, which count the partitions of a set; the ordered Bell numbers count partitions that have been equipped with a total order. Their alternative name, the Fubini numbers, comes from a connection to Guido Fubini and Fubini's theorem on equivalent forms of multiple integrals. Because weak orderings have many names, ordered Bell numbers may also be called by those names, for instance as the numbers of preferential arrangements or the numbers of asymmetric generalized weak orders. (Full article...)
Image 10
The Math Myth: And Other STEM Delusions is a 2016 nonfiction book by Queens College political scientist Andrew Hacker analyzing and critiquing the United States educational system's teaching of mathematics as a linear progression towards more advanced fields. Based on a 2012 New York Times op-ed by Hacker titled "Is Algebra Necessary", Hacker argues that the teaching of advanced algebra, trigonometry, and calculus is not useful to the majority of students. He further claims that the requirement of advanced mathematics courses in secondary education contributes to dropout rates and impedes socioeconomically disadvantaged students from pursuing further education. Hacker critiques the Common Core system and American focus on STEM education in lieu of social sciences, arguing that the educational system should prioritize numeracy over pure mathematics education.

The Math Myth received broadly critical coverage from critics and mathematicians, some citing Hacker's arguments as "disingenuous" and contributing to an elitist attitude towards mathematics, with many citing a lack of exploration on mathematics in early childhood and primary education. Others praised Hacker's work, describing the book as offering a convincing critique of STEM education in the United States and empowering to students struggling in mathematics. (Full article...)
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The biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of a Johnson solid and of a composite polyhedron. It can be found in stereochemistry as a structural chemical compound called the bicapped trigonal prismatic molecular geometry. (Full article...)
$Image 12The Laves graphIn geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance 2 {\displaystyle {\sqrt {2}}} . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K4 crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space. (Full article...)$
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The Laves graph

In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance ${\sqrt {2}}$ . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.
H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K₄ crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space. (Full article...)

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... that Peter Nagy based his song "Profesor Indigo" on his former math teacher?
... that you should "hang a gone fishin' notice on your office door" before reading Geometry of Quantum States?
... that the identity of Cleo, who provided online answers to complex mathematics problems without showing any work, was revealed over a decade later in 2025?
... that the Prison Mathematics Project brought a professor from Italy to an American prison to celebrate Pi Day?
... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
... that the father of Lithuanian scientific forestry also directed theatre, organized a school, compiled maps, and wrote a dictionary of plants, a mathematics textbook, and a collection of poetry?
... that Caltech students called their calculus books "Tommy 1" and "Tommy 2"?
... that Carmel Naughton, having been told that girls were "stupid and couldn't do maths", sponsored a STEM scholarship fund?

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Did you know...

...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
...that the Pythagorean Theorem generalizes to any three similar shapes on the three sides of a right-angled triangle?
...that the orthocenter, circumcenter, centroid and the centre of the nine-point circle all lie on one line, the Euler line?
...that an arbitrary quadrilateral will tessellate?
...that it has not been proven whether or not every even integer greater than two can be expressed as the sum of two primes?
...that the sum of the first n odd numbers divided by the sum of the next n odd numbers is always equal to one third?
...that i to the power of i, where i is the square root of -1, is a real number?

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The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages
Image credit: Matt Crypto

Cryptography (or cryptology) is derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write". In modern times, it has become a branch of information theory, as the mathematical study of information and especially its transmission from place to place. The noted cryptographer Ron Rivest has observed that "cryptography is about communication in the presence of adversaries." It is a central contributor to several fields: information security and related issues, particularly, authentication, and access control. One of cryptography's primary purposes is hiding the meaning of messages, not usually the existence of such messages. In modern times, cryptography also contributes to computer science. Cryptography is central to the techniques used in computer and network security for such things as access control and information confidentiality. Cryptography is also used in many applications encountered in everyday life; the security of ATM cards, computer passwords, and electronic commerce all depend on cryptography. (Full article...)

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