A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox.
Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems, they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics.
In general, to use mathematics for solving a real-world problem, the first step is to construct a mathematical model of the problem. This involves abstraction from the details of the problem, and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one. After the problem has been solved in the world of mathematics, the solution must be translated back into the context of the original problem.
Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been a rich source of inspiration.
Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines.
Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, and the Poincaré conjecture.
Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. [1] Formal definitions and computer-checkable deductions are absolutely central to mathematical science.
Mathematics educators using problem solving for evaluation have an issue phrased by Alan H. Schoenfeld:
The same issue was faced by Sylvestre Lacroix almost two centuries earlier:
Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for the Cambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote:
An imaginary number is a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.
Gilles Personne de Roberval, French mathematician, was born at Roberval near Beauvais, France. His name was originally Gilles Personne or Gilles Personier, with Roberval the place of his birth.
Claude Gaspar Bachet Sieur de Méziriac was a French mathematician and poet born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy. He wrote Problèmes plaisans et délectables qui se font par les nombres, Les éléments arithmétiques, and a Latin translation of the Arithmetica of Diophantus. He also discovered means of solving indeterminate equations using continued fractions, a method of constructing magic squares, and a proof of Bézout's identity.
François Viète, Seigneur de la Bigotière, commonly known by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.
Bernard Frénicle de Bessy, was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for Des quarrez ou tables magiques, a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4. The Frénicle standard form, a standard representation of magic squares, is named after him. He solved many problems created by Fermat and also discovered the cube property of the number 1729, later referred to as a taxicab number. He is also remembered for his treatise Traité des triangles rectangles en nombres published (posthumously) in 1676 and reprinted in 1729.
Émile-Auguste Chartier, commonly known as Alain, was a French philosopher, journalist, and pacifist. He adopted his pseudonym as an allusion to the 15th-century Norman poet Alain Chartier.
Martin David Davis was an American mathematician and computer scientist who made significant contributions to the fields of computability theory and mathematical logic. He is best known for his work on Hilbert's tenth problem leading to the MRDP theorem. He also advanced the Post–Turing model and co-developed the Davis–Putnam–Logemann–Loveland (DPLL) algorithm, which is foundational for Boolean satisfiability solvers.
Yusuf Dia Pasha al-Khalidi was a prominent Ottoman politician who served as mayor of Jerusalem during several non-consecutive terms in the nineteenth and early twentieth centuries. Al Khalidi served as mayor of Jerusalem from the years 1870 to 1876, 1878 to 1879, and 1899 to 1906.
Clément Janequin was a French composer of the Renaissance. He was one of the most famous composers of popular chansons of the entire Renaissance, and along with Claudin de Sermisy, was hugely influential in the development of the Parisian chanson, especially the programmatic type. The wide spread of his fame was made possible by the concurrent development of music printing.
Georges Politzer was a French philosopher and Marxist theoretician of Hungarian Jewish origin, affectionately referred to by some as the "red-headed philosopher". He was a native of Oradea, a city in present-day Romania. He was murdered in the Holocaust.
Marc Vulson de la Colombière (†1658) or Sieur de la Colombière was a French heraldist, historian, poet, minion of the royal court. His name is sometimes spelt as Wulson and also as Volson.
Jules Vuillemin was a French philosopher, Professor of Philosophy of Knowledge at the prestigious Collège de France, in Paris, from 1962 to 1990, succeeding Maurice Merleau-Ponty, and Professor emeritus from 1991 to 2001. He was an Invited Professor at the Institute for Advanced Study, in Princeton, New Jersey (1968).
Georges Toussaint Léon Palante was a French philosopher and sociologist.
André Rivet was a French Huguenot theologian.
Edmund Stone was an autodidact Scottish mathematician who lived in London and primarily worked as an editor of mathematical and scientific texts and translator from French and Latin into English. He is especially known for his translations of Nicholas Bion's Mathematical Instruments and the Marquis de l'Hospital's Analyse des Infiniment Petits (1730), and for his New Mathematical Dictionary. Stone was celebrated for having risen from uneducated gardener's son to accomplished scholar.
Jehan Bretel (c.1210 – 1272) was a trouvère. Of his known oeuvre of probably 97 songs, 96 have survived. Judging by his contacts with other trouvères he was famous and popular. Seven works by other trouvères (Jehan de Grieviler, Jehan Erart, Jaques le Vinier, Colart le Boutellier, and Mahieu de Gant) are dedicated to Bretel and he was for a time the "Prince" of the Puy d'Arras.
Michel Maffesoli is a French sociologist.
A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers. Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions.
The Neirab steles are two 8th-century BC steles with Aramaic inscriptions found in 1891 in Al-Nayrab near Aleppo, Syria. They are currently in the Louvre. They were discovered in 1891 and acquired by Charles Simon Clermont-Ganneau for the Louvre on behalf of the Commission of the Corpus Inscriptionum Semiticarum. The steles are made of black basalt, and the inscriptions note that they were funerary steles. The inscriptions are known as KAI 225 and KAI 226.
The Foucault gyroscope was a gyroscope created by French physicist Léon Foucault in 1852, conceived as a follow-up experiment to his pendulum in order to further demonstrate the Earth's rotation.