He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.
Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.
His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained “many technical errors of varying degree”; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the continuum hypothesis. The errors were nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the corrections. Like Albert Einstein, Hilbert had closest contacts with the Berlin Group whose leading founders had studied under Hilbert in Göttingen.
By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been Jewish or married to Jews.
*23 January 1862, Königsberg (now Kaliningrad, Russia) or Wehlau, Prussia
†14 February 1943, Göttingen, Nazi Germany
David Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).
Hilbert adopted and defended Georg Cantor’s set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century.
Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic.
In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (i.e., an associate professor). An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers.
Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen (“On the invariant properties of special binary forms, in particular the spherical harmonic functions”).
Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world.He remained there for the rest of his life.
Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally “report on numbers”). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.
He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.
Although Fibonacci’s Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century.
Fibonacci was a guest of Emperor Frederick II, who enjoyed mathematics and science.
In 1240, the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo) by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens.
Fibonacci is thought to have died between 1240 and 1250, in Pisa.
There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period.
Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis.
*c. 1170, Pisa, Republic of Pisa
†c. 1250 , Pisa, Republic of Pisa
Fibonacci also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano (‘Leonardo the Traveller from Pisa’), was an Italian mathematician from the Republic of Pisa, considered to be “the most talented Western mathematician of the Middle Ages”.
The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci (‘son of Bonacci’). However, even earlier in 1506 a notary of the Holy Roman Empire, Perizolo mentions Leonardo as “Lionardo Fibonacci”. He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.
Fibonacci travelled with his father as a young boy, and it was in Bugia where he was educated that he learned about the Hindu–Arabic numeral system.
Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic. He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system.
In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation), which popularized Hindu–Arabic numerals in Europe.
In the Liber Abaci, Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system. The manuscript book advocated numeration with the digits 0–9 and place value. The book showed the practical use and value of the new Hindu-Arabic numeral system by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications.
Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers.
Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father’s approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching.
Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein’s general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen.
In 1859, he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality.
His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca.
*17 September 1826, Breselenz, Kingdom of Hanover (modern-day Germany)
†20 July 1866, Selasca, Kingdom of Italy
Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.
Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.
During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics.
His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor’s knowledge.
In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family’s finances.
During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in Theology. However, once there, he began studying mathematics under Carl Friedrich Gauss .
He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as Euler and Jacobi.
In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan’s return to India, where he died in 1920 at the age of 32.
His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His “lost notebook”, containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.
A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess Namagiri Thayar. He once said, “An equation for me has no meaning unless it expresses a thought of God.”
*22 December 1887, Erode, Madras Presidency, British India
†26 April 1920, Kumbakonam, Madras Presidency, British India
Srinivasa Ramanujan was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.
Ramanujan initially developed his own mathematical research in isolation: according to Hans Eysenck: “He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered”.
Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing Ramanujan’s work as extraordinary, Hardy arranged for him to travel to Cambridge.
In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that “defeated me completely; I had never seen anything in the least like them before”, and some recently proven but highly advanced results.
During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.
Of his thousands of results, all but a dozen or two have now been proven correct. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas.
As late as 2012, researchers continued to discover that mere comments in his writings about “simple properties” and “similar outputs” for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.
In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Lexell when he collapsed and died from a brain hemorrhage.
Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler’s influence on mathematics: “Read Euler, read Euler, he is the master of us all. Carl Friedrich Gauss remarked: “The study of Euler’s works will remain the best school for the different fields of mathematics, and nothing else can replace it.”
Euler is credited for popularizing the Greek letter π (lowercase pi), as well as first employing the term f(x) to describe a function’s y-axis, the letter i to express the imaginary unit √-1, and the Greek letter Σ (capital sigma). He gave the current definition of the constant e, the base of the natural logarithm, still known as Euler’s number.
Euler was also the first practitioner of graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg). He became famous – among others – for solving the Basel Problem, after proving that the sum of the infinite series of squared integer reciprocals equaled exactly, π 2/6 and for discovering that the sum of the numbers of edges and faces minus vertices of a polyhedron equals 2, a number now commonly known as the Euler characteristic.
*15 April 1707, Basel, Switzerland
†18 September 1783, Saint Petersburg, Russian Empire
Leonhard Euler was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.
In 1720, at only thirteen years of age, he enrolled at the University of Basel. In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of René Descartes and Isaac Newton. Afterwards he enrolled in the theological faculty of the University of Basel.
He was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered Euler’s talent for mathematics. It was during this time that Euler, encouraged by the results of Johann Bernoulli’s tutorial, obtained his father’s consent to become a mathematician instead of a pastor.
In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono with which he unsuccessfully attempted to obtain a position at the University of Basel.
Then Euler spent 15 years in St. Petersburg where he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
In 1741, he requested for permission to leave to Berlin, arguing he was in need for a milder climate for his eyesight. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote several hundred articles.
In 1766 Euler accepted an invitation to return to the St. Petersburg Academy.
Pierre de Fermat
In Methodus ad disquirendam maximam et minimam and in De tangentibus linearum curvarum, Fermat developed a method (adequality) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.
Fermat was the first person known to have evaluated the integral of general power functions. With his method, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.
Fermat and Blaise Pascal helped lay the foundation for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory.
Pierre de Fermat died on January 12, 1665, at Castres, in the present-day department of Tarn.
*Between 31 October and 6 December 1607, Beaumont-de-Lomagne, France
†12 January 1665, Castres, France
Pierre de Fermat was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.
In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics.
He is best known for his Fermat’s principle for light propagation and his Fermat’s Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus’ Arithmetica.
He was also a lawyer at the Parlement of Toulouse, France.
He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux, he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius’s De Locis Planis to one of the mathematicians there.
Certainly, in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d’Espagnet who clearly shared mathematical interests with Fermat.
In 1630, he bought the office of a councilor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life.
He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz.
Fermat’s pioneering work in analytic geometry was circulated in manuscript form in 1636 (based on results achieved in 1629), predating the publication of Descartes’ famous La géométrie (1637), which exploited the work. This manuscript was published posthumously in 1679 in Varia opera mathematica, as Ad Locos Planos et Solidos Isagoge (Introduction to Plane and Solid Loci).
By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world.
The following year, Germany’s Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania where she taught, among others, doctoral and post-graduate women including Marie Johanna Weiss, Ruth Stauffer, Grace Shover Quinn and Olga Taussky-Toddone.
At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey.
In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology.
In 1935, she underwent surgery for an ovarian cyst and, despite signs of a recovery, died four days later at the age of 53.
*23 March 1882, Erlangen, Bavaria, German Empire
†14 April 1935, Bryn Mawr, Pennsylvania, United States
Amalie Emmy Noether was a German mathematician who made many important contributions to abstract algebra. She discovered Noether’s theorem, which is fundamental in mathematical physics.
She 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 some theories of rings, fields, and algebras. In physics, Noether’s theorem explains the connection between symmetry and conservation laws.
She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen, 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, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research.
The philosophical faculty objected, however, and she spent four years lecturing under Hilbert’s name. Her habilitation was approved in 1919.
Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the “Noether boys”.
In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether’s ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra.
He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).
Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875, is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable.
He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers.
Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor’s theorem.
The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris.
*3 March 1845, Saint Petersburg, Russian Empire
†6 January 1918, Halle, Province of Saxony, German Empire
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.
In fact, Cantor’s method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor’s work is of great philosophical interest, a fact he was well aware of.
Cantor’s work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle.
No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and “the infinite” (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.
Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.
In one of his earliest papers, Cantor proved that the set of real numbers is “more numerous” than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted “1-to-1 correspondence”) in set theory.
In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form.
Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell’s equations, an important step in the formulation of the theory of special relativity.
In 1905, Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.
Early in the 20th century he formulated the Poincaré conjecture that became over time one of the famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman.
*29 April 1854, Nancy, Meurthe-et-Moselle, France
†17 July 1912, Paris, France
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science.
After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions.
Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians.
In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. During the years 1883 to 1897, he taught mathematical analysis in the École Polytechnique.
In 1881–1882, Poincaré created a new branch of mathematics: qualitative theory of differential equations. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in celestial mechanics and mathematical physics.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.
Carl Friedrich Gauss
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid’s axioms were the only way to make geometry consistent and non-contradictory.
Research on these geometries led to, among other things, Einstein’s theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn “brotherhood and the banner of truth” as a student, had tried in vain for many years to prove the parallel postulate from Euclid’s other axioms of geometry.
In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula.
*30 April 1777, Brunswick, Principality of Brunswick-Wolfenbüttel
†23 February 1855, Göttingen, Kingdom of Hanover, German Confederation
Johann Carl Friedrich Gauss was a German mathematician and physicist who made significant contributions to many fields in mathematics and science.
Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen wrote that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, solved an arithmetic series problem faster than anyone else in his class of 100 pupils.
Sometimes referred to as “the greatest mathematician since antiquity”, Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history’s most influential mathematicians.
Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid’s axioms were the only way to make geometry consistent and non-contradictory.
Gauss also made important contributions to number theory with his 1801 book Arithmetical Investigations, which, among other things, introduced the triple bar symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. It appears that Gauss already knew the class number formula in 1801.