76-408: Julius Wilhelm Richard Dedekind ( German: [ˈdeːdəˌkɪnt] ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory , abstract algebra (particularly ring theory ), and the axiomatic foundations of arithmetic . His best known contribution is the definition of real numbers through the notion of Dedekind cut . He is also considered
152-420: A {\displaystyle p=2b-a} and q = a − b {\displaystyle q=a-b} such that p 2 = 2 q 2 {\displaystyle p^{2}=2q^{2}} . Since it can be seen geometrically that p < a {\displaystyle p<a} and q < b {\displaystyle q<b} , this contradicts
228-548: A 2 = 2 b 2 {\displaystyle a^{2}=2b^{2}} . Now consider two squares with sides a {\displaystyle a} and b {\displaystyle b} , and place two copies of the smaller square inside the larger one as shown in Figure 1. The area of the square overlap region in the centre must equal the sum of the areas of the two uncovered squares. Hence there exist positive integers p = 2 b −
304-408: A cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus | 2 b − a | ≥ 1 . Multiplying the absolute difference | √ 2 − a / b | by b ( √ 2 + a / b ) in the numerator and denominator, we get the latter inequality being true because it
380-493: A one-to-one correspondence between them. He invoked similarity to give the first precise definition of an infinite set : a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets . Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N , ( N → N ): Dedekind's work in this area anticipated that of Georg Cantor , who
456-424: A cut is that an irrational number divides the rational numbers into two classes ( sets ), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into
532-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
608-567: A finite number of terms, 2 {\displaystyle {\sqrt {2}}} appears in various trigonometric constants : It is not known whether 2 {\displaystyle {\sqrt {2}}} is a normal number , which is a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two . The identity cos π / 4 = sin π / 4 = 1 / √ 2 , along with
684-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
760-401: A perfect square) or irrational. The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent . A simple proof is attributed to Stanley Tennenbaum when he
836-585: A pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism . Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig . His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used
SECTION 10
#1732765122586912-788: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
988-459: A smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational . Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For
1064-612: A subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether . Ideals generalize Ernst Eduard Kummer 's ideal numbers , devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem . (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind and Heinrich Martin Weber applied ideals to Riemann surfaces , giving an algebraic proof of
1140-660: A thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals "). This thesis did not display the talent evident in Dedekind's subsequent publications. At that time, the University of Berlin , not Göttingen , was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries; they were both awarded
1216-481: A while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it, though this has little to any substantial evidence in traditional historian practice. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant . In ancient Roman architecture , Vitruvius describes
1292-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
1368-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
1444-475: Is a rational number must be false. This means that 2 {\displaystyle {\sqrt {2}}} is not a rational number; that is to say, 2 {\displaystyle {\sqrt {2}}} is irrational. This proof was hinted at by Aristotle , in his Analytica Priora , §I.23. It appeared first as a full proof in Euclid 's Elements , as proposition 117 of Book X. However, since
1520-512: Is accurate to about six decimal digits, and is the closest possible three-place sexagesimal representation of 2 {\displaystyle {\sqrt {2}}} , representing a margin of error of only –0.000042%: Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras ( c. 800 –200 BC), as follows: Increase the length [of the side] by its third and this third by its own fourth less
1596-477: Is also a right isosceles triangle. It also follows that FC = n − ( m − n ) = 2 n − m . Hence, there is an even smaller right isosceles triangle, with hypotenuse length 2 n − m and legs m − n . These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m : n is in lowest terms. Therefore, m and n cannot be both integers; hence, 2 {\displaystyle {\sqrt {2}}}
SECTION 20
#17327651225861672-547: Is an algebraic number , and therefore not a transcendental number . Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length ; this follows from the Pythagorean theorem . It was probably the first number known to be irrational . The fraction 99 / 70 (≈ 1.4142 857)
1748-432: Is assumed that 1< a / b < 3/2 , giving a / b + √ 2 ≤ 3 (otherwise the quantitative apartness can be trivially established). This gives a lower bound of 1 / 3 b for the difference | √ 2 − a / b | , yielding a direct proof of irrationality in its constructively stronger form, not relying on
1824-610: Is commonly considered the founder of set theory . Likewise, his contributions to the foundations of mathematics anticipated later works by major proponents of logicism , such as Gottlob Frege and Bertrand Russell . Dedekind edited the collected works of Lejeune Dirichlet , Gauss , and Riemann . Dedekind's study of Lejeune Dirichlet's work led him to his later study of algebraic number fields and ideals . In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: Although
1900-436: Is irrational. While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let a and b be positive integers such that 1< a / b < 3/2 (as 1<2< 9/4 satisfies these bounds). Now 2 b and
1976-402: Is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when p ( x ) {\displaystyle p(x)} is a monic polynomial with integer coefficients ; for such a polynomial, all roots are necessarily integers (which 2 {\displaystyle {\sqrt {2}}} is not, as 2 is not
2052-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
2128-408: Is rational. Therefore, Here, ( b , b , a ) is a primitive Pythagorean triple, and from the lemma a is never even. However, this contradicts the equation 2 b = a which implies that a must be even. The multiplicative inverse (reciprocal) of the square root of two is a widely used constant , with the decimal value: It is often encountered in geometry and trigonometry because
2204-560: Is sometimes used as a good rational approximation with a reasonably small denominator . Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: The Babylonian clay tablet YBC 7289 ( c. 1800 –1600 BC) gives an approximation of 2 {\displaystyle {\sqrt {2}}} in four sexagesimal figures, 1 24 51 10 , which
2280-472: Is the Babylonian method for computing square roots, an example of Newton's method for computing roots of arbitrary functions. It goes as follows: First, pick a guess, a 0 > 0 {\displaystyle a_{0}>0} ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through
2356-533: The Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of
Richard Dedekind - Misplaced Pages Continue
2432-538: The Riemann–Roch theorem . In 1888, he published a short monograph titled Was sind und was sollen die Zahlen? ("What are numbers and what are they good for?" Ewald 1996: 790), which included his definition of an infinite set . He also proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function . The next year, Giuseppe Peano , citing Dedekind, formulated an equivalent but simpler set of axioms , now
2508-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
2584-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
2660-452: The habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent , giving courses on probability and geometry . He studied for a while with Peter Gustav Lejeune Dirichlet , and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions . Yet he was also the first at Göttingen to lecture concerning Galois theory . About this time, he became one of
2736-462: The law of excluded middle . This proof constructively exhibits an explicit discrepancy between 2 {\displaystyle {\sqrt {2}}} and any rational. This proof uses the following property of primitive Pythagorean triples : This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square. Suppose the contrary that 2 {\displaystyle {\sqrt {2}}}
2812-460: The limit of x n as n → ∞ will be called (if this limit exists) f ( c ) . Then 2 {\displaystyle {\sqrt {2}}} is the only number c > 1 for which f ( c ) = c . Or symbolically: 2 {\displaystyle {\sqrt {2}}} appears in Viète's formula for π , which is related to the formula Similar in appearance but with
2888-430: The square root and arithmetic operations , if the square root symbol is interpreted suitably for the complex numbers i and − i : 2 {\displaystyle {\sqrt {2}}} is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1 , x 1 = c and x n +1 = c for n > 1 ,
2964-427: The unit vector , which makes a 45° angle with the axes in a plane , has the coordinates Each coordinate satisfies One interesting property of 2 {\displaystyle {\sqrt {2}}} is since This is related to the property of silver ratios . 2 {\displaystyle {\sqrt {2}}} can also be expressed in terms of copies of the imaginary unit i using only
3040-455: The Babylonian method after starting with a 0 = 1 ( 665,857 / 470,832 ) is too large by about 1.6 × 10 ; its square is ≈ 2.000 000 000 0045 . In 1997, the value of 2 {\displaystyle {\sqrt {2}}} was calculated to 137,438,953,444 decimal places by Yasumasa Kanada 's team. In February 2006, the record for the calculation of 2 {\displaystyle {\sqrt {2}}}
3116-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
Richard Dedekind - Misplaced Pages Continue
3192-486: The arcs BD and CE with centre A . Join DE . It follows that AB = AD , AC = AE and ∠ BAC and ∠ DAE coincide. Therefore, the triangles ABC and ADE are congruent by SAS . Because ∠ EBF is a right angle and ∠ BEF is half a right angle, △ BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n . By symmetry, DF = m − n , and △ FDC
3268-576: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2 . It may be written in mathematics as 2 {\displaystyle {\sqrt {2}}} or 2 1 / 2 {\displaystyle 2^{1/2}} . It
3344-544: The book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death. The 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory . (The word "Ring", introduced later by Hilbert , does not appear in Dedekind's work.) Dedekind defined an ideal as
3420-427: The correct value by less than 1 / 10,000 (approx. +0.72 × 10 ). The next two better rational approximations are 140 / 99 (≈ 1.414 1414...) with a marginally smaller error (approx. −0.72 × 10 ), and 239 / 169 (≈ 1.4142 012) with an error of approx −0.12 × 10 . The rational approximation of the square root of two derived from four iterations of
3496-429: The desired contradiction. As with the proof by infinite descent, we obtain a 2 = 2 b 2 {\displaystyle a^{2}=2b^{2}} . Being the same quantity, each side has the same prime factorization by the fundamental theorem of arithmetic , and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on
3572-500: The earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established
3648-838: The early 19th century, historians have agreed that this proof is an interpolation and not attributable to Euclid. Assume by way of contradiction that 2 {\displaystyle {\sqrt {2}}} were rational. Then we may write 2 + 1 = q p {\displaystyle {\sqrt {2}}+1={\frac {q}{p}}} as an irreducible fraction in lowest terms, with coprime positive integers q > p {\displaystyle q>p} . Since ( 2 − 1 ) ( 2 + 1 ) = 2 − 1 2 = 1 {\displaystyle ({\sqrt {2}}-1)({\sqrt {2}}+1)=2-1^{2}=1} , it follows that 2 − 1 {\displaystyle {\sqrt {2}}-1} can be expressed as
3724-788: The first people to understand the importance of the notion of groups for algebra and arithmetic . In 1858, he began teaching at the Polytechnic school in Zürich (now ETH Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia. Dedekind
3800-442: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
3876-410: The following recursive computation: Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with a 0 = 1 {\displaystyle a_{0}=1} , the subsequent iterations yield: A simple rational approximation 99 / 70 (≈ 1.4142 857) is sometimes used. Despite having a denominator of only 70, it differs from
SECTION 50
#17327651225863952-1060: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
4028-460: The greater class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"); in modern terminology, Vollständigkeit , completeness . Dedekind defined two sets to be "similar" when there exists
4104-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
4180-470: The irreducible fraction p q {\displaystyle {\frac {p}{q}}} . However, since 2 − 1 {\displaystyle {\sqrt {2}}-1} and 2 + 1 {\displaystyle {\sqrt {2}}+1} differ by an integer, it follows that the denominators of their irreducible fraction representations must be the same, i.e. q = p {\displaystyle q=p} . This gives
4256-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
4332-474: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
4408-700: The names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at Braunschweig Main Cemetery . He first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern . Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for
4484-408: The number's irrationality is the following proof by infinite descent . It is also a proof of a negation by refutation : it proves the statement " 2 {\displaystyle {\sqrt {2}}} is not rational" by assuming that it is rational and then deriving a falsehood. Since we have derived a falsehood, the assumption (1) that 2 {\displaystyle {\sqrt {2}}}
4560-452: The only possible rational roots are ± 1 {\displaystyle \pm 1} and ± 2 {\displaystyle \pm 2} . As 2 {\displaystyle {\sqrt {2}}} is not equal to ± 1 {\displaystyle \pm 1} or ± 2 {\displaystyle \pm 2} , it follows that 2 {\displaystyle {\sqrt {2}}}
4636-413: The original assumption. Tom M. Apostol made another geometric reductio ad absurdum argument showing that 2 {\displaystyle {\sqrt {2}}} is irrational. It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially
SECTION 60
#17327651225864712-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
4788-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
4864-498: The right, but an even number of times on the left—a contradiction. The irrationality of 2 {\displaystyle {\sqrt {2}}} also follows from the rational root theorem , which states that a rational root of a polynomial , if it exists, must be the quotient of a factor of the constant term and a factor of the leading coefficient . In the case of p ( x ) = x 2 − 2 {\displaystyle p(x)=x^{2}-2} ,
4940-444: The same algebraic proof as in the previous paragraph, viewed geometrically in another way. Let △ ABC be a right isosceles triangle with hypotenuse length m and legs n as shown in Figure 2. By the Pythagorean theorem , m n = 2 {\displaystyle {\frac {m}{n}}={\sqrt {2}}} . Suppose m and n are integers. Let m : n be a ratio given in its lowest terms . Draw
5016-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
5092-557: The standard ones. Dedekind made other contributions to algebra . For instance, around 1900, he wrote the first papers on modular lattices . In 1872, while on holiday in Interlaken , Dedekind met Georg Cantor . Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with Leopold Kronecker , who
5168-449: The thirty-fourth part of that fourth. That is, This approximation, diverging from the actual value of 2 {\displaystyle {\sqrt {2}}} by approximately +0.07%, is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers , which can be derived from the continued fraction expansion of 2 {\displaystyle {\sqrt {2}}} . Despite having
5244-442: The use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato . The system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion
5320-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
5396-419: Was a student in the early 1950s. Assume that 2 = a / b {\displaystyle {\sqrt {2}}=a/b} , where a {\displaystyle a} and b {\displaystyle b} are coprime positive integers. Then a {\displaystyle a} and b {\displaystyle b} are the smallest positive integers for which
5472-406: Was also used to design atria by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width. There are many algorithms for approximating 2 {\displaystyle {\sqrt {2}}} as a ratio of integers or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators,
5548-492: Was eclipsed with the use of a home computer. Shigeru Kondo calculated one trillion decimal places in 2010. Other mathematical constants whose decimal expansions have been calculated to similarly high precision include π , e , and the golden ratio . Such computations provide empirical evidence of whether these numbers are normal . This is a table of recent records in calculating the digits of 2 {\displaystyle {\sqrt {2}}} . One proof of
5624-532: Was elected to the Academies of Berlin (1880) and Rome, and to the French Academy of Sciences (1900). He received honorary doctorates from the universities of Oslo , Zurich , and Braunschweig . While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (German: Schnitt ), now a standard definition of the real numbers. The idea of
5700-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
5776-677: Was philosophically opposed to Cantor's transfinite numbers . Primary literature in English: Primary literature in German: There is an online bibliography of the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996). Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of
#585414