92-750: The Euler Committee of the Swiss Academy of Sciences (also known as the Euler Committee or the Euler Commission ) was founded in July 1907 with the objective of publishing the entire scientific production of Leonhard Euler in four series collectively called Opera Omnia ( Collected Works in Latin). The project represented a colossal challenge, as Euler is one of the most prolific scientists in history. The edition of Euler's Collected Works
184-473: A + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i , ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i . {\displaystyle {\begin{aligned}(a+bi)+(c+di)&=(a+c)+(b+d)i,\\[5mu](a+bi)(c+di)&=(ac-bd)+(ad+bc)i.\end{aligned}}} When multiplied by
276-673: A + b i ) = − b + a i , − i ( a + b i ) = b − a i . {\displaystyle i(a+bi)=-b+ai,\quad -i(a+bi)=b-ai.} The powers of i repeat in a cycle expressible with the following pattern, where n is any integer: i 4 n = 1 , i 4 n + 1 = i , i 4 n + 2 = − 1 , i 4 n + 3 = − i . {\displaystyle i^{4n}=1,\quad i^{4n+1}=i,\quad i^{4n+2}=-1,\quad i^{4n+3}=-i.} Thus, under multiplication, i
368-1029: A l l a c y ( − 1 ) ⋅ ( − 1 ) = 1 = 1 (incorrect). {\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}\mathrel {\stackrel {\mathrm {fallacy} }{=}} {\textstyle {\sqrt {(-1)\cdot (-1)}}}={\sqrt {1}}=1\qquad {\text{(incorrect).}}} Generally, the calculation rules x t y ⋅ y t y = x ⋅ y t y {\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}} and x t y / y t y = x / y {\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}} are guaranteed to be valid only for real, positive values of x and y . When x or y
460-478: A + bi can be represented by: a I + b J = ( a − b b a ) . {\displaystyle aI+bJ={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.} More generally, any real-valued 2 × 2 matrix with a trace of zero and a determinant of one squares to − I , so could be chosen for J . Larger matrices could also be used; for example, 1 could be represented by
552-588: A brain hemorrhage . Jacob von Staehlin [ de ] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss , one of Euler's disciples, wrote a more detailed eulogy, which he delivered at a memorial meeting. In his eulogy for the French Academy , French mathematician and philosopher Marquis de Condorcet , wrote: il cessa de calculer et de vivre — ... he ceased to calculate and to live. Euler
644-522: A convex polyhedron , and hence of a planar graph . The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huilier , is at the origin of topology . Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of
736-480: A number line , the imaginary axis , which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally. Integer sums of the real unit 1 and the imaginary unit i form a square lattice in the complex plane called the Gaussian integers . The sum, difference, or product of Gaussian integers is also a Gaussian integer: (
828-416: A candidate for its presidency by Jean le Rond d'Alembert , Frederick II named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways,
920-563: A century, as well as the financial support of the Swiss National Science Foundation , the Swiss Academy of Sciences and numerous donations from Swiss corporations. Leonhard Euler Leonhard Euler ( / ˈ ɔɪ l ər / OY -lər ; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər] ; 15 April 1707 – 18 September 1783)
1012-575: A dissertation that compared the philosophies of René Descartes and Isaac Newton . Afterwards, he enrolled in the theological faculty of the University of Basel. 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. In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by
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#17327811064711104-795: A house in Charlottenburg , in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt , the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess . This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It
1196-656: A medic in the Russian Navy . The academy at Saint Petersburg, established by Peter the Great , was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I , who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon
1288-469: A new field of study, analytic number theory . In breaking ground for this new field, Euler created the theory of hypergeometric series , q-series , hyperbolic trigonometric functions , and the analytic theory of continued fractions . For example, he proved the infinitude of primes using the divergence of the harmonic series , and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to
1380-404: A quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century. Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f ( x ) to denote the function f applied to the argument x . He also introduced
1472-414: A representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is isomorphic to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects. More generally, in the geometric algebra of any higher-dimensional Euclidean space ,
1564-462: A result otherwise known as the Euclid–Euler theorem . Euler also conjectured the law of quadratic reciprocity . The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss , particularly Disquisitiones Arithmeticae . By 1772 Euler had proved that 2 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained
1656-421: A right angle between them. Addition by a complex number corresponds to translation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every similarity transformation of the plane can be represented by a complex-linear function z ↦ a z + b . {\displaystyle z\mapsto az+b.} In the geometric algebra of
1748-460: A solution to several unsolved problems in number theory and analysis, including the famous Basel problem . Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a polyhedron equals 2, a number now commonly known as the Euler characteristic . In the field of physics, Euler reformulated Newton 's laws of physics into new laws in his two-volume work Mechanica to better explain
1840-646: A strong connection to the academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin. In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for
1932-449: A unit bivector of any arbitrary planar orientation squares to −1 , so can be taken to represent the imaginary unit i . The imaginary unit was historically written − 1 , {\textstyle {\sqrt {-1}},} and still is in some modern works. However, great care needs to be taken when manipulating formulas involving radicals . The radical sign notation x {\textstyle {\sqrt {x}}}
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#17327811064712024-501: A way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis . He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations . Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced
2116-525: A younger brother, Johann Heinrich. Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen , Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel . Around
2208-430: Is isomorphic to the complex numbers, and the variable x {\displaystyle x} expresses the imaginary unit. The complex numbers can be represented graphically by drawing the real number line as the horizontal axis and the imaginary numbers as the vertical axis of a Cartesian plane called the complex plane . In this representation, the numbers 1 and i are at the same distance from 0 , with
2300-837: Is a generator of a cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication. Written as a special case of Euler's formula for an integer n , i n = exp ( 1 2 π i ) n = exp ( 1 2 n π i ) = cos ( 1 2 n π ) + i sin ( 1 2 n π ) . {\displaystyle i^{n}={\exp }{\bigl (}{\tfrac {1}{2}}\pi i{\bigr )}^{n}={\exp }{\bigl (}{\tfrac {1}{2}}n\pi i{\bigr )}={\cos }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}+{i\sin }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}.} With
2392-425: Is a negative scalar. The quotient of a vector with itself is the scalar 1 = u / u , and when multiplied by any vector leaves it unchanged (the identity transformation ). The quotient of any two perpendicular vectors of the same magnitude, J = u / v , which when multiplied rotates the divisor a quarter turn into the dividend, Jv = u , is a unit bivector which squares to −1 , and can thus be taken as
2484-405: Is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics . Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of
2576-402: Is close to completion, with a total of 84 volumes comprising about 35,000 pages planned for the entire collection. A total of 80 volumes have been published so far. By 2010, three of the last four volumes of series IV were in active preparation. The gigantic effort of editing and publishing Euler's Opera Omnia required the continuous contribution of internationally acclaimed scientists for over
2668-482: Is inherently positive or negative in the sense that real numbers are. A more formal expression of this indistinguishability of + i and − i is that, although the complex field is unique (as an extension of the real numbers) up to isomorphism , it is not unique up to a unique isomorphism. That is, there are two field automorphisms of the complex numbers C {\displaystyle \mathbb {C} } that keep each real number fixed, namely
2760-434: Is real but negative, these problems can be avoided by writing and manipulating expressions like i 7 {\textstyle i{\sqrt {7}}} , rather than − 7 {\textstyle {\sqrt {-7}}} . For a more thorough discussion, see the articles Square root and Branch point . As a complex number, the imaginary unit follows all of the rules of complex arithmetic . When
2852-500: Is reserved either for the principal square root function, which is defined for only real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results: − 1 = i ⋅ i = − 1 ⋅ − 1 = f
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2944-460: Is said to have an argument of − π 2 , {\displaystyle -{\tfrac {\pi }{2}},} related to the convention of labelling orientations in the Cartesian plane relative to the positive x -axis with positive angles turning anticlockwise in the direction of the positive y -axis. Also, despite the signs written with them, neither + i nor − i
3036-631: Is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos . He spent most of his adult life in Saint Petersburg , Russia, and in Berlin , then
3128-430: Is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit . This solution is considered to be the first theorem of graph theory . Euler also discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices, edges, and faces of
3220-758: Is well known in analysis for his frequent use and development of power series , the expression of functions as sums of infinitely many terms, such as e x = ∑ n = 0 ∞ x n n ! = lim n → ∞ ( 1 0 ! + x 1 ! + x 2 2 ! + ⋯ + x n n ! ) . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).} Euler's use of power series enabled him to solve
3312-565: The Institutiones calculi differentialis was published. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences and of the French Academy of Sciences . Notable students of Euler in Berlin included Stepan Rumovsky , later considered as the first Russian astronomer. In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought
3404-510: The 4 × 4 identity matrix and i could be represented by any of the Dirac matrices for spatial dimensions. Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose coefficients are real numbers form a ring , denoted R [ x ] , {\displaystyle \mathbb {R} [x],} an algebraic structure with addition and multiplication and sharing many properties with
3496-1040: The Basel problem , finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized by Jacob Bernoulli and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that: ∑ n = 1 ∞ 1 n 2 = lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.} Euler introduced
3588-459: The Bernoulli numbers , Fourier series , Euler numbers , the constants e and π , continued fractions, and integrals. He integrated Leibniz 's differential calculus with Newton's Method of Fluxions , and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as
3680-458: The Euclidean plane , the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector
3772-519: The Euler approximations . The most notable of these approximations are Euler's method and the Euler–Maclaurin formula . Imaginary unit The imaginary unit or unit imaginary number ( i ) is a solution to the quadratic equation x + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication . A simple example of
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3864-550: The Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. When Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726, Euler eagerly accepted
3956-519: The St. Petersburg Academy , which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income. He was even involved in the design of
4048-433: The complex plane , which is a special interpretation of a Cartesian plane , i is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis ). Being a quadratic polynomial with no multiple root , the defining equation x = −1 has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. Although
4140-470: The imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma ) to express summations , the Greek letter Δ {\displaystyle \Delta } (capital delta ) for finite differences , and lowercase letters to represent the sides of a triangle while representing
4232-527: The largest known prime until 1867. Euler also contributed major developments to the theory of partitions of an integer . In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg . The city of Königsberg , Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem
4324-540: The Euler–Mascheroni constant, and studied its relationship with the harmonic series , the gamma function , and values of the Riemann zeta function . Euler introduced the use of the exponential function and logarithms in analytic proofs . He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers , thus greatly expanding
4416-403: The academy beginning in 1720) for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer , who became known as "the father of naval architecture", won and Euler took second place. Over the years, Euler entered this competition 15 times, winning 12 of them. Johann Bernoulli's two sons, Daniel and Nicolaus , entered into service at
4508-539: The age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics. In 1720, at thirteen years of age, Euler enrolled at the University of Basel . Attending university at such a young age was not unusual at the time. The course on elementary mathematics
4600-485: The angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm , now known as Euler's number . Euler is also credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg , which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, providing
4692-543: The ascension of the twelve-year-old Peter II . The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities. Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed power. Euler swiftly rose through
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#17327811064714784-417: The capital of Prussia . Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi ) to denote the ratio of a circle's circumference to its diameter , as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express
4876-498: The constant γ = lim n → ∞ ( 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 n − ln ( n ) ) ≈ 0.5772 , {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right)\approx 0.5772,} now known as Euler's constant or
4968-1045: The construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i with −1 ). Higher integral powers of i are thus i 3 = i 2 i = ( − 1 ) i = − i , i 4 = i 3 i = ( − i ) i = 1 , i 5 = i 4 i = ( 1 ) i = i , {\displaystyle {\begin{alignedat}{3}i^{3}&=i^{2}i&&=(-1)i&&=-i,\\[3mu]i^{4}&=i^{3}i&&=\;\!(-i)i&&=\ \,1,\\[3mu]i^{5}&=i^{4}i&&=\ \,(1)i&&=\ \ i,\end{alignedat}}} and so on, cycling through
5060-576: 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 1748 his text on functions called the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called
5152-486: The damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia. During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published. This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to
5244-537: The development of the prime number theorem . Euler's interest in number theory can be traced to the influence of Christian Goldbach , his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat . Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form 2 2 n + 1 {\textstyle 2^{2^{n}}+1} ( Fermat numbers ) are prime. Euler linked
5336-428: The entirety of the poem, along with stating the first and last sentence on each page of the edition from which he had learnt it. Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains
5428-668: The force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci . My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry! However, the disappointment was almost surely unwarranted from a technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional. Throughout his stay in Berlin, Euler maintained
5520-548: The four values 1 , i , −1 , and − i . As with any non-zero real number, i = 1. As a complex number, i can be represented in rectangular form as 0 + 1 i , with a zero real component and a unit imaginary component. In polar form , i can be represented as 1 × e (or just e ), with an absolute value (or magnitude) of 1 and an argument (or angle) of π 2 {\displaystyle {\tfrac {\pi }{2}}} radians . (Adding any integer multiple of 2 π to this angle works as well.) In
5612-431: The identity and complex conjugation . For more on this general phenomenon, see Galois group . Using the concepts of matrices and matrix multiplication , complex numbers can be represented in linear algebra. The real unit 1 and imaginary unit i can be represented by any pair of matrices I and J satisfying I = I , IJ = JI = J , and J = − I . Then a complex number a + bi can be represented by
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#17327811064715704-806: The imaginary unit i , any arbitrary complex number in the complex plane is rotated by a quarter turn ( 1 2 π {\displaystyle {\tfrac {1}{2}}\pi } radians or 90° ) anticlockwise . When multiplied by − i , any arbitrary complex number is rotated by a quarter turn clockwise. In polar form: i r e φ i = r e ( φ + π / 2 ) i , − i r e φ i = r e ( φ − π / 2 ) i . {\displaystyle i\,re^{\varphi i}=re^{(\varphi +\pi /2)i},\quad -i\,re^{\varphi i}=re^{(\varphi -\pi /2)i}.} In rectangular form, i (
5796-413: The imaginary unit is normally denoted by j instead of i , because i is commonly used to denote electric current . Square roots of negative numbers are called imaginary because in early-modern mathematics , only what are now called real numbers , obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so
5888-580: The imaginary unit is repeatedly added or subtracted, the result is some integer times the imaginary unit, an imaginary integer ; any such numbers can be added and the result is also an imaginary integer: a i + b i = ( a + b ) i . {\displaystyle ai+bi=(a+b)i.} Thus, the imaginary unit is the generator of a group under addition, specifically an infinite cyclic group . The imaginary unit can also be multiplied by any arbitrary real number to form an imaginary number . These numbers can be pictured on
5980-414: The integer n that are coprime to n . Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem . He contributed significantly to the theory of perfect numbers , which had fascinated mathematicians since Euclid . He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one,
6072-463: The left eye as well. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; and, in 1775, he produced, on average, one mathematical paper every week. 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 Anders Johan Lexell when he collapsed and died from
6164-610: The matrix aI + bJ , and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic. The most common choice is to represent 1 and i by the 2 × 2 identity matrix I and the matrix J , I = ( 1 0 0 1 ) , J = ( 0 − 1 1 0 ) . {\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} Then an arbitrary complex number
6256-483: The modern notation for the trigonometric functions , the letter e for the base of the natural logarithm (now also known as Euler's number ), the Greek letter Σ for summations and the letter i to denote the imaginary unit . The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones . The development of infinitesimal calculus
6348-502: The motion of rigid bodies . He also made substantial contributions to the study of elastic deformations of solid objects. Leonhard Euler was born on 15 April 1707, in Basel to Paul III Euler, a pastor of the Reformed Church , and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and
6440-455: The nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges . In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function . Euler invented the totient function φ( n ), the number of positive integers less than or equal to
6532-541: The notion of a mathematical function . He is also known for his work in mechanics , fluid dynamics , optics , astronomy , and music theory . Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he
6624-609: The offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg in May 1727. 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. Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as
6716-403: The polar opposite of Voltaire , who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating: I wanted to have a water jet in my garden: Euler calculated
6808-408: The property that its square is −1: i 2 = − 1. {\displaystyle i^{2}=-1.} With i defined this way, it follows directly from algebra that i and − i are both square roots of −1. Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number,
6900-520: The ranks in the academy and was made a professor of physics in 1731. He also left the Russian Navy, refusing a promotion to lieutenant . Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell . Frederick II had made an attempt to recruit
6992-522: The ring of integers . The polynomial x 2 + 1 {\displaystyle x^{2}+1} has no real-number roots , but the set of all real-coefficient polynomials divisible by x 2 + 1 {\displaystyle x^{2}+1} forms an ideal , and so there is a quotient ring R [ x ] / ⟨ x 2 + 1 ⟩ . {\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .} This quotient ring
7084-504: The scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions . For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies e i φ = cos φ + i sin φ {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi } which
7176-677: The services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Empress Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in need of a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members. Concerned about
7268-399: The square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to René Descartes , and Isaac Newton used the term as early as 1670. The i notation was introduced by Leonhard Euler . A unit is an undivided whole, and unity or the unit number is the number one ( 1 ). The imaginary unit i is defined solely by
7360-480: The subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " Cyclops ". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a cataract in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in
7452-466: The term "imaginary" is used because there is no real number having a negative square . There are two complex square roots of −1: i and − i , just as there are two complex square roots of every real number other than zero (which has one double square root ). In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in electrical engineering and control systems engineering ,
7544-530: The two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled + i (or simply i ) and the other is labelled − i , though it is inherently ambiguous which is which. The only differences between + i and − i arise from this labelling. For example, by convention + i is said to have an argument of + π 2 {\displaystyle +{\tfrac {\pi }{2}}} and − i
7636-443: The use of i in a complex number is 2 + 3 i . Imaginary numbers are an important mathematical concept; they extend the real number system R {\displaystyle \mathbb {R} } to the complex number system C , {\displaystyle \mathbb {C} ,} in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra ). Here,
7728-465: The water fountains at Sanssouci , the King's summer palace. The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he
7820-502: Was Johann Albrecht Euler , whose godfather was Christian Goldbach . Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783. His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy. Early in his life, Euler memorized the entirety of the Aeneid by Virgil , and by old age, could recite
7912-405: 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
8004-553: Was assisted by his student Anders Johan Lexell . While living in St. Petersburg, a fire in 1771 destroyed his home. On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell , a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by the Neva River . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son
8096-525: Was at the forefront of 18th-century mathematical research, and the Bernoullis —family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances. Euler
8188-803: Was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island . In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery . Euler worked in almost all areas of mathematics, including geometry , infinitesimal calculus , trigonometry , algebra , and number theory , as well as continuum physics , lunar theory , and other areas of physics . He
8280-427: Was called "the most remarkable formula in mathematics" by Richard Feynman . A special case of the above formula is known as Euler's identity , e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations . He found
8372-423: Was given by Johann Bernoulli , the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography: It was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor. In 1723, Euler received a Master of Philosophy with
8464-544: Was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist. Despite Euler's immense contribution to the academy's prestige and having been put forward as
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