Galois

Evariste Galois

Evariste Galois was a French mathematician who made significant contribution to the theory of functions, the theory of equations and number theory. His work became the basis for Galois Theory and Group theory. He was the first to use the word Group as a technical term in mathematics to represent a group of permutations. 

 The mathematical genius Evariste Galois was born on 25^th October 1811 in France. He was the second son of Nicholas Gabriel Galois and Adelaide Marie. His father was the director of a boarding school and later mayor of Bourg – la- Reive. Evariste had a happy childhood. He seemed to have a phenomenal memory.  While still in teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long standing problem. Up to the age of 12, he was educated by his mother who instilled in him knowledge of classics and a sceptical attitude towards religion.

He entered the College Royale de Louis le Grand in Paris in 1823 where his precocious mathematical genius first emerged. He studied the Legendre’s text book Elements de Geometrie and Lagrange’s work on Differential Equation and Analytic Function. By the age of 16, he had published many papers. He appeared for the examination of Ecole Polytechnic but failed. The next year Galois had the good fortune to be studying maths under a distinguished teacher Louis Paul Emile Richard who recognised his exceptional gifts. Richard was enthusiastic about Galois mathematical work and this led to his publication of first paper on continued fraction. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences but the Academy refused to accept them for publication.

On 28 July 1829 Galois's father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority and Louis Philippe became the king. His anti monarchist view led him to imprisonment leading his expulsion from the college.

Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government. They were later acquitted of all charges.

During his imprisonment, he continued developing his mathematical ideas. On the advice of Simon Poisson he submitted his work on the Theory of Equation for publication but Poisson decleared his work incomprehensible. Galois reacted violently to the rejection and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. 

During 1830s Galois published several papers in Bulletin des Sciences mathematiques of Baren de Ferussace, a review that normally only published work by established scientist. These articles contained most of his work on Theory of Equation which is now known as Galois Theory.

In 1831, he was arrested again for keeping illegal weapon. He was released on April 29, 1832. On May 25 he wrote to Chevalier, a close friend of his from the Ecole Normale, expressing his complete disenchantment with life, and hinting that a broken love affair was the reason. The woman in question was Stephanie Dumotel, the daughter of the resident physician at the hostel where Galois stayed during last months of his life.

On May 30, Galois fought a duel with pistol and was shot in the abdomen. Galois remained unattended for hours until a passerby took him to the hospital. He refused the services of a priest and died of peritonitis the following day, at the age of 20. A night before the duel was scheduled he recorded his mathematical idea in a later to his former school master Auguste Chevalier where he had out lined his work on elliptical integrals and permutations of groups.

Hermann Weyl, a mathematician, said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

Galois' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not solvable by radicals.

 Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

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Euclid

  Euclid

Euclid of Alexandria is better known as the Father of Geometry. He is one of the most prominent mathematicians of antiquity best known for his treatise The Elements. Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown. But whatever speculated data are available shows that he was born about 325 BC in Alexandria and died about 265 BC.

An Arabian author, al-Qifti (d. 1248), recorded that Euclid's father was Naucrates and his grandfather was Zenarchus, that he was a Greek, born in Tyre and lived in Damascus. But there is no real proof that this is the same Euclid. During the reign of Ptolemy I he taught at Alexandria. Ptolemy had created the great library at Alexandria, which was known as the Museum, because it was considered a house of the muses for the arts and sciences. Many scholars worked and taught there, and that is where Euclid wrote The Elements.

There is a very interesting story about Euclid---

A student who had begun to learn geometry with Euclid asked him “What shall I get by learning all these things in Geometry?” Euclid called his slave and told him to give three pence since he must make gain out of what he learns.

Euclid said to his students that --- There is no royal road to Geometry.

             A page from Euclid’s Elements

Euclid's most famous for his treatise on mathematics called The Elements. The book is a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid. The Elements is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry.

Euclid organized the known geometrical ideas, starting with simple definitions, axioms; formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. One of the proofs was for the theorem of Pythagoras, proving that the equation is always true for every right triangle. The Elements was the most widely used textbook of all time. It has appeared in more than 1,000 editions since it was first printed in 1482 and is thought to have sold more copies than any book other than the Bible. Original of Euclid's Elements have not been preserved, but Arabic mathematicians obviously had a full copy as an Arabic version of The Elements appeared at the end of the 8th century BC.

 Euclid believed that we can't be sure of any axioms without proof, so he devised logical steps to prove them. There are 5 axioms and 5 postulates found in the book of Euclid. He called Axiom "Common Notions," because they were common to all sciences.

Axioms

1.    Things which are equal to the same thing are also equal to one another.

2.    If equals are added to equals, the sums are equal.

3.    If equals are subtracted from equals, the remainders are equal.

4.    Things which coincide with one another are equal to one another.

5.    The whole is greater than the part.

Postulates

1.      You can draw a straight line between any two points.

2.      You can extend the line indefinitely.

3.      You can draw a circle using any line segment as the radius and one end point as the center.

4.      All right angles are equal.

5.      Given a line and a point, you can draw only one line through the point that is parallel to the first line.

The fifth postulates later became the cause of discovery of new branch of geometry known as Non Euclidian Geometry.

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time.

Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it; you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists. Mathematicians since Euclid have attempted without success to find a pattern to the sequence of prime numbers.

Although best known for its geometric results, the Elements also include number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Euclid also wrote Data, which contains 94 propositions, Phaenomena, concerning spherical astronomy, Caloptrics, about mirrors, Optics, the theory of perspective, and a work of music theory. In his works about optics, Euclid made light rays part of geometry, working with them as if they were straight lines. Many of the works ascribed to Euclid are no longer in existence or are incomplete.

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Emmy Noether

Emmy Noether

Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Albert Einstein described her as the most important woman in mathematical history. She lived and worked in Germany during the same period that Ramanujan worked in England and India, but whereas Ramanujan’s contributions were in analysis and number theory, hers were in abstract algebra and in the application of algebra to theoretical physics. She revolutionized the theories of rings, fields and algebras. In physics Noether’s theorem explains the fundamental connection between symmetry and conservation laws. The innovative approach to modern abstract algebra of Emmy Noether not only produced major new results, but also inspired highly productive work by students and colleagues who emulated her techniques.

Amalie Emmy Noether to give her the full name was born on March 23, 1882 in Erlangen in Germany. Her father Max Noether was professor of mathematics at the University of Erlangen while her mother Ida Amalia was the daughter of a wealthy Jewish family of Cologne. She was better known as Emmy. She loved to dance and enjoyed music. She attended the Municipal school for the Higher Education of Daughters until she was 18. Emmy Noether showed early proficiency in French and English. In the spring of 1900 she took the examination for teachers of these languages and received an overall score of very good. Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen. This was an unconventional decision because as late as 1900, women were not allowed to enroll in Universities in Germany. Professors frequently refused permission for women event to attend their lectures, and only very rarely women allowed taking university examination. The obstacle in a way could not deter her to enroll in a university, she had to get permission of the professors to take an entrance exam -- she did and she passed, after sitting in on mathematics lectures at the University of Erlangen. She was then allowed to audit courses -- first at the University of Erlangen and then the University of Göttingen, neither of which would permit a woman to attend classes for credit. Finally, in 1904, the University of Erlangen decided to permit women to enroll as regular students, and Emmy Noether returned there. She began to focus solely on mathematics. Under the supervision of Paul Gordan she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms). Her dissertation in algebraic math earned her a doctorate summa cum laude in 1908.

Having obtained her doctorate, Noether was well qualified for a position at a university, but the persistent sexiest atmosphere in Germany prevented the brilliant young woman from being able to even apply for a job. This depressed her greatly, but she began helping her father in his research. She also started publishing papers on her own, which were so well received that she was invited to join a number of European Mathematical Societies, including the German Mathematical Association, which had been founded by George Cantor, but still she could not obtain a paying position at a University in Germany. By 1915 Noether was a famous mathematician in her own right and her papers were read with interest throughout the world. They dealt primarily with algebra.

In 1915, Emmy Noether's mentors, Felix Klein and David Hilbert, invited her to join them at the Mathematical Institute in Gottingen, again without compensation. There, she pursued important mathematical work that confirmed key parts of the general theory of relativity. Noether arrived at Gottingen and began her work on invariance in mathematical physics. Meanwhile, Klein rallied to get her appointed a professor at Gottingen, but he had to struggle with the administration until 1919, when his request was finally granted. She became a privatdozent allowing her to teach students and students would pay her directly. In 1922, the University gave her a position as an adjunct professor with a small salary and no tenure or benefits. Soon after arriving at Göttingen, however, she demonstrated her capabilities by proving the theorem now known as Noether's theorem, which shows that a conservation law is associated with any differentiable symmetry of a physical system. 

American physicists Leon M. Lederman and Christopher T. Hill argue in their book Symmetry and the Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean Theorem". Her work went far beyond mathematical physics. She made important contributions to Galois Theory, to many other areas of abstract algebra and to topology. Noether was, in fact the greatest algebraist of her time.

Noether’s groundbreaking work in algebra began in 1920 with a paper on non commutative fields. Her work earned her enough recognition that she was invited as a visiting professor in 1928-1929 at the University of Moscow and in 1930 at the University of Frankfurt. In 1932 Emmy Noether and Emil Artin received the Ackermann Teubner Memorial Award for their contributions to mathematics.  Though she was never able to gain a regular faculty position at Göttingen, she was one of many Jewish faculty members who were purged by the Nazis in 1933.  

In America, the Emergency Committee to Aid Displaced German Scholars obtained for Emmy Noether an offer of a professorship at Bryn Mawr College in America, and they paid, with the Rockefeller Foundation, her first year's salary. The grant was renewed for two more years in 1934. This was the first time that Emmy Noether was paid a full professor's salary and accepted as a full faculty member. In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with and supervised Abraham Albert and Harry Vandiver. Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects. 

In April 1935 doctors discovered a tumor in Noether's pelvis. She was admitted to hospital for surgery to remove a uterine tumor. Although the operation was successful but on April 14, she fell unconscious and she developed a high fever resulting to her death.

Weyl said to her funeral – The memory of her work in science and of her personality among her fellows will not soon pass away. She was a great mathematician, the greatest, I firmly believe, her sex has ever produced, and a great woman.

Diophantus

Diophantus

Diophantus of Alexandria was a Greek mathematician. He was the first Greek mathematician who recognized fractions as numbers. He allowed positive rational numbers for the coefficients and solutions. He played a major role in the development of algebra and was a considerable influence on later number theorists. Diophantine analysis, which is closely related to algebraic geometry, has experienced a resurgence of interest in the past half century is due to him.

Little is known about Diophantus except the fact that he was born around 200 AD in Alexandria and died at the age of 84 in 284 AD. The claims that Diophantus lived from about 200 to 284 and spent time at Alexandria are based on detective work in finding clues to the times he flourished in his and others’ writing. Theon of Alexandra quoted Diophantus in 365 and his work was the subject of a commentary written by Theon’s daughter Hypatia at the beginning of 5^th century, which unfortunately is lost. Michael Psellus, an 11^th century Byzantine scholar mentioned that Diophantus dealt with Egyptian arithmetic which later led to other people working and developing the same branch of mathematics. The most details we have of Diophantus’s life come from the Greek Anthology which estimates his life span to be around 84 years. This anthology says----

 His boyhood lasted ¹/₆th of his life; he married after ¹/₇th more; his beard grew after ¹/₁₂th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son.

Diophantus is best known for his famous work Arithmetica, written on the solution of algebraic equations and on the theory of numbers. The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method of solving indeterminate equation is known as Diophantine analysis. An equation is indeterminate if it has more than one variable. The Pythagorean Triplets containing the integers x, y and z satisfying the equation x² + y² = z² is the example of Diophantine analysis. Historically, the problem of solving Diophantine equation has been to find expressions that show the relationship of the integral values of x and y that satisfy an indeterminate equation ax + by = c, where the coefficients a, b and c are integers. Such equations have infinite number of solutions.

The Arithmetica had 13 volumes but only six of the original 13 books were thought to have survived. However, an Arabic manuscript in the library Astan-i-Quds (The Holy Shrine Library) in Meshed Iran has a title claiming it is a translation of Arithmetica and was discovered in 1968. In Arithmetica he considered the solutions of linear and quadratic equation. Diophantus looked at three types of quadratic equations ax² + bx = c, ax² = bx + c and ax² + c = bx. Since there was no symbol of zero when the book was written so Diophantus took all coefficients to be positive. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. Besides that, he also solved problem from Simultaneous equations.

Looking at the work of Diophantus it was quite clear that he was fully aware of the fact that every number can be written as the sum of four squares which was later proved by Lagrange. Here are two examples of the problems solved in Arithmetica.

1.      Find four numbers, the sum of every arrangement three at a time being given: say 22, 24 , 27 and 20. (Answer:- 9, 7 , 4 and 11)

2.      Divide a number, such as 13 which is the sum of two squares 4 and 9, into two other squares. (Answer:- 324/25 and 1/25)

In the Arithmetica, Diophantus stated certain theorem without proofs. It is believed that his proofs were found in his lost book The Porism, a collection of lemmas. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers. For any given numbers a, b there exist numbers c, d such that a³ - b³= c³ + d³.

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the seventeenth and eighteenth centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. European mathematicians did not learn of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid...

Bombelli translated much of the work in 1570 but it was never published. Bombelli did borrow many of Diophantus's problems for his own Algebra. The most famous Latin translation of the Diophantus's Arithmetica is due to Bachet in 1621. It is that edition which Fermat studied. Certainly Fermat was inspired by this work which has become famous in recent years due to its connection with Fermat's Last Theorem. Diophantus earliest work was probably his short essay on polygonal numbers, containing ten propositions in which he employed the classical method in which numbers are represented by line segments.  The topic on polygonal numbers was of great interest to Pythagoreans.

Diophantus worked before the introduction of modern algebraic notation, but he moved from rhetorical algebra to syncopated algebra, where abbreviations are used. Diophantus made important advancement in mathematical notation. 

He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. 

Mathematical historian Kurt Vogel states: “The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”

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Augustin Louis Cauchy

Augustin Louis Cauchy

The greatest French Mathematician who has been described as the first revolution of rigour in mathematics is undoubtedly Augustin Louis Cauchy. He contributed in a major way to almost every branch of mathematics. It is humorously said—Cauchy did not so much master mathematics, mathematics mastered Cauchy.

Augustin Louis Cauchy was born in Paris on August 21, 1789. He was christened Augustin because he was born in August and Louis after his father’s name Louis Francois. His father Louis Francois was a parliamentary lawyer and a lieutenant of police in Paris. His mother Marie- Madeleine came from a very reputed Parisian family. Cauchy’s childhood fell in the bloodiest period of the revolution. In 1794, the Cauchy family fled to Arcueil to escape the terror. Louis Francois could hardly manage to feed his family and as a consequence Cauchy grew up delicate and underdeveloped physically.

Two years later the Cauchy family returned to Paris where Louis Francois began to rebuild his carrier under the new regime and was appointed as secretary general to the newly constituted Senate of which Laplace was chancellor. Louis had a great friendship with Lagrange’s and Laplace and they used to visit Cauchy’s house every now and then. Lagrange was impressed by the ability of Augustin and on his advice he was enrolled at the Ecole Centrale de Pantheon to study humanities. Pointing to Louis, Lagrange once said--- he will supplement all of us so far as we are mathematicians. Don’t let him touch a mathematical book till he is 17.

At the age of 13, Augustin Cauchy entered the Central school of Pantheon. He won the grand prize for the best student instituted by Napoleon. On leaving the school in 1804, Cauchy won the sweepstakes and a special prize in mathematics. For the next ten months he studied mathematics intensively with a good tutor and at the age of 16, he entered the Ecole Polytechnic. By 1810, he was a qualified Junior Engineer.

In March 1810, he left Paris to work on the construction of a naval base at Cherbourg, the Port Napoleon.  Cauchy took 4 books with him to Cherbourg-- The Mecanique celeste of Laplace, The Traite des fonctions analytiques of Lagrange, The Imitation of Christ by Thomas a Kempls and the work of Virgil. He remained there for 3 years where he used to do some mathematical research in part time.

In December 1810, he had begun to go over again all branches of mathematics, beginning with Arithmetic and finishing with Astronomy. Some of his discoveries were sufficiently significant to attract the attention of learned society in Paris particularly the memoir on polyhedral and that on symmetric functions, which included the germs of the fundamental ideas that eventually blossomed into Group Theory. He also extended the formula of Euler connecting the numbers(E), faces(F) and vertices(V) of a polyhedron given as E+2 =F+V

He also developed the theory of determinants but he got his due recognition as a brilliant mathematician when he submitted a work on Calculation of Definite Integrals to the Academy in 1814. In 1815, Cauchy created a sensation by proving one of the great theorems which Fermat has bequeathed to a baffled posteiry.

Every positive integers is a sum of three triangles, four squares, five pentagons, six hexagons and so on---

In 1816, he received the Grand Prize offered by the Academy for a theory of the propagation of waves on the surface of a heavy fluid of indefinite depth. In 1816 itself he was elected a member of the Paris Academy. This success made him the enemies of those great mathematician likes Monge and Carnot who were displaced from Academy. In the same year, he was appointed Assosiate Professor of Analysis at the Ecole Polytechnic.

The discovery with which Cauchy’s name is most firmly associated is his fundamental theorems in Complex Analysis. The first pivotal theorem proved by Cauchy, now known as Cauchy's integral theorem, was the following:

Where  f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. The contour integral is taken along the contour C.

 He is therefore known as the father of Complex Analysis.  In 1818, Cauchy married with Aloise de Bure. She was a close relative of the publisher who published most of Cauchy's works. They had two daughters Marie Francoise Alicia and Marie Mathilde born in 1819 and 1823 respectively. Encouraged by Laplace, Cauchy in 1821 wrote up for publication of his lecture on Analysis given in Ecole Polytechnic. This book contains the definition of limits and continuity and convergence of infinite series which we still read in the book of Calculus.

In 1825, Cauchy started his own monthly journal Exercises de mathemaliques. Its pages were exclusively filled by Cauchy himself. Cauchy's writings covered notable topics including: the theory of series, where he developed the notion of convergence and discovered many of the basic formulas for q-series. The theory of numbers and complex quantities; he was the first to define complex numbers as pairs of real numbers. 

The turmoil in 1830 in Paris had a great impact on the mind of Cauchy. Cauchy lost most of his positions at the institutes because he refused to take the oath of allegiance to the new king, Louis-Philippe.  He left for Switzerland and remain there in self imposed exile for eight years where he was offered the Post of Professor in Mahematical Physics by Charles Albert, the King of Sardinia.  He taught in Turin during 1832-1833. In 1831, he had been elected a foreign member of the Royal Swedish Academy of Sciences. In 1833, Cauchy was appointed tutor of Charles’ grandson.

Cauchy went back to Paris in 1838 when he finished his work with Charles X in Prague, and resumed his involvement with the Academy. At the time, because Cauchy was a mathematician, he was exempted from the oath of allegiance. After the establishment of the Second Republique in 1848, Cauchy resumed his position at the Sorbonne. Cauchy continued with his writings and publications through the remainder of his life. The long hour of work made him ill. On medical advice he left for Paris on May 12, 1857 but his health suddenly detorieted and he died on 23 rd May 1857 at the age of 67. His name is one of the 72 names inscribed on the Eiffel Tower.

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Casting Out Nines

                          Casting Out Nines

When you solve a sum whether it is addition, subtraction, multiplication or division, a doubt constantly haunts you. To avoid any mistake you begin to check every step and thus lose your precious time. Don’t you think that there should be a single method which is equally beneficial in all the basic mathematical operations? The answer to your question is YES, there is a method to check all the four fundamental mathematical operation i.e. addition, subtraction, multiplication or division and that too in few seconds. This wonder method is called Casting out Nines or Nines- Remainder Method.

How does this method work?

This is a big question. But you will be surprised to know that this method is so simple that even a primary school going kid can understand and check his/her calculation.

 Casting out Nines literally means to throw nines. Now let us focus on its working.

v  Add the digits of a number across, dropping out 9, to get a single figure. If it is not a single figure, add the digits obtained so as to get a single figure between  0 to 8.

v  9 is not taken into account in this process, as a digit sum of 9 is the same as a digit sum of zero.

Before coming to the actual problem, shouldn’t we understand the basic meaning of above two points? Let us take some examples to understand what meaning the above line has.

Example: Find the digit sum of 54653

Verification: Digit sum of 54653= 5 + 4 + 6 + 5 +3 = 23

                Since 23 is a double figure number so to get a single figure, we have to sum it again.

                 Digit sum of 23 = 2 + 3 = 5

The digit sum of 54653 can be done in other way very easily. As discussed above, we need not take 9 into account

Digit sum of 54653 = 5   +   4    +    6 +      5   +      3

The two group of number 5+ 4 and 6+ 3 can easily be left out while finding the digit sum of 54653, as their sum is equal to 9.

Example: Find the digit sum of 438219

Verification: Add all the digits

                        4 + 3 + 8 + 2 + 1 + 9 = 27

                 Digit sum of 27 = 2 + 7 = 9 

Check for Addition

Whatever we do to the numbers, we also do to their digit sum; then the result that we have from the digit sum of the numbers, must be equal to the digit sum of answer.

Example: Verify 47385 + 69384 + 58769+ 38173 + 29464 = 243144

Verification:                                                   Digit sum of number

                        47385                                                    0             

                        69384                                                    3

                        58769                                                    8

                        38173                                                    4

               +       29464                                                    3

                       243144                                               ?

LH.S. = Digit sum of 243144 = 2 + 4 + 3 + 1 + 4 + 4 = 0

R.H.S. = Sum of the Digit sum of number = 0 + 3 + 8 + 4 + 3 = 0

Since LHS = RHS

Result Verified

Example: Verify   87643 + 84397+ 38549 + 29765 = 240354

Verification                                                                    Digit sum of number

                        87643                                                                    1

                        84397                                                                    4

                        38549                                                                    2

                        29765                                                                    2

                        240354                                                                  ?

Check for subtraction

The process of subtraction is same as applied in addition. Remember that the value of the digit sum of minuend should be greater than that of subtrahend.

Example: Verify the result, 8934 – 6758 = 2176

Verification:                                                                Digit sum of number

                        8934                                                       6

                 -    6758                                                        8

                      2176                                                         ?

Since the digit sum of Minuend is less than the digit sum of Subtrahend, hence we need to replace the digit sum of 8934, in such a way that the final digit sum remains the same.

Here the value of the digit sum of 8934 = 6

This digit sum 6 can also be written in so many ways as the digit sum of 15, 24, 33, 42, 51 and 60 also gives the same value 6. Therefore, the need of the hour is to replace the digit sum of 8934 with any of the given value 15, 24, 33,42,51,60.

Digit sum of number

                        8934                                       15 or 24 or 33 or 42 or 51 or 60

                 -    6758                                                        8

                      2176                                                         ?

LHS = Digit sum of 2176 = 2 + 1 + 7+ 6 = 7

RHS = Digit sum of (15 – 8) = 7

          Or, Digit sum of (24 – 8 = 16) = 1 + 6 = 7

          Or, Digit sum of (33 – 8 = 25) = 2 + 5 = 7

          Or, Digit sum of (42 – 8 = 34) = 3 + 4 = 7

          Or, Digit sum of (51 – 8 = 43) = 4 + 3 = 7

          Or, Digit sum of (60 – 8 = 52) = 5+ 2 = 7

In all the above cases

LHS = RHS

LH.S. = Digit sum of 240354 = 2 + 4 + 0 + 3 + 5 + 4 = 0

R.H.S. = Sum of the Digit sum of number = 1 + 4 + 2 + 2 = 0

Check for Multiplication

Multiplication is the most error prone fundamental operation in mathematics. Students always have doubts about the accuracy of their result and waste time in re-checking every operation again and again. This method will prove a panacea for all those who are not very much sure about their result. Let us take few examples to understand the modus operandi of this method.

We know,

Multiplicand X Multiplier = Product

 Example: Verify 12876 x 43853 = 564651228

Verification:-          

 Digit sum of Multiplicand = 1 + 2 + 8 + 7 + 6 = 6

 Digit sum of Multipliers =4 + 3 + 8 + 5 + 3 = 5                              

LHS = Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS= Digit sum of Product =5 + 6 + 4 + 6 + 5+ 1 + 2 + 2 + 8 = 3

Since LHS = RHS

Result Verified

Example: Verify 5972 X 4853 = 29882116

Verification:-          

 Digit sum of Multiplicand = 5 + 9 + 7 + 2 = 5

 Digit sum of Multipliers = 4 + 8+ 5 + 3 = 2                     

LHS = Digit sum of Multiplicand and Multiplier taken together (5 X 2= 10)= 1 RHS=  Digit sum of Product = 2 + 9 + 8 + 8 + 2 + 1 + 1 + 6 = 1

Since LHS = RHS

Result Verified

Example: Verify 12589476 x 43256853 = 54058115267908

Verification:-          

 Digit sum of Multiplicand =1 + 2 + 5 + 8 + 9 + 4+ 7 + 6 = 6

 Digit sum of Multipliers =4 + 3 +2 + 5+ 6+ 8 + 5 + 3 = 5                             

LHS = Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS= Digit sum of Product = 5+ 4+ 0+ 5+ 8+1+1+ 5+ 2+ 6+ 7+ 9+ 0+ 8 = 7

Since LHS ≠ RHS

Result Incorrect

Check for Division

We Know

Dividend = Divisor x Quotient + Remainder

The casting out nines method described in the very beginning will suffice to check the division operation effectively. You have only one thing to do-

Find the digit sum of Dividend, Divisor, Quotient and Remainder and put the value of digits sum in LHS and RHS. If the same digit sum is obtained in both the side, it ultimately tells you that you have performed the right operation. Let us take some examples to understand how effectively this method work for Division.

Example: Verify 876543 ÷ 123, Q = 7126 and R = 45

Verification:-

Here Dividend = 876543

Digit sum of Dividend = 8 + 7 + 6 + 5 + 4 + 3 = 6

Divisor = 123

Digit sum of Divisor = 1 + 2 + 3 = 6

Quotient = 7126

Digit sum of quotient = 7 + 1 + 2 + 6= 7

Remainder = 45

Digit sum of Remainder = 4 + 5 = 0

Putting the digit sum value in the given formulae, we get,

L.H.S = Digit sum of Dividend = 6

R.H.S = Divisor x Quotient + Remainder

          = 6 X 7 + 0 = 42

Digit sum of 42 = 6

Hence LHS = RHS

Result Verified

Example: Verify 8765 ÷ 243, Q = 36 and R = 23