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86-504: Gaṇita Kaumudī contains about 475 verses of sūtra (rules) and 395 verses of udāharaṇa (examples). It is divided into 14 chapters ( vyavahāra ): Weights and measures, length, area, volume, etc. It describes addition, subtraction, multiplication, division, square, square root, cube and cube root. The problems of linear and quadratic equations described here are more complex than in earlier works. 63 rules and 82 examples Mathematics pertaining to daily life: “mixture of materials, interest on

172-624: A {\displaystyle a} is odd, x = a 2 ( a 2 − 3 ) , y = b 2 ( a 2 − 1 ) {\displaystyle x={\frac {a}{2}}(a^{2}-3),y={\frac {b}{2}}(a^{2}-1)} When k = − 4 {\displaystyle k=-4} , then ( a 2 ) 2 − N ( b 2 ) 2 = − 1 {\displaystyle ({\frac {a}{2}})^{2}-N({\frac {b}{2}})^{2}=-1} . Composing with itself yields (

258-441: A 2 − 1 , y = 2 a b {\displaystyle x=2a^{2}-1,y=2ab} Again using the equation, ( 2 a 2 − k ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (2a^{2}-k)^{2}-N(2ab)^{2}=k^{2}} ⇒ {\displaystyle \Rightarrow } ( 2

344-500: A 2 − 2 2 , a b 2 , 1 ) {\displaystyle ({\frac {a^{2}-2}{2}},{\frac {ab}{2}},1)} . Composing the triples gives ( a 2 ( a 2 − 3 ) ) 2 − N ( b 2 ( a 2 − 1 ) ) 2 = 1 {\displaystyle ({\frac {a}{2}}(a^{2}-3))^{2}-N({\frac {b}{2}}(a^{2}-1))^{2}=1} When

430-404: A 2 − 4 4 ) 2 − N ( 2 a b 4 ) 2 = 1 {\displaystyle ({\frac {2a^{2}-4}{4}})^{2}-N({\frac {2ab}{4}})^{2}=1} . Leading to the triples ( a 2 , b 2 , 1 ) {\displaystyle ({\frac {a}{2}},{\frac {b}{2}},1)} and (

516-573: A 2 − 61 b 2 = 1 {\displaystyle a^{2}-61b^{2}=1} ), issued as a challenge by Fermat many centuries later, was given by Bhaskara as an example. We start with a solution a 2 − 61 b 2 = k {\displaystyle a^{2}-61b^{2}=k} for any k found by any means. In this case we can let b be 1, thus, since 8 2 − 61 ⋅ 1 2 = 3 {\displaystyle 8^{2}-61\cdot 1^{2}=3} , we have

602-475: A 2 − k ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (2a^{2}-k)^{2}-N(2ab)^{2}=k^{2}} The new triple can be expressed as ( 2 a 2 − k , 2 a b , k 2 ) {\displaystyle (2a^{2}-k,2ab,k^{2})} . Substituting k = − 1 {\displaystyle k=-1} gives

688-398: A 2 − k k ) 2 − N ( 2 a b k ) 2 = 1 {\displaystyle \left({\frac {2a^{2}-k}{k}}\right)^{2}-N\left({\frac {2ab}{k}}\right)^{2}=1} Substituting k = 2 {\displaystyle k=2} , x = a 2 − 1 , y =

774-482: A 2 + 2 ) ( a 4 + 4 a 2 + 1 ) 2 ) 2 − N ( a b ( a 2 + 3 ) ( a 2 + 1 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)(a^{4}+4a^{2}+1)}{2}})^{2}-N({\frac {ab(a^{2}+3)(a^{2}+1)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( (

860-575: A 2 + 2 ) ( a 4 + 4 a 2 + 2 ) + N a 2 b 2 ( a 2 + 2 ) 4 ) 2 − N ( a b ( a 4 + 4 a 2 + 3 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)(a^{4}+4a^{2}+2)+Na^{2}b^{2}(a^{2}+2)}{4}})^{2}-N({\frac {ab(a^{4}+4a^{2}+3)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( (

946-462: A 2 + 2 ) [ ( a 2 + 1 ) ( a 2 + 3 ) − 2 ) ] 2 ) 2 − N ( a b ( a 2 + 3 ) ( a 2 + 1 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)[(a^{2}+1)(a^{2}+3)-2)]}{2}})^{2}-N({\frac {ab(a^{2}+3)(a^{2}+1)}{2}})^{2}=1} . This give us

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1032-397: A 2 + N b 2 4 ) 2 − N ( a b 2 ) 2 = 1 {\displaystyle ({\frac {a^{2}+Nb^{2}}{4}})^{2}-N({\frac {ab}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( a 2 + 2 2 ) 2 − N (

1118-416: A 4 + 4 a 2 + 2 2 ) 2 − N ( a b ( a 2 + 2 ) 2 ) 2 = 1 {\displaystyle ({\frac {a^{4}+4a^{2}+2}{2}})^{2}-N({\frac {ab(a^{2}+2)}{2}})^{2}=1} Finally, from the earlier equations, compose the triples ( a 2 + 2 2 ,

1204-432: A , b , k ) = ( 8 , 1 , − 3 ) {\displaystyle (a,b,k)=(8,1,-3)} . We want a positive integer m such that k divides a  +  bm , i.e. 3 divides 8 + m, and | m  − 67| is minimal. The first condition implies that m is of the form 3 t + 1 (i.e. 1, 4, 7, 10,… etc.), and among such m , the minimal value is attained for m = 7. Replacing (

1290-473: A b {\displaystyle x=a^{2}-1,y=ab} Substituting k = − 2 {\displaystyle k=-2} , x = a 2 + 1 , y = a b {\displaystyle x=a^{2}+1,y=ab} Substituting k = 4 {\displaystyle k=4} into the equation ( 2 a 2 − 4 4 ) 2 − N ( 2

1376-615: A b 2 ) 2 = 1 {\displaystyle ({\frac {a^{2}+2}{2}})^{2}-N({\frac {ab}{2}})^{2}=1} . Again composing itself yields ( ( a 2 + 2 ) 2 + N a 2 b 2 ) 4 ) 2 − N ( a b ( a 2 + 2 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)^{2}+Na^{2}b^{2})}{4}})^{2}-N({\frac {ab(a^{2}+2)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } (

1462-401: A b 2 , 1 ) {\displaystyle ({\frac {a^{2}+2}{2}},{\frac {ab}{2}},1)} and ( a 4 + 4 a 2 + 2 2 , a b ( a 2 + 2 ) 2 , 1 ) {\displaystyle ({\frac {a^{4}+4a^{2}+2}{2}},{\frac {ab(a^{2}+2)}{2}},1)} , to get ( (

1548-398: A b 4 ) 2 = 1 {\displaystyle ({\frac {2a^{2}-4}{4}})^{2}-N({\frac {2ab}{4}})^{2}=1} creates the triple ( a 2 − 2 2 , a b 2 , 1 ) {\displaystyle ({\frac {a^{2}-2}{2}},{\frac {ab}{2}},1)} . Which is a solution if a {\displaystyle a}

1634-531: A Lutheran , is also the author of 3:16 Bible Texts Illuminated , in which he examines the Bible by a process of systematic sampling , namely an analysis of chapter 3, verse 16 of each book. Each verse is accompanied by a rendering in calligraphic art, contributed by a group of calligraphers led by Hermann Zapf . Knuth was invited to give a set of lectures at MIT on the views on religion and computer science behind his 3:16 project, resulting in another book, Things

1720-423: A Society for Industrial and Applied Mathematics conference and someone asked what he did. At the time, computer science was partitioned into numerical analysis , artificial intelligence , and programming languages . Based on his study and The Art of Computer Programming book, Knuth decided the next time someone asked he would say, "Analysis of algorithms". In 1969, Knuth left his position at Princeton to join

1806-462: A  +  bm , and | m  − 67| is minimal. We then update a , b , and k to a m + N b | k | , a + b m | k | {\displaystyle {\frac {am+Nb}{|k|}},{\frac {a+bm}{|k|}}} and m 2 − N k {\displaystyle {\frac {m^{2}-N}{k}}} respectively. We have (

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1892-781: A ,  b ,  k ) with ( a m + N b | k | , a + b m | k | , m 2 − N k ) {\displaystyle \left({\frac {am+Nb}{|k|}},{\frac {a+bm}{|k|}},{\frac {m^{2}-N}{k}}\right)} , we get the new values a = ( 8 ⋅ 7 + 67 ⋅ 1 ) / 3 = 41 , b = ( 8 + 1 ⋅ 7 ) / 3 = 5 , k = ( 7 2 − 67 ) / ( − 3 ) = 6 {\displaystyle a=(8\cdot 7+67\cdot 1)/3=41,b=(8+1\cdot 7)/3=5,k=(7^{2}-67)/(-3)=6} . That is, we have

1978-537: A Computer Scientist Rarely Talks About , where he published the lectures God and Computer Science . Knuth strongly opposes granting software patents to trivial solutions that should be obvious, but has expressed more nuanced views for nontrivial solutions such as the interior-point method of linear programming . He has expressed his disagreement directly to both the United States Patent and Trademark Office and European Patent Organisation . In

2064-407: A National Science Foundation Fellowship and Woodrow Wilson Foundation Fellowship but they had the condition that you could not do anything else but study as a graduate student so he would not be able to continue as a consultant to Burroughs. He chose to turn down the fellowships and continued with Burroughs. In summer 1962, he wrote a FORTRAN compiler for Univac, but considered that “I sold my soul to

2150-632: A PhD in mathematics from the California Institute of Technology , with a thesis titled Finite Semifields and Projective Planes . In 1963, after receiving his PhD, Knuth joined Caltech's faculty as an assistant professor. While at Caltech and after the success of the Burroughs B205 ALGOL compiler, he became consultant to Burroughs Corporation, joining the Product Planning Department. At Caltech he

2236-443: A book on computer programming language compilers . While working on this project, he decided that he could not adequately treat the topic without first developing a fundamental theory of computer programming, which became The Art of Computer Programming . He originally planned to publish this as a single book, but as he developed his outline for the book, he concluded that he required six volumes, and then seven, to thoroughly cover

2322-404: A correspondent, "Beware of bugs in the above code; I have only proved it correct, not tried it." Knuth published his first "scientific" article in a school magazine in 1957 under the title "The Potrzebie System of Weights and Measures". In it, he defined the fundamental unit of length as the thickness of Mad No. 26, and named the fundamental unit of force "whatmeworry". Mad published

2408-429: A general way to find x {\displaystyle x} and y {\displaystyle y} of x 2 = N y 2 + 1 , {\displaystyle x^{2}=Ny^{2}+1,} when given a 2 = N b 2 + k {\displaystyle a^{2}=Nb^{2}+k} , when k is ±1, ±2, or ±4. Using Brahmagupta's identity to compose

2494-588: A mythical range of mountains which orbits around the Earth like a wall and not limited by light and darkness. Brahmagupta in 628 CE studied indeterminate quadratic equations, including Pell's equation for minimum integers x and y . Brahmagupta could solve it for several N , but not all. Jayadeva and Bhaskara offered the first complete solution to the equation, using the chakravala method to find for x 2 = 61 y 2 + 1 , {\displaystyle \,x^{2}=61y^{2}+1,}

2580-467: A new triple In the general method, the main idea is that any triple ( a , b , k ) {\displaystyle (a,b,k)} (that is, one which satisfies a 2 − N b 2 = k {\displaystyle a^{2}-Nb^{2}=k} ) can be composed with the trivial triple ( m , 1 , m 2 − N ) {\displaystyle (m,1,m^{2}-N)} to get

2666-464: A novel approach that Newsweek and CBS Evening News later reported on. Knuth was one of the founding editors of the Case Institute's Engineering and Science Review , which won a national award as best technical magazine in 1959. He then switched from physics to mathematics, and received two degrees from Case in 1960: his Bachelor of Science, and simultaneously a master of science by

Ganita Kaumudi - Misplaced Pages Continue

2752-465: A principal, payment in instalments, mixing gold objects with different purities and other problems pertaining to linear indeterminate equations for many unknowns” 42 rules and 49 examples Arithmetic and geometric progressions, sequences and series. The generalization here was crucial for finding the infinite series for sine and cosine. 28 rules and 19 examples. Geometry. 149 rules and 94 examples. Includes special material on cyclic quadratilerals, such as

2838-514: A small printing business and taught bookkeeping. While a student at Milwaukee Lutheran High School , Knuth thought of ingenious ways to solve problems. For example, in eighth grade, he entered a contest to find the number of words that the letters in "Ziegler's Giant Bar" could be rearranged to create; the judges had identified 2,500 such words. With time gained away from school due to a fake stomachache, Knuth used an unabridged dictionary and determined whether each dictionary entry could be formed using

2924-433: A solution a 2 − 67 b 2 = k {\displaystyle a^{2}-67b^{2}=k} for any k found by any means; in this case we can let b be 1, thus producing 8 2 − 67 ⋅ 1 2 = − 3 {\displaystyle 8^{2}-67\cdot 1^{2}=-3} . At each step, we find an m  > 0 such that k divides

3010-427: A solution: x = 2 a 2 + 1 , y = 2 a b {\displaystyle x=2a^{2}+1,y=2ab} For k = 1 {\displaystyle k=1} , the original ( a , b ) {\displaystyle (a,b)} was already a solution. Substituting k = 1 {\displaystyle k=1} yields a second: x = 2

3096-525: A special award of the faculty, who considered his work exceptionally outstanding. At the end of his senior year at Case in 1960, Knuth proposed to Burroughs Corporation to write an ALGOL compiler for the B205 for $ 5,500. The proposal was accepted and he worked on the ALGOL compiler between graduating from Case and going to Caltech . In 1963, with mathematician Marshall Hall as his adviser, he earned

3182-508: Is Gao Dena ( simplified Chinese : 高德纳 ; traditional Chinese : 高德納 ; pinyin : Gāo Dénà ). He was given this name in 1977 by Frances Yao shortly before making a three-week trip to China . In the 1980 Chinese translation of Volume 1 of The Art of Computer Programming ( simplified Chinese : 计算机程序设计艺术 ; traditional Chinese : 計算機程式設計藝術 ; pinyin : Jìsuànjī chéngxù shèjì yìshù ), Knuth explains that he embraced his Chinese name because he wanted to be known by

3268-610: Is −4, we can use Brahmagupta's idea: it can be scaled down to the rational solution ( 39 / 2 , 5 / 2 , − 1 ) {\displaystyle (39/2,5/2,-1)\,} , which composed with itself three times, with m = 7 , 11 , 9 {\displaystyle m={7,11,9}} respectively, when k becomes square and scaling can be applied, this gives ( 1523 / 2 , 195 / 2 , 1 ) {\displaystyle (1523/2,195/2,1)\,} . Finally, such procedure can be repeated until

3354-463: Is a cyclic algorithm to solve indeterminate quadratic equations , including Pell's equation . It is commonly attributed to Bhāskara II , (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE). Jayadeva pointed out that Brahmagupta 's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his Bijaganita treatise. He called it

3440-617: Is an organist and a composer . He and his father served as organists for Lutheran congregations. Knuth and his wife have a 16-rank organ in their home. In 2016 he completed a piece for organ, Fantasia Apocalyptica , which he calls a "translation of the Greek text of the Revelation of Saint John the Divine into music". It was premièred in Sweden on January 10, 2018. Knuth's Chinese name

3526-457: Is even: x = a 2 − 2 2 , y = a b 2 {\displaystyle x={\frac {a^{2}-2}{2}},y={\frac {ab}{2}}} If a is odd, start with the equations ( a 2 ) 2 − N ( b 2 ) 2 = 1 {\displaystyle ({\frac {a}{2}})^{2}-N({\frac {b}{2}})^{2}=1} and ( 2

Ganita Kaumudi - Misplaced Pages Continue

3612-460: Is of the form 6 t  + 5 (i.e. 5, 11, 17,… etc.), and among such m , | m  − 67| is minimal for m  = 5. This leads to the new solution a  = (41⋅5 + 67⋅5)/6, etc.: For 7 to divide 90 + 11 m , we must have m = 2 + 7 t (i.e. 2, 9, 16,… etc.) and among such m , we pick m = 9. At this point, we could continue with the cyclic method (and it would end, after seven iterations), but since

3698-529: Is one hexadecimal dollar", and $ 0.32 for "valuable suggestions". According to an article in the Massachusetts Institute of Technology 's Technology Review , these Knuth reward checks are "among computerdom's most prized trophies". Knuth had to stop sending real checks in 2008 due to bank fraud, and now gives each error finder a "certificate of deposit" from a publicly listed balance in his fictitious "Bank of San Serriffe ". He once warned

3784-470: Is scaled down (or Bhaskara's lemma is directly used) to get: For 3 to divide 8 + m {\displaystyle 8+m} and | m 2 − 61 | {\displaystyle |m^{2}-61|} to be minimal, we choose m = 7 {\displaystyle m=7} , so that we have the triple ( 39 , 5 , − 4 ) {\displaystyle (39,5,-4)} . Now that k

3870-460: Is to instruct a computer what to do, let us concentrate rather on explaining to human beings what we want a computer to do. Knuth embodied the idea of literate programming in the WEB system. The same WEB source is used to weave a TeX file, and to tangle a Pascal source file. These in their turn produce a readable description of the program and an executable binary respectively. A later iteration of

3956-844: Is useful to find a solution to Pell's Equation , but it is not always the smallest integer pair. e.g. 36 2 − 52 ∗ 5 2 = − 4 {\displaystyle 36^{2}-52*5^{2}=-4} . The equation will give you x = 1093436498 , y = 151632270 {\displaystyle x=1093436498,y=151632270} , which when put into Pell's Equation yields 1195601955878350801 − 1195601955878350800 = 1 {\displaystyle 1195601955878350801-1195601955878350800=1} , which works, but so does x = 649 , y = 90 {\displaystyle x=649,y=90} for N = 52 {\displaystyle N=52} . The n  = 61 case (determining an integer solution satisfying

4042-529: The ACM Turing Award , informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms ". Knuth is the author of the multi-volume work The Art of Computer Programming . He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In

4128-567: The MIX / MMIX instruction set architectures . He strongly opposes the granting of software patents , and has expressed his opinion to the United States Patent and Trademark Office and European Patent Organisation . Donald Knuth was born in Milwaukee , Wisconsin , to Ervin Henry Knuth and Louise Marie Bohning. He describes his heritage as "Midwestern Lutheran German". His father owned

4214-506: The Stanford University faculty, where he became Fletcher Jones Professor of Computer Science in 1977. He became Professor of The Art of Computer Programming in 1990, and has been emeritus since 1993. Knuth is a writer as well as a computer scientist. "The best way to communicate from one human being to another is through story." In the 1970s, Knuth called computer science "a totally new field with no real identity. And

4300-585: The Theta Chi fraternity . While studying physics at Case, Knuth was introduced to the IBM 650 , an early commercial computer . After reading the computer's manual, Knuth decided to rewrite the assembly and compiler code for the machine used in his school because he believed he could do it better. In 1958, Knuth created a program to help his school's basketball team win its games. He assigned "values" to players in order to gauge their probability of scoring points,

4386-725: The square root of 61, while the chakravala method is much simpler. Selenius, in his assessment of the chakravala method, states Hermann Hankel calls the chakravala method From Brahmagupta's identity , we observe that for given N , For the equation x 2 − N y 2 = k {\displaystyle x^{2}-Ny^{2}=k} , this allows the "composition" ( samāsa ) of two solution triples ( x 1 , y 1 , k 1 ) {\displaystyle (x_{1},y_{1},k_{1})} and ( x 2 , y 2 , k 2 ) {\displaystyle (x_{2},y_{2},k_{2})} into

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4472-560: The sum of unit fractions had previously been given in the Gaṇita-sāra-saṅgraha of Mahāvīra ( c.  850 ). Nārāyaṇa's Gaṇita-kaumudi gave a few more rules: the section bhāgajāti in the twelfth chapter named aṃśāvatāra-vyavahāra contains eight rules. The first few are: Combinatorics. 97 rules and 45 examples. Generating permutations (including of a multiset), combinations, integer partitions , binomial coefficients, generalized Fibonacci numbers. Narayana Pandita noted

4558-509: The 1960s, and was acknowledged as a major contributor in Joseph Madachy 's Mathematics on Vacation . Knuth also appears in a number of Numberphile and Computerphile videos on YouTube , where he discusses topics from writing Surreal Numbers to why he does not use email. Knuth had proposed the name " algorithmics " as a better name for the discipline of computer science. In addition to his writings on computer science, Knuth,

4644-576: The 1970s, the publishers of TAOCP abandoned Monotype in favor of phototypesetting . Knuth became so frustrated with the inability of the latter system to approach the quality of the previous volumes, which were typeset using the older system, that he took time out to work on digital typesetting and created TeX and Metafont . While developing TeX, Knuth created a new methodology of programming, which he called literate programming , because he believed that programmers should think of programs as works of literature: Instead of imagining that our main task

4730-617: The Chakravala method: chakra meaning "wheel" in Sanskrit , a reference to the cyclic nature of the algorithm. C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity. This method is also known as the cyclic method and contains traces of mathematical induction . Chakra in Sanskrit means cycle. As per popular legend, Chakravala indicates

4816-510: The TUG 2010 Conference, Knuth announced a satirical XML -based successor to TeX, titled "iTeX" ( pronounced [iː˨˩˦tɛks˧˥] , performed with a bell ringing), which would support features such as arbitrarily scaled irrational units, 3D printing , input from seismographs and heart monitors, animation, and stereophonic sound. In 1971, Knuth received the first ACM Grace Murray Hopper Award . He has received various other awards, including

4902-455: The Vedic period: the Śulba Sūtras give an approximation of √ 2 equivalent to 1 + 1 3 + 1 3 ⋅ 4 − 1 3 ⋅ 4 ⋅ 34 {\displaystyle 1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}} . Systematic rules for expressing a fraction as

4988-400: The article in issue No. 33 (June 1957). To demonstrate the concept of recursion , Knuth intentionally referred "Circular definition" and "Definition, circular" to each other in the index of The Art of Computer Programming , Volume 1 . The preface of Concrete Mathematics has the following paragraph: When DEK taught Concrete Mathematics at Stanford for the first time, he explained

5074-482: The book to prepare students for doing original, creative research. In 1995, Knuth wrote the foreword to the book A=B by Marko Petkovšek , Herbert Wilf and Doron Zeilberger . He also occasionally contributes language puzzles to Word Ways: The Journal of Recreational Linguistics . Knuth has delved into recreational mathematics . He contributed articles to the Journal of Recreational Mathematics beginning in

5160-399: The chosen value. This results in a new triple ( a , b , k ). The process is repeated until a triple with k = 1 {\displaystyle k=1} is found. This method always terminates with a solution (proved by Lagrange in 1768). Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases. In AD 628, Brahmagupta discovered

5246-453: The devil” to write a FORTRAN compiler. After graduating, Knuth returned to Burroughs in June 1961 but did not tell them he had graduated with a master's degree, rather than the expected bachelor's degree. Impressed by the ALGOL syntax chart, symbol table, recursive-descent approach and the separation of the scanning, parsing and emitting functions of the compiler Knuth suggested an extension to

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5332-670: The equivalence of the figurate numbers and the formulae for the number of combinations of different things taken so many at a time. The book contains a rule to determine the number of permutations of n objects and a classical algorithm for finding the next permutation in lexicographic ordering though computational methods have advanced well beyond that ancient algorithm. Donald Knuth describes many algorithms dedicated to efficient permutation generation and discuss their history in his book The Art of Computer Programming . Magic squares. 60 rules and 17 examples. Chakravala method The chakravala method ( Sanskrit : चक्रवाल विधि )

5418-512: The growing numbers of computer programmers in China at the time. In 1989, his Chinese name was placed atop the Journal of Computer Science and Technology 's header, which Knuth says "makes me feel close to all Chinese people although I cannot speak your language". Knuth used to pay a finder's fee of $ 2.56 for any typographical errors or mistakes discovered in his books, because "256 pennies

5504-495: The knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type n x 2 + k 2 = y 2 {\displaystyle nx^{2}+k^{2}=y^{2}} . Contains factorization method, 11 rules and 7 examples. Contains rules for writing a fraction as a sum of unit fractions. 22 rules and 14 examples. Unit fractions were known in Indian mathematics in

5590-537: The letters in the phrase. Using this algorithm, he identified over 4,500 words, winning the contest. As prizes, the school received a new television and enough candy bars for all of his schoolmates to eat. Knuth received a scholarship in physics to the Case Institute of Technology (now part of Case Western Reserve University ) in Cleveland , Ohio, enrolling in 1956. He also joined the Beta Nu Chapter of

5676-443: The new solution: At this point, one round of the cyclic algorithm is complete. We now repeat the process. We have ( a , b , k ) = ( 41 , 5 , 6 ) {\displaystyle (a,b,k)=(41,5,6)} . We want an m  > 0 such that k divides a  +  bm , i.e. 6 divides 41 + 5 m , and | m  − 67| is minimal. The first condition implies that m

5762-413: The new triple ( a m + N b , a + b m , k ( m 2 − N ) ) {\displaystyle (am+Nb,a+bm,k(m^{2}-N))} for any m . Assuming we started with a triple for which gcd ( a , b ) = 1 {\displaystyle \gcd(a,b)=1} , this can be scaled down by k (this is Bhaskara's lemma ): Since

5848-612: The process, he also popularized the asymptotic notation . In addition to fundamental contributions in several branches of theoretical computer science , Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems designed to encourage and facilitate literate programming , and designed

5934-460: The right-hand side is among ±1, ±2, ±4, we can also use Brahmagupta's observation directly. Composing the triple (221, 27, −2) with itself, we get Donald Knuth Donald Ervin Knuth ( / k ə ˈ n uː θ / kə- NOOTH ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University . He is the 1974 recipient of

6020-435: The signs inside the squares do not matter, the following substitutions are possible: When a positive integer m is chosen so that ( a  +  bm )/ k is an integer, so are the other two numbers in the triple. Among such m , the method chooses one that minimizes the absolute value of m  −  N and hence that of ( m  −  N )/ k . Then the substitution relations are applied for m equal to

6106-456: The solution This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat , using continued fractions. A method for the general problem was first completely described rigorously by Lagrange in 1766. Lagrange's method, however, requires the calculation of 21 successive convergents of the simple continued fraction for

6192-441: The solution is found (requiring 9 additional self-compositions and 4 additional square-scalings): ( 1766319049 , 226153980 , 1 ) {\displaystyle (1766319049,\,226153980,\,1)} . This is the minimal integer solution. Suppose we are to solve x 2 − 67 y 2 = 1 {\displaystyle x^{2}-67y^{2}=1} for x and y . We start with

6278-498: The solutions x = ( a 2 + 2 ) [ ( a 2 + 1 ) ( a 2 + 3 ) − 2 ) ] 2 y = a b ( a 2 + 3 ) ( a 2 + 1 ) 2 {\displaystyle x={\frac {(a^{2}+2)[(a^{2}+1)(a^{2}+3)-2)]}{2}}y={\frac {ab(a^{2}+3)(a^{2}+1)}{2}}} (Note, k = − 4 {\displaystyle k=-4}

6364-513: The somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of his colleagues, he was not going to teach the Theory of Aggregates, nor Stone's Embedding Theorem , nor even the Stone–Čech compactification . (Several students from the civil engineering department got up and quietly left the room.) At

6450-468: The standard of available publications was not that high. A lot of the papers coming out were quite simply wrong. ... So one of my motivations was to put straight a story that had been very badly told." From 1972 to 1973, Knuth spent a year at the University of Oslo among people such as Ole-Johan Dahl . This is where he had originally intended to write the seventh volume in his book series, which

6536-564: The state-of-the-art, co-designed with J. McNeeley. He attended a conference in Norway in May, 1967 organised by the people who invented the Simula language. Knuth influenced Burroughs to use Simula. Knuth had a long association with Burroughs as a consultant from 1960 to 1968 until his move into more academic work at Stanford in 1969. In 1962, Knuth accepted a commission from Addison-Wesley to write

6622-609: The subject. He published the first volume in 1968. Just before publishing the first volume of The Art of Computer Programming , Knuth left Caltech to accept employment with the Institute for Defense Analyses' Communications Research Division , then situated on the Princeton campus, which was performing mathematical research in cryptography to support the National Security Agency . In 1967, Knuth attended

6708-571: The symbol table that one symbol could stand for a string of symbols. This became the basis of the DEFINE in Burroughs ALGOL, which has since been adopted by other languages. However, some really disliked the idea and wanted DEFINE removed. The last person to think it was a terrible idea was Edsger Dijkstra on a visit to Burroughs. Knuth worked on simulation languages at Burroughs producing SOL ‘Simulation Oriented Language’, an improvement on

6794-497: The system, CWEB , replaces Pascal with C , C++ , and Java . Knuth used WEB to program TeX and METAFONT, and published both programs as books, both originally published the same year: TeX: The Program (1986); and METAFONT: The Program (1986). Around the same time, LaTeX , the now-widely adopted macro package based on TeX, was first developed by Leslie Lamport , who later published its first user manual in 1986. Donald Knuth married Nancy Jill Carter on 24 June 1961, while he

6880-413: The triple ( a , b , k ) {\displaystyle (a,b,k)} with itself: ( a 2 + N b 2 ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (a^{2}+Nb^{2})^{2}-N(2ab)^{2}=k^{2}} ⇒ {\displaystyle \Rightarrow } ( 2

6966-475: The triple ( a , b , k ) = ( 8 , 1 , 3 ) {\displaystyle (a,b,k)=(8,1,3)} . Composing it with ( m , 1 , m 2 − 61 ) {\displaystyle (m,1,m^{2}-61)} gives the triple ( 8 m + 61 , 8 + m , 3 ( m 2 − 61 ) ) {\displaystyle (8m+61,8+m,3(m^{2}-61))} , which

7052-479: The “third diagonal”. Excavations. 7 rules and 9 examples. Stacks. 2 rules and 2 examples. Mounds of grain. 2 rules and 3 examples. Shadow problems. 7 rules and 6 examples. Linear integer equations. 69 rules and 36 examples. Quadratic. 17 rules and 10 examples. Includes a variant of the Chakravala method . Ganita Kaumudi contains many results from continued fractions . In the text Narayana Pandita used

7138-640: Was a graduate student at the California Institute of Technology. They have two children: John Martin Knuth and Jennifer Sierra Knuth. Knuth gives informal lectures a few times a year at Stanford University , which he calls "Computer Musings". He was a visiting professor at the Oxford University Department of Computer Science in the United Kingdom until 2017 and an Honorary Fellow of Magdalen College . Knuth

7224-516: Was operating as a mathematician but at Burroughs as a programmer working with the people he considered to have written the best software at the time in the ALGOL compiler for the B220 computer (successor to the B205). He was offered a $ 100,000 contract to write compilers at Green Tree Corporation but turned it down making a decision not to optimize income and continued at Caltech and Burroughs. He received

7310-483: Was published in 1994. In April 2020, Knuth said he anticipated that Volume 4 will have at least parts A through F. Volume 4B was published in October 2022. Knuth is also the author of Surreal Numbers , a mathematical novelette on John Conway 's set theory construction of an alternate system of numbers. Instead of simply explaining the subject, the book seeks to show the development of the mathematics. Knuth wanted

7396-460: Was to deal with programming languages. But Knuth had finished only the first two volumes when he came to Oslo, and thus spent the year on the third volume, next to teaching. The third volume came out just after Knuth returned to Stanford in 1973. By 2011, Volume 4A had been published. Concrete Mathematics: A Foundation for Computer Science 2nd ed., which originated with an expansion of the mathematical preliminaries section of Volume 1 of TAoCP ,

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