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Deepdive Into Mathematics

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Deepdive Into Mathematics ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ ԳՐԻԳՈՐՅԱՆ Ա. ԴԱՐԲԻՆՅԱՆ Ա. ՍՏԵՓԱՆՅԱՆ Ն. ՄԱԹԵՄԱՏԻԿԱՅԻ ԱՇԽԱՐՀՈՒՄ Դասագիրք Մաթեմատիկայի և մեխանիկայի, Ինֆորմատիկայի և կիրառական մաթեմատիկայի ֆակուլտետների ուսանողների համար ԵՐԵՎԱՆ ԵՊՀ ՀՐԱՏԱՐԱԿՉՈՒԹՅՈՒՆ 2019 YEREVAN STATE UNIVERSITY GRIGORIAN A. DARBINYAN A. STEPANYAN N. DEEPDIVE INTO MATHEMATICS Textbook for the students of the Faculty of Mathematics and Mechanics, Informatics and Applied Mathematics YEREVAN YSU PRESS 2019 ՀՏԴ 51(075.8) ԳՄԴ 22.1ց73 Գ 888 ՀՀ ԿԳ նախարարության կողմից հաստատվել է որպես բուհական դասագիրք: Հրատարակության է երաշխավորել ԵՊՀ եվրոպական լեզուների և հաղորդակցության ֆակուլտետի գիտական խորհուրդը: Դասագիրքը կազմել են՝ Գրիգորյան Ա. – Lesson 2, 8, 14, 20, 21, Miscellaneous, Proper Names, Դարբինյան Ա. – Lesson 1, 3, 4, 5, 11, 12, 18, 22, Proper Names, Ստեփանյան Ն. – Lesson 6, 7, 9, 10, 13, 15, 16, 17, 19, Proper Names, Guide to Mathematical Notation. Գրիգորյան Ա., Դարբինյան Ա., Ստեփանյան Ն. Գ 888 Մաթեմատիկայի աշխարհում: Դասագիրք / Գրիգորյան Ա., Դարբինյան Ա., Ստեփանյան Ն.: -Եր., ԵՊՀ հրատ., 2019, 328 էջ: Դասագիրքը նախատեսված է բուհերի Մաթեմատիկայի և մեխանի- կայի, Ինֆորմատիկայի և կիրառական մաթեմատիկայի ֆակուլտետների բակալավրիատի ուսանողների, ինչպես նաև ճշգրիտ գիտություններով զբաղվող մասնագետների համար: ՀՏԴ 51(075.8) ԳՄԴ 22.1ց73 ISBN 978-5-8084-2376-3 ԵՊՀ հրատ., 2019 Գրիգորյան Ա., Դարբինյան Ա., Ստեփանյան Ն., 2019 4 ԱՌԱՋԱԲԱՆ “Deepdive into Mathematics” անգլերեն լեզվի սույն դասա- գիրքը բաղկացած է 22 դասից: Յուրաքանչյուր դաս սկսվում է բնաբանով, որը նպատակաուղղված է տվյալ նյութի նկատմամբ հետաքրքրություն առաջացնելուն: Դասագրքի նպատակն է ու- սուցանել մասնագիտական բառապաշար, ամրապնդել մի շարք կարևոր քերականական դրույթներ, զարգացնել ուսանողների բանավոր խոսքը մասնագիտության բնագավառում, օժանդակել գիտական զեկույցներ, հոդվածներ գրելուն և հետազոտություն- ներ կատարելուն: Դասում ներկայացված հիմնական նյութն ընտրված է բարձրագույն մաթեմատիկայի տարբեր բնագավառներից: Կան նաև տեքստեր մաթեմատիկոս-գիտնականների կյանքի և մաս- նագիտական գործունեության վերաբերյալ: Տեքստի բովանդակությունն ընկալելու և ուսանողի բառա- պաշարը հարստացնելու համար յուրաքանչյուր դասի տեքստին հաջորդում է մասնագիտական բառապաշարից բառացանկ, որ- տեղ տրվում է բառի հնչյունական տառադարձումը, խոսքիմա- սային պատկանելությունը և թարգմանությունը: Հաջորդիվ տեղ են գտել ավանդական տարաբնույթ վար- ժություններ, որոնց նպատակն է զարգացնել սովորողների գրա- վոր և բանավոր խոսքը, խորացնել գիտական բառապաշարը, կատարելագործել քերականական հմտությունները: Դասագրքում ընդգրկված վարժությունները միտված են կանխարգելել համընդհանուր բնույթի սխալները, ապահովել լեզվական նյութի ընկալումը: Հեղինակները դասագիրքը կազմե- լիս հիմնվել են նաև իրենց բազմամյա դասավանդման փորձի 5 վրա՝ այն համապատասխանեցնելով դասավանդման ժամանա- կակից պահանջներին: Տրված են նաև մաթեմատիկական խնդիրներ, որոնք հնա- րավորություն կտան ուսանողներին անգլերենով կատարել քննարկումներ իրենց մասնագիտության շուրջ: Դասերում հանդիպող հատուկ անուններն, իրենց տառա- դարձումներով, ամփոփված են տեղեկատու-հավելվածում՝ տվյալ գիտնականի կամ նշված վայրի վերաբերյալ համառոտ տեղեկությամբ: Հավելվածում առանձին բաժնով ներկայացված է մաթեմա- տիկական սիմվոլները, նշանները, կոտորակներն ու բանա- ձևերն անգլերենով արտահայտելու ուղեցույցը: Դասագրքում տեղ է գտել նաև լրացուցիչ ընթերցանության և ինքնուրույն աշխատանքի համար նախատեսված “Miscellaneous” խորագրով բաժինը, որը բաղկացած է մաթեմա- տիկային առնչվող տարբեր թեմաներով չափածո գրվածքներից, հանրահայտ գիտնականների կյանքի զվարճալի պատմություն- ներից և հումորից: Վերոհիշյալ բաժինը ամփոփվում է գիտնականների ասույթ- ներով: Սույն աշխատանքը խորհուրդ է տրվում օգտագործել անգլերեն լեզվի ուսուցանման Մաթեմատիկայի և մեխանիկայի, Ինֆորմատիկայի և կիրառական մաթեմատիկայի ֆակուլտետ- ների բակալավրիատի ուսանողների կրթության համար: Դասագրքի համահեղինակներն իրենց խորին երախտագի- տությունն են հայտնում գրախոսներ ԵՊՀ եվրոպական լեզու- ների և հաղորդակցության ֆակուլտետի անգլերեն լեզվի No 1 ամբիոնի վարիչ, բ.գ.թ., պրոֆեսոր Մ. Հ. Ապրեսյանին, ԵՊՀ մի- 6 ջազգային հարաբերությունների ֆակուլտետի դիվանագիտա- կան ծառայության և մասնագիտական հաղորդակցման ամբիո- նի բ.գ.թ., դոցենտ Ա. Մ. Բաբայանին, ՀԱՊՀ օտար լեզուների ամբիոնի վարիչ, բ.գ.թ. դոցենտ Հ. Զ. Ղազարյանին, ԵՊՀ մա- թեմատիկայի և մեխանիկայի ֆակուլտետի ֆինանսական մաթե- մատիկայի ամբիոնի վարիչ, ֆ.մ.գ.թ., դոցենտ Մ. Պ. Պողոս- յանին` օգտակար դիտարկումների, ինչպես նաև գրքի կայաց- մանը նպաստելու համար: 7 PREFACE The creation of the textbook “Deepdive into Mathematics” stemmed from the need to teach English on Bachelor’s courses and develop students’ study skills necessary to function effectively in a university context. Materials of the textbook are thoroughly selected to provide a good background of English for students-mathematicians and people who are interested in mathematics. The final objective of the textbook is to develop students’ confidence and competence, to make them autonomous learners in order to successfully carry out research and complete assignments, such as essays, reports, theses and oral presentations. The structure of the textbook: The textbook comprises 4 parts. The first part consists of 22 lessons. Each lesson begins with an epigraph, which is aimed at arousing students’ interest toward the contents of the lesson. The material of the texts is taken from different fields of higher mathematics. There are also texts about mathematicians and their contributions to the mathematical science, and the history of mathematics. The text to each lesson is provided with topical words, their transcriptions and translations into Armenian. The texts are followed by various lexical and grammatical exercises to develop students’ reading, writing, speaking skills, improve their grammar knowledge and enrich professional vocabulary. In the final part of each lesson some mathematical problems and puzzles are given to advance students’ skills to read mathematical formulas, prove theorems, memorize definitions in English. “Proper Names” contains valuable information about scientists and the geographical places mentioned in the textbook. 8 “Guide to Mathematical Symbols” possesses big tables of mathe- matical signs, symbols, formulas, equations and the way they are read in English. The last part of the textbook entitled “Miscellaneous” is an additional reading material for students’ individual work. It consists of poems, jokes, humorous stories and quotes on mathematics, as well as interesting facts about prominent scientists and real stories from their lives. 9 CONTENTS ԱՌԱՋԱԲԱՆ.…………………………………………………... 5 PREFACE………………………………………….…………. 8 CONTENTS….………………………………………………….. 10 LESSON 1 WHAT IS MATHEMATICS?.…………………… 12 LESSON 2 THE FIELDS OF MATHEMATICS…………….. 22 LESSON 3 FINANCIAL MATHEMATICS…………………. 37 LESSON 4 WHAT IS AN ACTUARY? …………………….. 47 LESSON 5 GEOMETRY THROUGH THE AGES………..... 57 LESSON 6 ALGEBRA AS A SCIENCE……………… 65 LESSON 7 THE THEORY OF EQUATIONS………… 74 LESSON 8 AMALIE EMMY NOETHER…………………… 82 LESSON 9 THE DIGIT THAT MEANS NOTHING………... 93 LESSON 10 MATHEMATICS AND ART……………. 105 LESSON 11 PROBABILITY………………………………… 113 LESSON 12 THE THEORY OF GAMES…………………… 123 LESSON 13 ALGORITHMS…………………………… 133 LESSON 14 JOHN VON NEUMANN………………………. 144 LESSON 15 THE GAME OF CHESS……………………….. 155 LESSON 16 GEORG CANTOR…………………………….. 167 10 LESSON 17 SET THEORY………………………………….. 177 LESSON 18 THE PRINCE OF AMATEURS…………..…… 187 LESSON 19 NEWTON AND LEIBNIZ ………………..…… 197 LESSON 20 THE THEORY OF ERRORS …………….……. 208 LESSON 21 THOMAS SIMPSON ……………………….…. 219 LESSON 22 ADA LOVELACE…………………………..…. 230 PROPER NAMES………………………………………….... 239 GUIDE TO MATHEMATICAL NOTATION……………... 275 MISCELLANEOUS………………………………………….. 302 BIBLIOGRAPHY…………………………………………….. 323 11 Lesson 1 WHAT IS MATHEMATICS? Mathematics is the science that draws necessary conclusions. BENJAMIN PIERCE`1 The word mathematics comes from Greek. It means “something that must be learnt or understood”. The plural form of the word mathematics goes back to the Latin mathematica based on the Greek plural used by Aristotle meaning “all things mathematical”. Following the pattern of word formation, according to which the words physics, politics, economics, etc. were formed, the English word mathematics came to be used for mathematical study. It is often shortened to maths and takes singular verb forms. Mathematics is the study of quantity, structure, space and change. As Karl Friedrich Gauss had said mathematics is the queen of all sciences. It helps the researchers to formulate new conjectures and establish new disciplines by rigorous deduction from axioms and definitions. Mathematics developed from counting, calculation, measurement and the study of the shapes and motions of physical objects by means of abstraction and logical reasoning. The first uses of mathematics were seen in trading and commerce, land measurement, painting and weaving patterns, the recording of time, taxation, building and construction. The development of mathematics began as early as 3000 BC by the Babylonians and Egyptians, but the systematic study of it began between 600 and 300 BC by the ancient Greeks. Later mathematics continued to develop in China, India and Arabia until the Renaissance when mathematical innovations led to new scientific discoveries. Mathematics has been supplying methods and conclusions for science 12 enabling scientists to predict results, describe phenomena and prepare their minds to new way of thinking. One of the basic characteristics of mathematics is its symbolic language which is commonly acknowledged as “the language of science”. It consists mostly of signs and symbols which are universal and have remained almost unchanged throughout the time. Unlike the common language
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