LEADER 00000cam 2200697 i 4500 001 on1107450146 003 OCoLC 005 20220607213020.0 006 m o d 007 cr n 008 190617t20192019enka o 001 0 eng d 015 GBB9C86632bnb 016 7 0194435302Uk 019 1117884270a1126178817a1135601942a1167704711a1202558352 a1228590603a1235833314a1292985248a1295860346 a1295992746a1303481852 020 9781783747016q(electronic book) 020 1783747013q(electronic book) 020 1783746998 020 9781783746996 020 1783747005 020 9781783747009 020 z9781783747009q(Hardback) 020 z9781783746996q(Paperback) 024 3 9781783746996 035 (OCoLC)1107450146z(OCoLC)1117884270z(OCoLC)1126178817 z(OCoLC)1135601942z(OCoLC)1167704711z(OCoLC)1202558352 z(OCoLC)1228590603z(OCoLC)1235833314z(OCoLC)1292985248 z(OCoLC)1295860346z(OCoLC)1295992746z(OCoLC)1303481852 037 b01575197 040 AUDbengerdaepncAUDdAUDdEBLCPdOCLCOdN$TdUKMGB dOCLCFdOCLCQdLVTdU3GdSFBdVT2dOAPENdKUKdOCLCQ dOPENTdDLCdUEJdYDXdOCLCQdUKKNUdLUNdVFLdOCLCQ dOCLCOdOCLCQdVHCdOCLCOdOCLCQ 049 STJJ 050 4 QA39.3b.B67 2019 082 04 510223 100 1 Borovik, Alexandre,eauthor. 245 14 The essence of mathematics :bthrough elementary problems /cAlexandre Borovik and Tony Gardiner. 264 1 Cambridge :bOpen Book Publishers,c[2019] 264 4 c©2019 300 1 online resource (389 pages) :billustrations. 336 textbtxt2rdacontent 337 computerbc2rdamedia 338 online resourcebcr2rdacarrier 490 1 OBP series in mathematics,x23871134 ;vvol. 3 500 Available through Open Book Publishers. 500 Includes index. 505 0 1. Mental Skills  2. Arithmetic  3. Word problems  4. Algebra  5. Geometry. 505 0 Preface  About this text  Chapter 1. Mental Skills. 1.1. Mental arithmetic and algebra ; 1.1.1. Times tables ; 1.1.2. Squares, cubes, and powers of 2 ; 1.1.3. Primes ; 1.1.4. Common factors and common multiples ; 1.1.5. The Euclidean algorithm ; 1.1.6. Fractions and ratio ; 1.1.7. Surds  1.2. Direct and inverse procedures ; 1.2.1. Factorisation  1.3. Structural arithmetic  1.4. Pythagoras' Theorem ; 1.4.1. Pythagoras' Theorem, trig for special angles, and CAST ; 1.4.2. Converses and Pythagoras' Theorem ; 1.4.3. Pythagorean triples ; 1.4.4. Sums of two squares  1. 5. Visualisation  1.6. Trigonometry and radians ; 1.6.1. Sine Rule ; 1.6.2. Radians and spherical triangles ; 1.6.3. Polar form and sin(A+B)  1.7. Regular polygons and regular polyhedra ; 1.7.1. Regular polygons are cyclic ; 1.7.2. Regular polyhedra  1.8. Chapter 1: Comments and solutions  Chapter 2. Arithmetic. 2.1. Place value and decimals: basic structure ; 2.2. Order and factors ; 2.3. Standard written algorithms ; 2.4. Divisibility tests ; 2.5. Sequences ; 2.5.1. Triangular numbers ; 2.5.2. Fibonacci numbers ; 2.6. Commutative, associative and distributive laws ; 2.7. Infinite decimal expansions ; 2.8. The binary numeral system ; 2.9. The Prime Number Theorem  2.10. Chapter 2: Comments and solutions  Chapter 3. Word Problems. 3.1. Twenty problems which embody "3  1 = 2" ; 3.2. Some classical examples ; 3.3. Speed and acceleration ; 3.4. Hidden connections  3.5. Chapter 3. Comments and solutions  Chapter 4. Algebra. 4.1. Simultaneous linear equations and symmetry ; 4.2. Inequalities and modulus ; 4.2.1. Geometrical interpretation of modulus, of inequalities, and of modulus inequalities ; 4.2.2. Inequalities ; 4.3. Factors, roots, polynomials and surds ; 4.3.1. Standard factorisations ; 4.3.2. Quadratic equations ; 4.4. Complex numbers ; 4.5. Cubic equations ; 4.6. An extra ; 4.7. Chapter 4: Comments and solutions  Chapter 5. Geometry. 5.1. Comparing geometry and arithmetic ; 5.2. Euclidean geometry: a brief summary ; 5.3. Areas, lengths and angles ; 5.4 Regular and semi regular tilings in the plan ; 5.5. Ruler and compasses constructions for regular polygons ; 5.6. Regular and semi regular polyhedra ; 5.7. The Sine Rule and the Cosine Rule ; 5.8. Circular arcs and circular sectors ; 5.9. Convexity ; 5.10. Pythagoras' Theorem in three dimensions ; 5.11. Loci and conic sections ; 5.12. Cubes in higher dimensions ; 5.13. Chapter 5: Comments and solutions  Chapter 6. Infinity: recursion, induction, infinite descent. 6.1. Proof by mathematical induction ; 6.2. 'Mathematical induction' and 'scientific induction' ; 6.3. Proof by mathematical induction II ; 6.4. Infinite geometric series ; 6.5. Some classical inequalities ; 6.6. The harmonic series ; 6.7. Induction in geometry, combinatorics and number theory ; 6.8. Two problems ; 6.9. Infinite descent ; 6.10. Chapter 6: Comments and solutions. 520 It is increasingly clear that the shapes of reality  whether of the natural world, or of the built environment  are in some profound sense mathematical. Therefore, it would benefit students and educated adults to understand what makes mathematics itself 'tick', and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. This book explores the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. 520 "It is increasingly clear that the shapes of reality  whether of the natural world, or of the built environment  are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself 'tick', and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. The Essence of Mathematics consists of a sequence of 270 problems  with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline. The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 1418, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works."Publisher's website 542 This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). 590 Promoted: Local to Global CooperativebOpen Textbook Library 650 0 Mathematics. 650 0 MathematicsvMiscellanea. 650 2 Mathematics. 650 7 Mathematics & science.2bicssc 650 7 Mathematics.2fast0(OCoLC)fst01012163 700 1 Gardiner, A.q(Anthony),d1947eauthor. 710 2 Open Book Publishers,epublisher. 776 08 iPaperback version :z9781783746996 776 08 iHardback version :z9781783747009 830 0 OBP series in mathematics ;vvolume 3. 914 on1107450146 994 92bSTJ
