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049    STJJ 
050  4 QA39.3|b.B67 2019 
082 04 510|223 
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    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
490 1  OBP series in mathematics,|x2387-1134 ;|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 14-18, 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 Cooperative|bOpen Textbook 
       Library 
650  0 Mathematics. 
650  0 Mathematics|vMiscellanea. 
650  2 Mathematics. 
650  7 Mathematics & science.|2bicssc 
650  7 Mathematics.|2fast|0(OCoLC)fst01012163 
700 1  Gardiner, A.|q(Anthony),|d1947-|eauthor. 
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    92|bSTJ