Description 
1 online resource (389 pages) : illustrations. 
Series 
OBP series in mathematics, 23871134 ; vol. 3 

OBP series in mathematics ; volume 3.

Note 
Available through Open Book Publishers. 

Includes index. 
Contents 
1. Mental Skills  2. Arithmetic  3. Word problems  4. Algebra  5. Geometry. 

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 semiregular tilings in the plan ; 5.5. Ruler and compasses constructions for regular polygons ; 5.6. Regular and semiregular 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. 
Summary 
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. 

"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 
Note 
This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). 
Local Note 
Promoted: Local to Global Cooperative Open Textbook Library 
Subject 
Mathematics.


Mathematics  Miscellanea.


Mathematics.


Mathematics & science.


Mathematics. (OCoLC)fst01012163

Added Author 
Gardiner, A. (Anthony), 1947 author.


Open Book Publishers, publisher.

Other Form: 
Paperback version : 9781783746996 

Hardback version : 9781783747009 
ISBN 
9781783747016 (electronic book) 

1783747013 (electronic book) 

1783746998 

9781783746996 

1783747005 

9781783747009 

9781783747009 (Hardback) 

9781783746996 (Paperback) 
Standard No. 
9781783746996 
