The Essence of Mathematics Through Elementary Problems

The Essence of Mathematics Through Elementary Problems

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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, pat...

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Bibliographic Details
Main Authors: Borovik, Alexandre (Author), Gardiner, Tony (Author)
Format: Electronic eBook
Language:English
Published: [Place of publication not identified] Open Book Publishers [2019]
Series:Open textbook library.
Subjects:
Mathematics > Textbooks
Online Access:Access online version
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Table of Contents:
  • I. Mental Skills
  • 1.1 Mental arithmetic and algebra
  • 1.2 Direct and inverse procedures
  • 1.3 Structural arithmetic
  • 1.4 Pythagoras' Theorem
  • 1.5 Visualisation
  • 1.6 Trigonometry and radians
  • 1.7 Regular polygons and regular polyhedra
  • 1.8 Chapter 1: Comments and solutions
  • II. 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
  • 2.6 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
  • III. 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
  • IV. Algebra
  • 4.1 Simultaneous linear equations and symmetry
  • 4.2 Inequalities and modulus
  • 4.3 Factors, roots, polynomials and surds
  • 4.4 Complex numbers
  • 4.5 Cubic equations
  • 4.6 An extra
  • 4.7 Chapter 4: Comments and solutions
  • V. 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 plane
  • 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 coonic sections
  • 5.12 Cubes in higher dimensions
  • 5.13 Chapter 5: Comments and solutions
  • VI. Infinity: recursions, induction, infinite descent
  • 6.1 Proof by mathematical induction I
  • 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

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