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The Essence of Mathematics Through Elementary Problems

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The Essence of Mathematics Through Elementary Problems

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.

Applied MathematicsMathematics
Author
Alexandre Borovik & Tony Gardiner
Publisher
University of Manchester & University of Birmingham
Source
LibreTexts
License
CC BY
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 59 sections
  1. 01: Mental Skills
  2. 1.01: Mental Arithmetic and Algebra
  3. 1.02: Direct and Inverse Procedures
  4. 1.03: Structural Arithmetic
  5. 1.04: Pythagoras' Theorem
  6. 1.05: Visualisation
  7. 1.06: Trigonometry and Radians
  8. 1.07: Regular Polygons and Regular Polyhedra
  9. 1.08: Chapter 1- Comments and Solutions
  10. 02: Arithmetic
  11. 2.01: Place value and decimals- basic structure
  12. 2.02: Order and factors
  13. 2.03: Standard written algorithms
  14. 2.04: Divisibility tests
  15. 2.05: Sequences
  16. 2.06: Commutative associative and distributive laws
  17. 2.07: Infinite decimal expansions
  18. 2.08: The Binary Mumeral System
  19. 2.09: The Prime Number Theorem
  20. 2.10: Comments and solutions
  21. 03: Word Problems
  22. 3.01: Twenty problems which embody 3 1 2
  23. 3.02: Some classical examples
  24. 3.03: Speed and acceleration
  25. 3.04: Hidden connections
  26. 3.05: Comments and solutions
  27. 04: Algebra
  28. 4.01: Simultaneous linear equations and symmetry
  29. 4.02: Inequalities and modulus
  30. 4.03: Factors roots polynomials and surds
  31. 4.04: Complex numbers
  32. 4.05: Cubic equations
  33. 4.06: An extra
  34. 4.07: Comments and solutions
  35. 05: Geometry
  36. 5.01: Comparing geometry and arithmetic
  37. 5.02: Euclidean geometry- a brief summary
  38. 5.03: Areas lengths and angles
  39. 5.04: Regular and semi-regular tilings in the plane
  40. 5.05: Ruler and compasses constructions for regular polygons
  41. 5.06: Regular and semi-regular polyhedra
  42. 5.07: The Sine Rule and the Cosine Rule
  43. 5.08: Circular arcs and circular sectors
  44. 5.09: Convexity
  45. 5.10: Pythagoras Theorem in three dimensions
  46. 5.11: Loci and conic sections
  47. 5.12: Cubes in higher dimensions
  48. 5.13: Comments and solutions
  49. 06: Infinity - Recursion Induction Infinite Descent
  50. 6.01: Comparing geometry and arithmetic
  51. 6.02: Mathematical induction and scientific induction
  52. 6.03: Proof by mathematical induction II
  53. 6.04: Infinite geometric series
  54. 6.05: Some classical inequalities
  55. 6.06: The harmonic series
  56. 6.07: Induction in geometry combinatorics and number theory
  57. 6.08: Two problems
  58. 6.09: Infinite descent
  59. 6.10: Comments and solutions