Contents

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The Essence of Mathematics Through Elementary Problems
More info and resources at: https://doi.org/10.11647/obp.0168

Table of Contents

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I. Mental Skills

1.1. Mental arithmetic and algebra

1.1.1 Times tables.

1.1.2 Squares, cubes, and powers of 2.

1.1.4 Common factors and common multiples

1.1.5 The Euclidean algorithm

1.1.6 Fractions and ratio

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

2.1. Place value and decimals: basic structure

2.2. Order and factors

2.3. Standard written algorithms

2.4. Divisibility tests

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

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

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

4.7. Chapter 4: Comments and solutions

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

5.10. Pythagoras’ Theorem in three dimensions

5.11. Loci and conic sections

5.12. Cubes in higher dimensions

5.13. Chapter 5: Comments and solutions

VI. Infinity: recursion, 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

6.8. Infinite descent

6.10. Chapter 6: Comments and solutions

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