**Table of Contents****1. The World of Numbers**

1.1 The unending supply of numbers

1.2 Roman numerals

1.3 Hindu–Arabic numerals

1.4 Positive and negative

1.5 Addition and subtraction

1.6 Multiplication and division

1.7 Number patterns

1.8 Palindromic entertainment

1.9 Miscellaneous problems I

1.10 Miscellaneous problems II

1.11 Miscellaneous problems III

**2. Beginnings of Algebra**

2.1 Framing sentences

2.2 Expressing patterns

2.3 Laws of algebra

2.4 Using algebra to explain patterns

2.5 Using algebra to discover properties

2.6 Factorisation

2.7 Equations

2.8 Closing quotes

**3. Families of Integers**

3.1 Odd and even

3.2 Multiples of 3

3.3 Multiples of 5

3.4 Prime and composite

3.5 GCD and LCM

3.6 Euclid’s algorithm

3.7 Perfect numbers

3.8 Squares and cubes

3.9 Triangular numbers

3.10 Powers of 2

3.11 Miscellaneous problems

**4. Fractions, Decimals and Percentages**

4.1 Fractions as numbers

4.2 Equivalent fractions

4.3 Arithmetic with fractions

4.4 Decimal fractions

4.5 Percentages

4.6 Fractions on the number line

4.7 Rational numbers

4.8 Miscellaneous exercises

**5. Fractions: Special Topics**

5.1 Recurring decimals

5.2 Powers of integers---on a clothes line!

5.3 Cyclic numbers

5.4 Decimal expansion of 1*/**p*

5.5 Irrational numbers

5.6 Unit fractions

5.7 Mischievous fractions

5.8 Farey sequences

5.9 Miscellaneous exercises

**6. Ratio and Proportion**

6.1 Ratios for unit conversion

6.2 Rate

6.3 Dimensionless ratios

6.4 Proportion

6.5 Results concerning proportion

6.6 Continued proportion

6.7 Historical note

6.8 Approximations to pi

6.9 The tangram puzzle

**7. Indices and Exponents**

7.1 Introduction

7.2 The laws of indices

7.3 Big numbers

7.4 The four twos

7.5 Visualising big numbers

7.6 Negative exponents

7.7 Fractional exponents

7.8 Growth patterns

7.9 Exponential notation in science

7.10 Three diversions

**8. Polynomials**

8.1 Notion of a polynomial

8.2 Degree of the zero polynomial

8.3 Alternate definition

8.4 Multiplication

8.5 Division

8.6 GCD

8.7 Roots of a polynomial

8.8 The remainder theorem

8.9 Symmetric polynomials

**9. Expansions in Algebra**

9.1 Homogeneous expressions

9.2 Expansions using shortcuts

9.3 Binomial expansions I

9.4 Using the Pascal triangle

9.5 Binomial expansions II

9.6 Squares of multinomials

9.7 Sum and product formulas

9.8 Difference of two squares

9.9 Visual representations

9.10 Miscellaneous expansions

9.11 A claim of Ramanujan’s

**10. Factorisation of Polynomials**

10.1 Difference of two squares

10.2 Application to integer factorisation

10.3 Factorisation by breaking up an expression

10.4 Factorisation of quadratic trinomials

10.5 A more complex example

10.6 Using the factor theorem

10.7 A square investigation

**11. Linear Equations**

11.1 Introduction to equations

11.2 Genesis of equations

11.3 Solution sets

11.4 Solving an equation

11.5 Linear equations in two unknowns

11.6 Application I: Clock problems

11.7 Application II: Pythagorean triples

11.8 Application III: Squares in AP

11.9 Application IV: Unit fractions

**12. Problem Corner**

12.1 Appetisers

12.2 For the junior mathlete

**13. Investigations**

13.1 Mischievous cancellations

13.2 Mischievous square roots

13.3 Prime numbers

13.4 Digital sums

13.5 Digital reversals

13.6 Squares and 2006 . . .

13.7 Squares and cubes

13.8 Still more squares

13.9 And yet more squares . . .

13.10 Accommodating numbers

**Answers to Exercises**

**Index**