PHƯƠNG PHÁP dạy TOÁN BẰNG TIẾNG ANH

Contents
0.1

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

0.2

Notations and Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1

Notations and Symbols in mathematics . . . . . . . . . . . . . . . . . . . . .
1.1.1

1.2

1.3

1.4

5

Some differences in the math symbols in English (Eng.) and Vietnamese (Vie.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.2

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.3

Abbreviations and Notations . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.4

Notations for Numbers, Sets and Logic Relations . . . . . . . . . . .

7

1.1.5

The Greek alphabet is commonly used in mathematics . . . . . . . .

7

1.1.6

Mathematical Symbols

. . . . . . . . . . . . . . . . . . . . . . . . . .

8

Pronunciation of mathematical expressions . . . . . . . . . . . . . . . . . . .

9

1.2.1

Logic and Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.2

Real numbers and operations . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.3

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.2.4

Some notation shortcuts are used in written English . . . . . . . . .

12

1.2.5

Some notation shortcuts are often used in mathematics . . . . . . .

13

Some Common Mathematical Symbols and Abbreviations (with History) .

14

1.3.1

Binary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3.2

Some Symbols from Mathematical Logic . . . . . . . . . . . . . . . .

14

1.3.3

Some Notation from Set Theory . . . . . . . . . . . . . . . . . . . . .

16

1.3.4

Some Important Numbers in Mathematics . . . . . . . . . . . . . . .

17

1.3.5

Appendix: Common Latin Abbreviations and Phrases . . . . . . . .

18

Skills Needed for Mathematical Problem Solving

. . . . . . . . . . . . . . .

20

1.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.4.2

Mathematical problem solving as a process. . . . . . . . . . . . . . .

21

1.4.3

Factors and skills involved in problem solving.

22

i

. . . . . . . . . . . .


ii

Contents
1.4.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 Mathematical Writing

32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

1.5.1

Some notes when writing mathematical . . . . . . . . . . . . . . . . .

32

1.5.2

A Guide to Writing Mathematics . . . . . . . . . . . . . . . . . . . . .

39

1.5.3

Mathematical Ideas into Writing . . . . . . . . . . . . . . . . . . . . .

43

Chapter 2. Basic of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.1 Sets and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.1.1

Notation and Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.1.2

Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.1.3

Equivalence Relations and Classes . . . . . . . . . . . . . . . . . . . .

56

2.1.4

Natural Numbers, Integers, and Rational Numbers . . . . . . . . . .

58

2.2 Infinity and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

2.2.1

Countable Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

2.2.2

Uncountable Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.2.3

The Principle of Induction . . . . . . . . . . . . . . . . . . . . . . . . .

67

2.2.4

The Real Number System . . . . . . . . . . . . . . . . . . . . . . . . .

71

2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

2.3.1

Negation of a Statement . . . . . . . . . . . . . . . . . . . . . . . . . .

74

2.3.2

Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

2.3.3

Disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

2.3.4

Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

2.3.5

Compound Statements

. . . . . . . . . . . . . . . . . . . . . . . . . .

83

2.3.6

Biconditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . .

85

2.3.7

Tautologies

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

2.3.8

Equivalence

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Chapter 3. Methods of mathematical proof . . . . . . . . . . . . . . . . . . . . .

92

3.1 Mathematical Induction - Problems With Solutions . . . . . . . . . . . . . .

93

3.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.2.2

Applications of the Pigeonhole Principle . . . . . . . . . . . . . . . . 101

3.2.3

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.2.4

More difficult examples and exercises . . . . . . . . . . . . . . . . . . 113

3.2.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.3 Direct Proof

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118


iii

Contents
3.3.1
3.3.2

3.4

3.5

Theorems
Definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3.3

Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.3.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Contrapositive Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.1

Contrapositive Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.4.2

Congruence of Integers

3.4.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

. . . . . . . . . . . . . . . . . . . . . . . . . . 136

Proof by Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.5.1

Proving Statements with Contradiction

. . . . . . . . . . . . . . . . 141

3.5.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter 4. Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.1

Elementary properties of integers . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1.1

Fundamental Notions and Laws . . . . . . . . . . . . . . . . . . . . . . 151

4.1.2

Definition of Divisibility. The Unit . . . . . . . . . . . . . . . . . . . . 152

4.1.3

Prime Numbers. The Sieve of Eratosthenes . . . . . . . . . . . . . . . 154

4.1.4

The Number of Primes is Infinite . . . . . . . . . . . . . . . . . . . . . 155

4.1.5

The Fundamental Theorem of Euclid . . . . . . . . . . . . . . . . . . 157

4.1.6

Divisibility by a Prime Number . . . . . . . . . . . . . . . . . . . . . . 157

4.1.7

The Unique Factorization Theorem . . . . . . . . . . . . . . . . . . . 158

4.1.8

The Divisors of an Integer . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.1.9

The Greatest Common Factor of Two or More Integers . . . . . . . 161

4.1.10 The Least Common Multiple of Two or More Integers . . . . . . . . 164
4.1.11 Scales of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.1.12 Highest Power of a Prime p Contained in n!. . . . . . . . . . . . . . . 168
4.1.13 Remarks Concerning Prime Numbers . . . . . . . . . . . . . . . . . . 172
4.2

4.3

On the indicator of an integers . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.2.1

Definition. Indicator of a Prime Power . . . . . . . . . . . . . . . . . 173

4.2.2

The Indicator of a Product . . . . . . . . . . . . . . . . . . . . . . . . 173

4.2.3

The Indicator of any Positive Integer . . . . . . . . . . . . . . . . . . 175

4.2.4

Sum of the Indicators of the Divisors of a Number . . . . . . . . . . 177

Elementary properties of congruences . . . . . . . . . . . . . . . . . . . . . . 179
4.3.1

Congruences Modulo m . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.3.2

Solutions of Congruences by Trial . . . . . . . . . . . . . . . . . . . . 181


iv

Contents

4.4

4.3.3

Properties of Congruences Relative to Division . . . . . . . . . . . . 182

4.3.4

Congruences with a Prime Modulus . . . . . . . . . . . . . . . . . . . 183

4.3.5

Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

The theorems of Fermat and Wilson . . . . . . . . . . . . . . . . . . . . . . . 187
4.4.1

Fermat’s General Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.4.2

Euler’s Proof of the Simple Fermat Theorem . . . . . . . . . . . . . . 188

4.4.3

Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.4.4

The Converse of Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . 191

4.4.5

Impossibility of 1 ⋅ 2 ⋅ 3⋯n − 1 + 1 = nk for n > 5. . . . . . . . . . . . . 191

4.4.6

Extension of Fermat’s Theorem . . . . . . . . . . . . . . . . . . . . . . 192

4.4.7

On the Converse of Fermat’s Simple Theorem . . . . . . . . . . . . . 195

4.4.8

Application of Previous Results to Linear Congruences . . . . . . . . 196

4.4.9

Application of the Preceding Results to the Theory of Quadratic
Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Chapter 5. Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.1.1

Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.1.2

Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.1.3

Cube Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.1.4

Geometrical Representation . . . . . . . . . . . . . . . . . . . . . . . . 204

5.1.5

Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.1.6

Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.1.7

De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.1.8

Cube Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

5.1.9

Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 207

5.1.10 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.1.11 Primitive Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.2 Polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.2.1

Definitions and basic operations . . . . . . . . . . . . . . . . . . . . . 212

5.2.2

Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.2.3

Analytic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

5.2.4

More identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.2.5

Integers and rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.2.6

Polynomial problems with detailed solutions. . . . . . . . . . . . . . 236


v

Contents

5.2.7
5.3

5.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3.3

Basic Methods For Solving Functional Equations . . . . . . . . . . . 248

5.3.4

Cauchy Equation and Equations of the Cauchy type . . . . . . . . . 250

5.3.5

Problems with Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 250

5.3.6

More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.3.7

Problems for Independent Study . . . . . . . . . . . . . . . . . . . . . 279

A Tour of Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
5.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

5.4.2

Isogonal conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

5.4.3

Simson line and line of reflections . . . . . . . . . . . . . . . . . . . . 295

5.4.4

Rectangular circum-hyperbolas . . . . . . . . . . . . . . . . . . . . . . 298

5.4.5

Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

5.4.6

Further examples of reflections . . . . . . . . . . . . . . . . . . . . . . 309

5.4.7

A metric relation and its applications . . . . . . . . . . . . . . . . . . 313

5.4.8

The Apollonian Circles and Isodynamic Points . . . . . . . . . . . . 316

Chapter 6. Exercises with Solutions and Answers . . . . . . . . . . . . . . . . . 336
6.1

6.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.1.1

Grade 10 algebra excercises . . . . . . . . . . . . . . . . . . . . . . . . 337

6.1.2

Grade 10 math word exercises . . . . . . . . . . . . . . . . . . . . . . 339

6.1.3

Grade 10 geometry excercises

6.1.4

Grade 10 trigonometry excercises . . . . . . . . . . . . . . . . . . . . . 342

6.1.5

Grade 10 math algebra excercises (advanced). . . . . . . . . . . . . . 344

6.1.6

Grade 10 math word excercises (advanced). . . . . . . . . . . . . . . 347

6.1.7

Grade 10 geometry excercises (advanced). . . . . . . . . . . . . . . . 349

6.1.8

Grade 10 trigonometry excercises (advanced). . . . . . . . . . . . . . 354

6.1.9

Grade 10 excercises on complex (advanced). . . . . . . . . . . . . . . 357

. . . . . . . . . . . . . . . . . . . . . . 340

Solutions and Answers to the Above Excercises . . . . . . . . . . . . . . . . 359
6.2.1

Grade 10 algebra excercises . . . . . . . . . . . . . . . . . . . . . . . . 359

6.2.2

Grade 10 math word exercises . . . . . . . . . . . . . . . . . . . . . . 360

6.2.3

Grade 10 geometry excercises

6.2.4

Grade 10 trigonometry excercises . . . . . . . . . . . . . . . . . . . . . 364

. . . . . . . . . . . . . . . . . . . . . . 362


vi

Contents
6.2.5

Grade 10 math algebra excercises (advanced). . . . . . . . . . . . . . 367

6.2.6

Grade 10 math word excercises (advanced). . . . . . . . . . . . . . . 373

6.2.7

Grade 10 geometry excercises (advanced). . . . . . . . . . . . . . . . 379

6.2.8

Grade 10 trigonometry excercises (advanced). . . . . . . . . . . . . . 387

6.2.9

Grade 10 excercises on complex (advanced). . . . . . . . . . . . . . . 395

Chapter 7. Examples and Exercises on Mathematical Training . . . . . . . . 397
7.0.10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
7.1 Operations on Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 399
7.1.1

Basic Rules on Addition, Subtraction, Multiplication, Division . . . 399

7.1.2

Rule for Removing Brackets . . . . . . . . . . . . . . . . . . . . . . . . 399

7.1.3

Ingenious Ways for Calculating . . . . . . . . . . . . . . . . . . . . . . 399

7.1.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

7.1.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

7.2 Monomials and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

7.2.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

7.2.2

Operations on Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 406

7.2.3

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

7.2.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

7.3 Linear Equations of Single Variable . . . . . . . . . . . . . . . . . . . . . . . . 410
7.3.1

Usual Steps for Solving Equations . . . . . . . . . . . . . . . . . . . . 410

7.3.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

7.3.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

7.4 System of Simultaneous Linear Equations . . . . . . . . . . . . . . . . . . . . 416

7.4.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

7.4.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

7.5 Multiplication Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
7.5.1

Basic Multiplication Formulae . . . . . . . . . . . . . . . . . . . . . . 424

7.5.2

Generalization of Formulae . . . . . . . . . . . . . . . . . . . . . . . . 424

7.5.3

Derived Basic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 424

7.5.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

7.6 Some Methods of Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 429
7.6.1

Basic Methods of Factorization . . . . . . . . . . . . . . . . . . . . . . 429

7.6.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

7.6.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432


Contents
7.7

7.8

7.9

vii

Absolute Value and Its Applications . . . . . . . . . . . . . . . . . . . . . . . 434
7.7.1

Basic Properties of Absolute Value . . . . . . . . . . . . . . . . . . . . 434

7.7.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

7.7.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

Linear Equations with Absolute Values . . . . . . . . . . . . . . . . . . . . . 439
7.8.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

7.8.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Sides and Angles of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
7.9.1

Basic Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

7.9.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

7.9.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

7.10 Pythagoras’ Theorem and Its Applications . . . . . . . . . . . . . . . . . . . 450
7.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
7.10.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
7.11 Congruence of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
7.11.1 Basic Criteria for Congruence of Two Triangles . . . . . . . . . . . . 458
7.11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
7.11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

7.12 Divisions of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
7.12.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
7.12.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
7.13 Congruence of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
7.13.1 Basic Properties of Congruence . . . . . . . . . . . . . . . . . . . . . . 473
7.13.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
7.13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
7.14 Decimal Representation of Integers . . . . . . . . . . . . . . . . . . . . . . . . 478
7.14.1 Decimal Expansion of Whole Numbers with Same Digits or Periodically Changing Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
7.14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
7.14.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
7.15 Perfect Square Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
7.15.1 Basic Properties of Perfect Square Numbers . . . . . . . . . . . . . . 484
7.15.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488


viii

Contents
7.16 [x] and {x} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

7.16.1 Some Basic Properties of x and{x} . . . . . . . . . . . . . . . . . . . . 490
7.16.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

7.16.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
7.17 Diophantine Equations (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
7.17.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
7.17.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
7.17.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

7.18 Diophantine Equations (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
7.18.1 Basic Methods for Solving Quadratic Equations on Z . . . . . . . . . 505
7.18.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
7.18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
7.19 Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
7.19.1 Basic Forms of Pigeonhole Principle . . . . . . . . . . . . . . . . . . . 512
7.19.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
7.19.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
7.20 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
7.20.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
7.20.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
7.21 Solutions to given exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534


1

Preface

0.1

Preface

In recent years, due to the requirements of international integration, the demand
for the knowledge of English in some professional field is becoming increasingly
urgent, especially for the class teachers, high school students and college students.
In Viet Nam, The Ministry of Education and Training has planned to set up the
programs for class teaching in bilingual Vietnamese - English, first for students of
natural science subjects and then for students of social science ones in Specializing
Upper Secondary Schools

However, this is an extremely hard work because the knowledge of English of
most teachers in this professional field is not good enough to carry out the task.
They need to be trained again to meet the demand. Students, also, need to be
taught in such a way that they can be able to understand the lessons in English.
Another obstacle is that teachers and students’ ability of English listening,
speaking and writing is rather poor which has been considered an inherent weakness
of foreign language learning and teaching in our country today.
To make some contribution to the ambitious program, I have decided to have
the lectures written and designed in English in order to help students specializing
in math understand and know the technical terms of solving exercises in English,
so that their general knowledge of English will be improved as well. Some of the
beginning chapters and sections have been made directly by the teachers who are
teaching in class, but most of the content of this research is for students to read
and practice under the help and guidance of the teachers.
This book includes 6 chapters and is divided into two parts.
Chapter I. Introduction: Provide the knowledge needed to understand the
book: The system of notation, Greek alphabet and the rules of mathematical word
in English
Chapter II. Basic of Mathematics: Present the basic knowledge of mathematics including set theory, logic, relations and functions.
Chapter III. Number Theory: Present some basic knowledge of number
theory. This chapter is only for students specializing in math.
Chapter IV. Methods of mathematical proof : Provide some common
methods of proof in mathematics: Direct and indirect proof, Contradiction and
Induction etc.


2

Preface

Chapter V. Other topics: Present the individual subjects in the school program as well as for students of math.
Chapter VI. Exercises with Solutions and Answers: Offer some simple
exercises as rehearsals for students to do and explain exercises in English.
Chapter VII. Examples and Exercises on Mathematical Training: Offer
some questions for students to practise, and test students’ ability to apply their
knowledge in solving real competition questions. These examples and exercises are
taken from a range of countries, e.g. China, Russia, the USA and Singapore, e.t.c.
The books shown in the Bibliography are mostly sent to the author from
students who are studying abroad and hard to find in Vietnam. With any luck,
readers can find a pdf- files online, but if possible, you can order at amazon.com
However, because of our limited level of writing, the limitation of time and the
length of the research, there are still some dissatisfaction in the discussion as the
following:
1. Some parts missing, such as: inequality, the transcendental equations, inequations, and systems of equations (exponential, logarithmic and trigonometric),
survey plot functions, applications of derivatives, etc.
2. Some topics for gifted students have not been put in, such as (Invariant theory,
game theory, extreme theory, combinatorial mathematics, etc.) or only superficial
presented in this textbook (Graph Theory, Principle Direchlet, etc.)
3. Also lack the classical geometry (plane and space), Vector and applications,
Transformation, Method coordinates in Space, etc.
In addition, the desired book is applied to three characters: mathematics teachers, students specializing in math, and students of Specializing Upper Secondary
Schools in general. Hence the book style is not consistent. While sections for students of math and for teacher are written with style accurately and scientifically,
while other parts of text are freely written.
With full of hope, the author’s colleagues will continue to improve he research on
the shortcomings and with the sincere comments of your readers, the next version
will be better.
The book is written by software Viettex 2.5 and PCTEXv5.2. The picture drawn
by WinTpic, WinFig, Sketchpad and Graph4.3.

3

0.1. Preface

A part of the lecture was set up by the author and MA Tran Thi Ha Phuong,
who have used it to teach students of math in Bac Giang Specializing Upper Secondary school for the last few years and has obtained some good results.
We would like to express our sincere thanks to:
Professor, Doctor of Science, People’s Teacher NGUYEN VAN MAU,
who has read and given many valuable comments on the content and the format
of the manuscript, and Doctor, Associate Professor Nguyen Vu Luong for
his strong support
The teachers: MA Bach Dang Khoa, MA Nguyen Anh Tuan, MA Tran
Thi Ha Phuong, BA Nguyen Van Thao and MA Tran Anh Duc for reading
and editing this manuscript.
In particular, thanks to MA Tran Thi Ha Phuong and a group of students of
math in mathematics courses K17, K18, K19, K20, K21 of Bac Giang Specializing
Upper Secondary school for their contributions to the manuscript.
And, the most sincere thanks to some teachers of English in Bac Giang Specializing Upper Secondary school, Ms. Do Thi Minh Hong, Ms. Mai Thu Giang,
Ms. Vu Thi Hue and especially Mr. Nguyen Danh Hao, who have checked
the text carefully.
Without timely support and help from these characters, the research could not
be completed successfully.
The author would like to receive feedback and contributions from readers. All
comments should be sent to Bac Giang Specializing Upper Secondary school, Hoang
Van Thu street, Bac Giang City, Vietnam or send email to mailboxes

Sincerely thanks.
Bac Giang, on 28/10/2012
Nguyen Van Tien

4

Preface

0.2

Notations and Symbol

In this book the authors use the notation system, and abbreviated as
1. ”:=” is the symbol of the phrases:
” called . . . ” , or ” is denoted by . . . ” , or ” instead . . . ” .
2. WLOG := Without loss of generality.
3. g.e. := given equation ; g.i. := given inequation ; g.s. := given system of equations.
4. QED or ⊠:= "quod erat demonstrandum" (in Latin) or "Question est démontrée" (in
France).
5. LHS := left hand side ; RHS := right hand side ; s.t.:= such that
6. "e.g." := "for example" (from "exempli gratia" in Latin) or "including"
7. i.e := it means that (from phrase "id est" in Latin) or "that is", or "in other word" or "it
is".
8. wrt := with respect to ; "etc" := (from phrase "et cetera" in Latin) (or Et les autres choses
in France).
9. R ∶= (−∞; +∞) := A set of real numbers ; C := A set of complex numbers ; R∗ ∶= R ∖ {0}.
10. R+ ∶= (0; +∞) ; R+ ∶= [0; +∞) ; R− ∶= (−∞; 0) ; R− ∶= (−∞; 0].
11. N ; Z ; Q ; I ; P ; Nc ; Nl corresponding are the symbols of the sets: the set of natural
numbers, integers, rational numbers, irrational numbers, prime numbers, the even natural
numbers, odd natural numbers
12. With X ⊆ R we denoted X ∗ ∶= X ∖ {0} ; X + ∶= X ∩ (0; +∞) ;
X+ ∶= X ∩ [0; +∞) ; X − ∶= X ∩ (−∞; 0) ; X− ∶= X ∩ (−∞; 0].
13. F(A):= Set of all functions f ∶ A → R with A ⊆ R.

14. With f (x) ∈ F(R) we denoted:
Df := The Domain of function f (x) ;
Gf := The Graph of function y = f (x) ;
Rf := Set the value of the function f (x) ; Rf = f (Df ) ;
Tf := Set of zero of function f (x), Tf = {x ∈ Df ∣ f (x) = 0}.
15. C(A) [D (A)] := Set of continuous functions on A [differentiable on A].
16. k..l ∶= {k; k + 1; k + 2; ⋯; l − 1; l} with k, l ∈ Z ; k < l.
(i)

17. = := "equality occurs under conditions (i)";

abcd

⇒:= "inferred according to claim abcd".

18. "=∶" instead of the phrase "equality occurs if and only if" ; "⇛” instead of the phrase
"become".
19. The other symbols will be indicated as first appeared.


Chapter 1
Introduction
There is a good chance that you have never written a paper in a math class before.
So you might be wondering why writing is required in your math class now.
The Greek word mathemas, from which we derive the word mathematics, embodies the
notions of knowledge, cognition, understanding, and perception. In the end, mathematics
is about ideas. In math classes at the university level, the ideas and concepts encountered
are more complex and sophisticated. The mathematics learned in college will include
concepts which cannot be expressed using just equations and formulas. Putting mathemas
on paper will require writing sentences and paragraphs in addition to the equations and

formulas.
Mathematicians actually spend a great deal of time writing. If a mathematician wants
to contribute to the greater body of mathematical knowledge, she must be able communicate her ideas in a way which is comprehensible to others. Thus, being able to write
clearly is as important a mathematical skill as being able to solve equations. Mastering
the ability to write clear mathematical explanations is important for nonmathematicians
as well. As you continue taking math courses in college, you will come to know more
mathematics than most other people. When you use your mathematical knowledge in the
future, you may be required to explain your thinking process to another person (like your
boss, a co-worker, or an elected official), and it will be quite likely that this other person will know less math than you do. Learning how to communicate mathematical ideas
clearly can help you advance in your career.

1.1
1.1.1

Notations and Symbols in mathematics
Some differences in the math symbols in English (Eng.)
and Vietnamese (Vie.)
No
1
4

Eng.
(a, b)
3.25

Vie.
(a; b)
3, 25

No

2
5

Eng.
N
→
→
a × b

Vie.
N∗
→
[→
a, b]

5

No
3
6

Eng.
If P, Q
a, and b

Vie.
Nếu P thì Q
a và b

6

Chapter 1. Introduction

1.1.2
N o.
1.

2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

1.1.3

Geometry
written as
∣x∣
→
OA, →
u

read as
the norm (or modulus) of x

OA
... ∼ ...
... ⊥ ...
... ∥ ...
∠...
→
→
→
a ⋅ b,→
a b
→
→
a × b
→ → →
i, j,k
∆ABC
... ≅ ...
→
∣OA∣ , ∣ →
u∣

vector OA vector u
OA / the length of the segment OA
... is similar to... /Indicates two objects are geometrically similar
... is perpendicular to...
... is parallel to...
Angle...

the scalar product of vectors a and b
the vector product of vectors a and b
unit vectors in the directions of the cartesian coordinate axes
the triangle ABC
... is congruent to ...
the magnitude of vector OA the magnitude of vector u

Abbreviations and Notations

AHSME
AIME
APMO
ASUMO
AUSTRALIA
BMO
CHNMO
CHNMOL
CMO
HUNGARY
IMO
JAPAN
KIEV
MOSCOW
NORTH EUROPE
RUSMO
SSSMO
SMO
SSSMO(J)
UKJMO
USAMO

VMO
HOMO
SOMO

American High School Mathematics Examination
American Invitational Mathematics Examination
Asia Pacific Mathematics Olympiad
Olympics Mathematical Competitions of All the Soviet Union
Australia Mathematical Competitions
British Mathematical Olympiad
China Mathematical Olympiad
China Mathematical Competition for Secondary Schools
Canada Mathematical Olympiad
Hungary Mathematical Competition
International Mathematical Olympiad
Japan Mathematical Olympiad
Kiev Mathematical Olympiad
Moscow Mathematical Olympiad
North Europe Mathematical Olympiad
All-Russia Olympics Mathematical Competitions
Singapore Secondary Schools Mathematical Olympiad
Singapore Mathematical Olympiad
SSSMO for Junior Section
United Kingdom Junior Mathematical Olympiad
United States of American Mathematical Olympiad
Vietnames Mathematical Olympiad
Hanoi Open Mathematical Olympiad
Singapore Open Mathematical Olympiad

7

1.1. Notations and Symbols in mathematics

1.1.4

Notations for Numbers, Sets and Logic Relations
N
N0
Z
Z+
k..m
Q
Q+
Q+0 ; Q+
R
R−
R−0 ; R−
[a, b]
(a, b)
⇔
⇒
A⊆B
A∖B
A∪B
A∩B
a∈A
Rt

1.1.5

the set of positive integers (natural numbers)
the set of non-negative integers
the set of integers
the set of positive integers
the set {k, k + 1, . . . , m − 1, m}, k, m ∈ Z, k < m
the set of rational numbers
the set of positive rational numbers
the set of non-negative rational numbers
the set of real numbers
the set of negative real numbers
the set of non-positive real numbers
the closed interval, i.e. all x such that a ⩽ x ⩽ b
the open interval, i.e. all x such that a < x < b
iff, if and only if
implies
A is a subset of B
the set formed by all the elements in A but not in B
the union of the sets A and B
the intersection of the sets A and B
the element a belongs to the set A
Right triangle

The Greek alphabet is commonly used in mathematics
(g.r.:= greek letter, Uppercase and lowercase letters)
g.r.

read

No

g.r.

read

No

g.r.

read

1

A, α

alpha

2

B, β

beta

3

Γ, γ

gamma

4

∆, δ

delta

5

E, ǫ

epsilon

6

H, η

eta

7

Θ, θ

theta

8

I, ι

iota

9

K, κ

kappa

10

Λ, λ

lambda

11

M, µ

mu

12

N, ν

nu

13

Ξ, ξ

xi

14

F, f

digamma

15

Π, π

pi

16

P, ρ

rho

17

Σ, σ

sigma

18

T, τ

tau

19

Υ, υ

upsilon

20

Φ, φ

phi

21

X, χ

chi

22

Z, ζ

zeta

23

Ψ, ψ

psi

24

Ω, ω

omega

No


8

Chapter 1. Introduction

1.1.6

Mathematical Symbols

You will encounter many mathematical symbols during your math courses. The table
below provides you with a list of the more common symbols, how to read them, and notes
on their meaning and usage. The following page has a series of examples of these symbols
in use.
Symbol
a≈b
(a, b)
(a, b)
[a, b]

(a, b]

How to read it
a is approximately equal to b

the point a b
the open interval from a to b
the closed interval from a to b
half-open interval from a to b.

R
C
Z
N
∀x

the real numbers
the complex numbers
the integers
the natural numbers
for all x

∃x
∃ !x
f ○g
n!
[x]
⌈x⌉

there exists x
there exists a unique x
f composed with g or f of g
n factorial
the floor of x
the ceiling of x

f = O(g)

f is big oh of g when x to a

f = o(g)
x→a

f is little oh of g when x to a

x → a+

x goes to a from the right

x→a

o

... degree(s)

∶

... is to... /... such that... /
... it is true that...
... such that... /...it is true
that...

∣

Notes on meaning and usage

Do not write = when you mean ≈.
A coordinate in R2 .
The values between a and b,
but not including the endpoints.
The values between a and b,
including the endpoints.
The values between a and b, excluding
a, and includingb. Similar for [a, b).
It can also be used for the plane as R2 .
C = {a + bi ∶ a, b ∈ R}, where i2 = −1.
Z = {. . . , −2, −1, 0, 1, 2, 3, . . .}.
N = {1, 2, 3, 4, . . .}.
Something is true for all (any) value
of x (usually with a side condition
like ∀ x > 0).
Used in proofs as a shorthand.
Used in proofs as a shorthand.
Denotes f (g(⋅)).
n! = n(n − 1)(n − 2)⋯ × 2 × 1.
The nearest integer ⩽ x.
The nearest integer ⩾ x.
f (x)
∣ = C = const ≠ 0.
lim ∣
x→a g(x)
f (x)
lim ∣
∣ = 0.
x→a g(x)
x is approaching a, but x is always

greater than a. Similar for x → a− .
Angular measure /Temperature /
Degree symbol
Colon,ratio sign
Symbol following logical
quantifier or used in defining a set


9

1.2. Pronunciation of mathematical expressions

1.2

Pronunciation of mathematical expressions

The pronunciations of the most common mathematical expressions are given in the list
below. In general, the shortest versions are preferred (unless greater precision is necessary).

1.2.1

Logic and Set
N o.
1.
2.
3.
4.

written as
∃

∀
p⇒q
p⇔q

5.
6.
7.
8.
9.

p&q
p∨q
p
x∈A
x∉A

10.
11.
12.
13.
14.

A⊂B
A⊃B
A∩B
A∪B
A∖B

15.
16.

17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.

A×B
∅
!
+ ; (−) ; ∞
∝
∶∶
(a, b)
⋂S
⋃S
{x ∶ t(x)}
∀ x, P (x)
∃ x, P (x)
a≡b
a ≡ b (mod n)

read as
there exists

for all
p implies q / if p, then q
p if and only if q /p is equivalent to q
/ p and q are equivalent
p and q
p or q
not p
x belongs to A / x is an element (or a member) of A
x does not belong to A
/x is not an element (or a member) of A
A is contained in B / A is a subset of B
A contains B / B is a subset of A
A cap B / A meet B / A intersection B
A cup B / A join B / A union B
A minus B / the difference between A and B
complement of B in A
A cross B / the cartesian product of A and B
The null set / The empty set
... factorial
plus ; (minus) ; infinity
... is proportional to...
... averaged with...
ordered pair
intersection of family S
union of family S
set of x having property t
For all x, P (x) holds
There exists an x such that P (x) holds
a is equivalent to b / a is indentically equal to b
a and b are congruent modulo n

10

1.2.2

Chapter 1. Introduction

Real numbers and operations

N o.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

written as
x+1
x−1

x±1
xy
x−y ; x+y
x
or x/y
y
=
x=5
x≠5
x≡y
x ≡/ y
x⩾y
x⩽y
a0⩽x⩽1

16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.

30.

n

∣x∣
x2
x3
x4
xn
√
x
√
3
x
√
4
x
√
n
x
(x + y)2
x 2
( )
y
n!
x
̂; x; ̃
x
xi

n
∑ ai or ∑i=1 ai

read as
x plus one
x minus one
x plus or minus one
xy / x multiplied by y
x minus y ;x plus y
x over y
the equals sign
x equals 5 / x is equal to 5
x (is) not equal to 5
x is equivalent to (or identical with) y
x is not equivalent to (or identical with) y
x is greater than or equal to y
x is less than or equal to y
a is less than x is less than b
zero is less than or equal
to x is less than or equal to 1
mod x / modulus x
x squared / x (raised) to the power 2
x cubed
x to the fourth / x to the power four
x to the nth / x to the power n
(square) root x / the square root of x
cube root (of) x
fourth root (of) x
nth root (of) x
x plus y all squared

x over y all squared
n factorial
x hat ; x bar ; x tilde
xi / x subscript i / x suffix i / x sub i
the sum from i equals one to n of a i

i=1

/ the sum as i runs from 1 to n of the a i


1.2. Pronunciation of mathematical expressions

1.2.3

11

Functions
N o.
1.
2.
3.
4.
5.
6.
7.
8.
9.

written as

f (x)
f ∶S→T
x↦y
f ′ (x)
f ′′ (x)

f ′′′ (x)

f (4) (x)

∫0 f (x) dx
∞

lim f (x)
x→0

the limit of f (x) as x tends to zero

x→0−

the limit of f (x) as x tends to a from above

12.

loga y

log y to the base a / log to the base a of y

13.

ln y

10.
11.

14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
24.
26.
27.
28.
29.

lim f (x)

read as
f x / f of x / the function f of x
a function f from S to T
x maps to y / x is sent (or mapped) to y
f prime x / f dash x
/ the (first) derivative of f with respect to x

f double − prime x
/ the second derivative of f with respect to x
f triple − prime x
/ the third derivative of f with respect to x
f four x / the fourth derivative
of f with respect to x
the integral of f (x) from zero to infinity

x→a+

lim f (x)

AT
A−1
x−n
x>y
xDom(f ) ∶= Df
Im(f )
Rng(f ) ∶= Rf
∫ f (x)dx
ex , exp(x)
g ○ f, gf
∆x, δx
lub(S) ; sup(S)
glb(S) ; inf(S)
yRx
Rez ; Imz

the limit of f (x) as x approaches zero from below

log y to the base e
/ log to the base e of y / natural log (of) y
A transpose / the transpose of A
A inverse / the inverse of A
x to the (power) minus n
x is greater than y
x is less than y
domain of f
image of a function f
range of a function f
indefinite integral of f (x) with respect to x
exponential function of x
the composite function of f and g
an increment of x
least upper bound of the set S ; supremum of S
greatest lower bound of the set S ; infimum of S
y is related to x by the ralation R
the real part of z ; the imaginary part of z

Individual mathematicians often have their own way of pronouncing mathematical
expressions and in many cases there is no generally accepted ”correct”pronunciation.


12

Chapter 1. Introduction

1.2.4

Some notation shortcuts are used in written English

,

Comma

;

Semicolon

:

Colon

.

Period; Dot, Full stop; Decimal point

‘

Apostrophe

"

Open double quote; Open quote;
Open inverted commas

’

Open single quote

"

Close double quote; Close quote;
Close inverted commas

’
/

Close single quote

>

Greater than sign; Close angle bracket

Forward slash

<

Less than sign; Open angle bracket

Backslash

!

Exclamation mark; Exclamation point

∼

Tilde

#

Number sign; Pound sign; Hash sign

@

At sign

?

Question mark

$

Dollar sign

%

Percent sign

ˆ

Carat

}

_

Underscore

[
]

Close brace; Close square brace

−−

Em dash

–

En dash

−

Hyphen; Minus sign; Dash

=

Equal sign

,

+

Plus sign

3
4

Three quarter sign

Vertical bar

&

Ampersand; And sign

1
4

One quarter sign

1
2

One half sign

∗

Asterisk

±

Plus or minus sign

Double slash

<<

Open angle quote

/

∣

//

(

Open parenthesis; Open paren

)

Close parenthesis; Close paren

{

Open brace; Open curly bracket
Close brace; Close curly bracket
Open brace; Open square brace

CopyRight
TM

Trademark sign

‘

Back quote

>>

Close angle quote

×

Multiplication sign

⋯

Ellipsis; Dot dot dot


1.2. Pronunciation of mathematical expressions

13

Distinctions made in writing are often not made explicit in speech; thus the sounds f x
→
may be interpreted as any of: f x, f (x), fx , (F X), F X, F X, F X?. The difference is usually
made clear by the context; it is only when confusion may occur, or where he/she wishes
to emphasise the point, that the mathematician will use the longer forms: f multiplied by
x, the function f of x, f subscript x, line F X,segment F X or the length of the segment
F X, the algebraic length of the segment F X, vector F X.
Similarly, a mathematician is unlikely to make any distinction in speech (except sometimes a difference in intonation or length of pauses) between pairs such as the following:
√
√
x + (y + z) and (x + y) + z ; ax + b and ax + b ; an − 1 and an−1

1.2.5

Some notation shortcuts are often used in mathematics

1. l : the first letter of the word length
2. S : the first letter of the word square
3. V : the first letter of the word volume
4. R : the first letter of the word radius of a circle
5. D : the first letter of the word diameter
6. C : the first letter of the word circle
7. M : the first letter of the word Midpoint
8. h : the first letter of the word height
9. N : the first letter of the word natural number
10. Q : the first letter of the word quotient number
11. R : the first letter of the word real number
12. C : the first letter of the word complex number
13. i : the first letter of the word imaginary unit
14. r : the first letter of the word remaider
15. p : the first letter of the word prime number
16. d : the first letter of the word distance
17. m : the first letter of the word median
18. P : the first letter of the word Perimeter
19. R : the first letter of the word Radius of circumscribed circle
20. r : the first letter of the word Radius of incircle
21. R: the first letter of the word Relation


14

1.3

1.3.1

Chapter 1. Introduction

Some Common Mathematical Symbols and Abbreviations (with History)
Binary Relations

◇ The symbol: = (the equals sign) means ”is the same as” and was first
introduced in the 1557 book The Whetstone of Witte by physician and mathematician Robert Recorde (c. 1510-1558). He wrote, ”I will sette as I doe often in woorke
use, a paire of parralles, or Gemowe lines of one lengthe, thus: =, bicause noe 2
thynges can be moare equalle.” (Recorde’s equals sign was significantly longer than
the one in modern usage.)

◇ The symbol: < (the less than sign) means ”is strictly less than”, and >
(the greater than sign) means ”is strictly greater than”. These first appeared in
Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (”The Analytical
Arts Applied to Solving Algebraic Equations”) by mathematician and astronomer
Thomas Harriot (1560-1621), which was published posthumously in 1631. Pierre
Bouguer (1698-1758) later refined these to ⩽ (”is less than or equals”) and ⩾
(”is greater than or equals”) in 1734. Bouger is sometimes called ”the father of
naval architecture” due to his foundational work in the theory of naval navigation.
◇ The symbol: ∶= (the equal by definition sign) means ”is equal by definition to”. This is a common alternate form of the symbol ”=Def ”, which first
appeared in the 1894 book Logica Matematica by the logician Cesare Burali-Forti
def.
(1861-1931). Other common alternate forms of the symbol ” =Def ” include ” = ”
and ”≡”, the latter being especially common in applied mathematics.

◇ The symbol: ≐ (the approximately equals sign) means ”is nearly equal
to” and was first used in 1875 by mathematician Anton Steinhauser (1802-1890)
in his Lehrbuch der Mathematik. (This symbol was also briefly used in 1832 by geometer Farkas Wolfgang Bolyai (1775-1856) to signify ”equal by definition”.) Other

modern symbols for ”approximately equals” include ”≈” (read as ”is approximately
equal to”), ”≅” (read as ”is congruent to”), ”≃” (read as ”is similar to”), ” ≍ ” (read as
”is asymptotically equal to”), and ”∝” (read as ”is proportional to”). Usage varies,
and these are sometimes used to denote varying degrees of ”approximate equality”
within some context.

1.3.2

Some Symbols from Mathematical Logic

◇ The symbol: ∴ (three dots) means ”therefore” and first appeared in print
in the 1659 book Teusche Algebra (”Teach Yourself Algebra”) by mathematician Jo-


1.3. Some Common Mathematical Symbols and Abbreviations (with History)

15

hann Rahn (1622-1676). (Teusche Algebra also contains the first use of the obelus,
”÷”, to denote division.)
◇ The symbol: ∵ (upside-down dots) means ”because” and seems to have
first appeared in the 1805 book The Gentleman’s Mathematical Companion. However, it is much more common (and less ambiguous) to just abbreviate ”because”
as ”b/c”.
◇ The symbol: ∋ (the such that sign) means ”under the condition that”
and first appeared in the 1906 edition of Formulaire de mathematiqu’es by the
logician Giuseppe Peano (1858-1932). However, it is much more common (and less
ambiguous) to just abbreviate ”such that”as ”s.t.”.
There are two good reasons to avoid using ”∋” in place of ”such that”. First of
all, the abbreviation ”s.t.” is significantly more suggestive of its meaning than is
”∋”. Perhaps more importantly, though, is that it has become increasingly common

for the symbol ”∋” to mean ”contains as an element”, which is a logical extension
of the usage of the unquestionably standard symbol ”∈” to mean ”is contained as
an element in”.
◇ The symbol: ⇒ (the implies sign) means ”logically implies that”, and
⇐ (the is implied by sign) means ”is logically implied by”. Both have an unclear
historical origin. (E.g., ”if it’s raining, then it’s pouring” is equivalent to saying
”it’s raining ? it’s pouring.”)

◇ The symbol: ⇔ (the iff symbol) means ”if and only if ” and is used to
connect logically equivalent statements. (E.g., ”it’s raining iff it’s really humid ”
means simultaneously that ”if it’s raining, then it’s really humid” and that ”if it’s
really humid, then it’s raining”. In other words, the statement ”it’s raining ⇔ it’s
really humid ” means simultaneously that ”it’s raining ⇒ it’s really humid ” and
”it’s raining ⇒ it’s really humid ”.) The abbreviation ”iff ” is attributed to the
mathematician Paul Halmos (1916-2006).

◇ The symbol: ∀ (the universal quantifier) means ”for all” and was first
used in the 1935 publication Untersuchungen ueber das logische Schliessen (”Investigations on Logical Reasoning”) by logician Gerhard Gentzen (1909-1945). He
called it the All-Zeichen (”all character ”) by analogy to the symbol ”∃ ”, which
means ”there exists”.
◇ The symbol: ∃ (the existential quantifier) means ”there exists” and
was first used in the 1897 edition of Formulaire de mathematiqu’es by logician
Giuseppe Peano (1858-1932).
◻ (the Halmos tombstone or Halmos symbol) means ”Q.E.D.”, which is
an abbreviation for the Latin phrase quod erat demonstrandum (”which was to be


16

Chapter 1. Introduction

proven”). ”Q.E.D.” has been the most common way to symbolize the end of a logical
argument for many centuries, but the modern convention of the ”tombstone’ ’ is now
generally preferred because it is easier to write and is also visually more compact.
The symbol ”∃” was first made popular by mathematician Paul Halmos (1916-2006).

1.3.3

Some Notation from Set Theory

◇ The symbol: ⊂ (the is included in sign) means ”is a subset of ” and ⊃
(the includes sign) means ”has as a subset”. Both symbols were introduced in the
1890 book Vorlesungen uăber die Algebra der Logik (”Lectures on the Algebra of
the Logic”) by logician Ernst Schrăoder (1841-1902).
◇ The symbol: ∈ (the is in sign) means ”is an element of ” and first appeared
in the 1895 edition of Formulaire de mathematiqu’es by logician Giuseppe Peano
(1858-1932). Peano origi- nally used the Greek letter ” e ” (viz. the first letter of
the Latin word est for ”is”), and it was the great logician and philosopher Betrand
Russell (1872-1970) who introduced the modern stylized version of this symbol in
his 1903 book Principles of Mathematics. It is also common to use the symbol
”∋” to mean ”contains as an element”, which is not to be confused with the more
archaic usage of ”∋” to mean ”such that”.
◇ The symbol: ∪ (the union sign) means ”take the elements that are in
either set”, and ∩ (the intersection sign) means ”take the elements that the
two sets have in common”. They were introduced in the 1888 book Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann preceduto dalle operazioni
della logica deduttiva (”Geometric Calculus based upon the teachings of H. Grassman, preceded by the operations of deductive logic”) by logician Giuseppe Peano
(1858-1932).
◇ The symbol: ∅ (the null set or empty set) means ”the set without any
elements in it” and was first used in the 1939 book E’l’ements de math’ematique
by Nicolas Bourbaki. (Bourbaki is the collective pseudonym for a group of primarily

European mathematicians who have written many mathematics books together.)
It was borrowed simultaneously from the Norwegian, Danish and Faroese alphabets
by group member Andr’e Weil (1906-1998).
◇ The symbol: ∞ (infinity) denotes ”a quantity or number of arbitrarily
large magnitude” and first appeared in print in the 1655 De Sectionibus Conicus
(”Tract on Conic Sections”) by mathematician John Wallis (1616-1703). Possible
explanations for Wallis’ choice of ”∞” include its resemblance to the symbol ”oo”
(used by ancient Romans to denote the number 1000), to the final letter of the
Greek alphabet ω (used symbolically to mean the ”final ” number), and to the ease
with which this simple curve (called a ”lemniscate”) can be endlessly traversed.


1.3. Some Common Mathematical Symbols and Abbreviations (with History)

1.3.4

17

Some Important Numbers in Mathematics

◇ The symbol: π (the ratio of the circumference to the diameter of a circle)
denotes the number 3.141592653589..., and was first used by mathematician William
Jones (1675-1749) in his 1706 book Synopsis palmariorum mathesios (”A New
Introduction to Mathematics”). It was then the great mathematician Leonhard
Euler (1707-1783) who popularized the use of π to denote this number in his 1748
book Introductio in Analysin Infinitorum. (It is speculated that Jones chose the
letter ” π ” because ” π ” is the first letter in the Greek word perimetron (πǫιµǫτ ρoν ),
which roughly means ”around ”.)
◇ The symbol: e = lim (1 + )
1

n

n→∞

n

(the natural logarithm base, also some-

times called Euler’s number) denotes the number 2.718281828459... , and was
first used by Leonhard Euler (1707-1783) in the manuscript Meditatio in Experimenta explosione tormentorum nuper instituta (”Meditation on experiments made
recently on the firing of cannon”), which was written when Euler was only 21 years
old. (It is speculated that Euler chose ” e” because ” e” is the first letter in the word
”exponential ”.) The mathematician Edmund Landau (1877-1938) once wrote that,
”The letter e may now no longer be used to denote anything other than this positive
universal constant.”
√
◇ The symbol: i = −1 (the imaginary unit) was first used by Leonhard
Euler (1707-1783) in his 1777 memoir Institutionum calculi integralis (”Foundations
of Integral Calculus”). The five most important numbers in mathematics are widely
considered to be (roughly in order) 0, 1, i, π, and e, which are remarkably linked by
the equation
eip + 1 = 0.
◇ The symbol: γ = lim ( ∑

1
− ln n) (the Euler-Mascheroni constant, also
k=1 k
n

n→∞

known as just Euler’s constant), denotes the number 0.577215664901..., and was
first used by geometer Lorenzo Mascheroni (1750-1800) in his 1792 book Adnotationes ad Euleri Calculum Integralem (”Annotations to Euler’s Integral Calculus”). The number γ is widely considered to be the sixth most important important number in mathematics due to its frequent appearance in formulas from
number theory and applied mathematics. However, no one knows whether ? is even
an irrational number.
√
1+ 5
◇ The symbol: φ =
(the golden ratio) denotes the number
2

φ ≈ 1.618033988749.... Its use was first attributed to the American Mathematician