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.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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

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 . . . . . . . . .


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


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


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


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

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


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


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


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

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


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



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

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


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

7.6.2

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

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

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


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
[email protected].
Sincerely thanks.
Bac Giang, on 28/10/2012
Nguyen Van Tien


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

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



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...

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




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


∆, δ

delta

5

E, ǫ

epsilon

6

H, η

eta

7

Θ, θ

theta

8

I, ι

iota

9


F, f

digamma

15

Π, π

pi

16

P, ρ

rho

17

Σ, σ

sigma

18

T, τ

tau

19


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


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

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
∃

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



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

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


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

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

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


Greater than sign; Close angle bracket

Forward slash



[
]

Close brace; Close square brace

−−

Em dash

–

En dash

−

Hyphen; Minus sign; Dash

=

Equal sign

,

+

Plus sign

3
4


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

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

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


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

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→∞