Language of Mathematics

1. Elements
Symbols of Maths
English Language vs Mathematics Language

2.  At the end of this chapter, the students
should be able to:
 discuss the language, symbols, and conventions
of mathematics
 translate an English expression into a math
expression
 translate an English sentence into a math
equation
 describe sets and the relations between them
 define or describe relations and functions of sets
 acknowledge that mathematics is a useful
language

3.  Ana est deux fois plus âgée que son frère et
la somme de leurs âges est de 36 ans. Quel
âge ont-ils?

4.  Language is a systematic way of
communicating with other people by the use
of sounds or conventional symbols.
 It is a system of word used in a particular
discipline.
 It is also a system of abstract codes which
represent antecedent events and concepts
and arranged in ordered sequence to form
words, with rules.

5.  Language is important to understand and
express one’s ideas, feelings or opinion.
 Language serves as the transmitter of
information and knowledge.
 It helps to construct social identity.
 Misunderstanding of one’s language leads to
confusion and misconcpetions.

6.  The language of Math was designed so we
can write about things such as numbers, sets,
functions, etc. and what we do with those
things like perform operations such as
addition, subtraction, multiplication and
division.

7. Precise means able to make very fine
distinctions or definitions
Concise means able to say things briefly
Powerful means able to express complex
thought with relative ease

8.  Nouns Negations
 Pronouns Sentence Structure
 Verbs Paragraph Structure
 Sentences Conventions
 Vocabulary Abbreviations
 Grammar
 Syntax
 Synonyms

9. ENGLISH MATHEMATICS
SYMBOLS English Alphabet and
punctuation
English Alphabet,
Numerals, Greek
Letters, Grouping
Symbols, Special
Symbols
Name Noun
Pronoun
Expressions
Variable
Complete thought Sentence Sentence
Action Verbs Operations and other
actions(e.g. simplify,
rationalize
What’s in a sentence Verbs Equality, inequality,
membership in a set
Attribute of a sentence Fact or fiction True or false
Synonyms Different words but
the same meaning
The same object but
different names

10.  English – Noun
= is used to name things we want to talk
about
 Math – Expression
= refers to the object of interest

11.  English – Noun
= is used to name things we want to talk
about
Example:
Carol loves Mathematics.

12.  English – Noun
= is used to name things we want to talk
about
Examples:
Carol loves Mathematics

13.  Math – Expression
= refers to the object of interest.
Examples:
5, 1.2 + 6, 3x – 3

14.  Math – Expression
= refers to the object of interest.
Examples:
5, 1.2 + 6, x, 3x – 3
Other types of expressions:
* numbers, sets, functions, ordered pairs,
matrices, vectors, groups, etc.

15.  English – Pronoun
= another way of calling a noun
 Math - Variable
= is the symbol that represents any
constant value

16.  English – Pronoun
= another way of calling a noun
Example:
She loves Mathematics.

17.  Math - variable
= is the symbol that represents any
constant value.
Examples :
x – 2

18.  English/Math – Sentence
= must show complete thought (noun and
verb)
= can express true, false or sometimes true
or sometimes false idea.
Examples:
Carol loves Mathematics 1.2 + 6 = 7.2

19.  English – Verbs
= action words
 Math = Verbs
= action words such as equals,
inequalities, simplify, rationalize…

20.  English – verbs
= action words
Example:
Ana computes for the value of x.

21.  English – verbs
= action words
Example:
Ana computes for the value of x.

22.  Math = Verbs
= action words such as equals,
inequalities, simplify, rationalize…
Examples:
2x + 5 = 7
Simplify the expression (x2 - 2x + 5) – 5(x – 4).

23.  Math = Verbs
= action words such as equal,
inequalities, simplify, rationalize…
Examples:
2x + 5 = 7
Simplify the expression (x2 - 2x + 5) – 5(x – 4).

24.  English – Synonyms
= different words with the same meaning
(have nearest meaning)
Example : Group - association

25.  Mathematics – Synonyms
= the same object but different names
Example : 1 + 2 + 5 and 8
½ + ½ , 2 - 1, 5/5,

26.  The language of mathematics has an
abundant vocabulary of specialist and
technical terms and also uses symbols
instead of words which are essential to the
power of modern mathematics.
 Some of the symbols commonly used in
Mathematics are the following:

27. R Set of Real numbers ∈ Element of (or member of)
N Set of Natural numbers ⊆ Subset of
Z Set of Integers ⊂ Proper subset
Z+ Set of Positive Integers → If - then
Z- Set of Negative Integers ↔ If and only if
Q Set of Rational Numbers Σ The sum of
 For every (for any) ∞ Infinity
∃ There exists

28.  The mathematical notation used for formulas
has its own grammar, not dependent on a
specific natural language, but shared
internationally by mathematicians regardless
of their mother tongues.
 This includes the conventions that the
formulas are written predominantly left to
right.
 The Latin alphabet is commonly used for
simple variables and parameters.
 There are structural rules governing the use
of symbols representing mathematical
objects.

29.  Some Difficulties in the Math Language
1. Different meaning/use of words in Math
and English
“and” is equivalent to plus
“is” may have different meaning
2. The different uses of numbers : cardinal,
ordinal or nominal
3. Mathematical objects may be expressed
in many ways such as sets and functions

30. Operation Symbol Words Algebraic
Expression
Word
Equivalent
Addition
+
Plus, sum,
more than,
increased by,
add to, total
x + 2
- two more
than a
number
- x increased
by 2
Subtraction
-
Subtracted
from, minus,
difference
of, less than,
decreased
by, less
z - 5
- z minus 5
- a number
z
subtracted
by 5
- five less
than a
number z

31. Operation Symbol Words Algebraic
expression
Word
equivalent
Multiplication
•, ( )
Times,
product,
multiply,
twice, of
7(k) or 7•k
or simply 7k
- 7 times a
number k
- The
product of
7 and k
Division
÷, /
Divided by,
quotient,
into, ratio of
w ÷ 8 or
w/8
- w divided
by 8
- The
quotient
of a
number w
and 8

32. Exercise:
Translate each of the following phrases into
mathematical expression. Use as few
variables as possible:
1. The sum of a number and 10
2. The product of two numbers
3. The product of -1 and a number
4. One-half times the sum of two numbers
5. Twice a number

33. Choose a quantity to be represented by a
variable, then write the mathematical
expression for each.
1. Lota’s age in 5 years
Answer : let x = be the present age of Lota
x + 5 = Lota’s age in 5 years

34. 2. A three-digit numbers whose hundreds digit
is half the tens digit and the tens digit is 2
more than the units digit.
Let x = be the unit’s digit
x+2 = tens digit
½ (x + 2) = hundreds digit

35. 3. The total interest earned after one year
when P 100 000 is invested part at 6 %
annual interest rate and the remaining part
at 7.5 % annual interest rate.
Let x = be the part to be invested at 7.5%
100,000 – x = the part to be invested at 6%
y = be the total earned interest
0.06(P 100 000 – x) + 0.075x = y

36.  A statement of equality of two algebraic
expressions which involves one or more
literals (variables) is called an equation.
 Ex.
x + 3 = 7

37.  Ex.) The quotient of 3 and a number is
1
3
.
 Remember that the word ‘quotient’ translates
into division. The phrase ‘a number’ will be
replaced by v and the word ‘is’ will be replaced
by the equality symbol =. Hence, putting them
together, we have

3
𝑣
=
1
3

38.  A universal statement says that a certain
property is true for all elements in a set.
Ex. All positive numbers are greater than
zero.
 A conditional statement says that if one
thing is true then some other things also has
to be true.
Ex. If 378 is divisible by 18, then 378 is
divisible by 6.

39.  An existential statement says that there is
at least one thing for which the property is
true.
Ex. There is a prime number that is even.

40. 1. The sum of any two real numbers is also a
real number.
Answer:
 a, b  , a + b  

41. 2. The square of any real number x is greater
than or equal to zero.
Answer :
 x  , x2  0

42. 1.  x , y  , x - y = 0
Answer:
The difference of any two real numbers
x and y is zero.

43. 2.  m, n  , m – n  m + n
Answer:
There exist integers m and n, such
that m minus n is less than or
equal to m plus n.

44.  Use of the word set as a formal
mathematical term was introduced in
1879 by Georg Cantor (1845 – 1918).
 Set is a well-defined collection of
objects, which may be concrete or
abstract.

45.  Sets are conventionally denoted by capital
letters. Small letters are used as names for
the objects.
 The object of a set is called its member or
element. The symbol “∈” denotes
membership “∉“ denotes non-membership to
a set.
 The number of elements of a set is called its
cardinal number and is denoted by n(A).

46. The set-Roster method
A method used to describe or
define a set by explicitly listing its
elements between braces.
Ex)
1. Set A is the set of distinct letters in
the word “paper”
A = {p, a, e, r}

47. The Set-builder notation
In this method, a set is defined by
enclosing in braces a descriptive phrase,
and agreeing that the elements of the
set have the described/common
property.
This method uses the symbols “x”
and “|”

48.  Let S denote a set and let P(x) be a
property that elements of S may or
may not satisfy. We may define a new
set to be the set of all elements x in
S such that P(x) is true.
{x є S | P(x) }

49.  Given that R denotes the set of all real
numbers, Z the set of all integers, and 𝑍+
the set of all positive integers, describe
each of the following sets.
 a. {x є R | -2 < x < 5}
 b. {x є Z | -2 < x < 5}
 c. {x є Z | -2 < x < 5}
+

50.  If A and B are sets, then A is called a subset
of B, written A ⊆ B, if and only if, every
element of A is also an element of B.
 A ⊆ B means that For all elements x, if x ∈ A
then x ∈ B.
 A is contained in B and B contains A.
 A ⊈ B means that, there is at least one
element x such that x ∈ A and x ∉ B.

51.  Let A and B be sets. A is a proper subset of B
if and only if every element of A is in B but
there is at least one element of B that is not
in A.
 A ⊂ B

52. Given elements a and b, the symbol (a,
b) denotes the ordered pair consisting
of a and b together with the
specification that a is the first element
of the pair and b is the second
element. Two ordered pairs (a, b) and
(c, d) are equal if, and only if, a = c
and b = d. Symbolically:
(a, b)=(c, d) means that a = c and b = d

53. RELATION
Let A and B be sets. A relation R from A to B
is a subset of A x B. Given an ordered pair
(x, y) in A x B, x is related to y by R, written
x R y, if, and only if, (x, y) is in R. The first
element is called the domain of R and the
second element is called the range.
x R y means that (x, y) є R.

54. Let A = {1, 2} and B = {1, 2, 3} and define a
relation R from A to B as follows:
Given any (x, y) є A x B,
(x, y) є R means that (x – y) / 2 is an
integer.
1. State explicitly which ordered pairs are in
A x B and which are in R.
1. Is 1 R 3? Is 2 R 3? Is 2 R 2?
2. What are the domain and range of R?

55. FUNCTION
A relation F from a set A to a set B is a
function if the following conditions are
satified:
1. For every element x in A, there is an
element y in B such that (x, y) є F.
2. For all elements x in A and y and z in B,
if (x, y) є F and (x, z) є F, and y = z.

56. FUNCTION
Properties (1) and (2) can be stated less
formally as follows: A relation F from A to B
is a function if, and only if,
1. Every element of A is the first element of
an ordered pair of F.
2. No two distinct ordered pairs in F have
the same first element.

57. Example:
1. Is the relation A = { (1, 9), (2, 2), (3, 5) } also
a function?
2. Is the relation B = { (1, 6), (4, 5), (4, 7) } also
a function?

58. NOTATION
If A and B are sets and F is a function
from A to B, then given any element
x in A, the unique element in B that
is related to x by F is denoted f(x),
which is read “f of x.”

59.  For any function f, the notation f(x), read as
“function of x” represents the value of y
when x is replaced by the number of
expression inside the parenthesis.
 To find the value of the function means to
evaluate a function.