Mathematics As A Language

CHAPTER 2
MATHEMATICS
AS A LANGUAGE
INTRODUCTION
• Language is very powerful. It is used to express our emotions,
thoughts, and ideas. However, if the recipient of the message cannot
understand you, then there is no communication at all. It is very
important that both of you understand the language. Mathematics is
very hard for others to study because they are very overwhelmed
with the numbers, operations, symbols and formulae.
EXPRESSION VERSUS SENTENCES
• You learned in your English subject that expressions do not state
a complete thought, but sentences do. Mathematical sentences
state a complete thought. On the other hand, mathematical
expressions do not.You cannot test if it is true or false.
MATHEMATICAL EXPRESSION
• In Mathematics, an expression or mathematical expression is a

finite combination of symbols that is well-formed according to
rules that depend on the context. Mathematical symbols can
designate numbers (constants), variables, operations, functions,
brackets, punctuation, and grouping to help determine order of
operations, and other aspects of logical syntax.
MATHEMATICAL EXPRESSION
• Expressions are mathematical statements that have a minimum

of two terms containing numbers or variables, or both,
connected by an operator in between. The mathematical
operators can be of addition, subtraction, multiplication or
division. In math, there are two types of expressions, arithmetic
expressions-that contain only numbers; and algebraic
expressions-that contain both numbers and variables.
MATHEMATICAL SENTENCE
• Mathematical sentence is an expression which is

either true or false.
• Open mathematical sentence is a sentence which
could be true or false depending on the value or
values of unknown variables.
CONVENTIONS IN THE MATHEMATICAL
LANGUAGE
• The common symbol, used for multiplication is x but it can be mistakenly
taken as the variable x. There are instances when the centered dot ( . ) is
a shorthand to be used for multiplication especially when variables are
involved. If there will be no confusion, the symbol may be dropped.
7 . b = 7b
e . f . g = efg
m . n. 6 = 6mn
It is conventional to write the number first before the letters. If in case the
letters are more than one, you have to arrange the letters alphabetically.
THE LANGUAGE OF SETS
• Use of the word set as a formal mathematical term was introduced in 1879 by Georg
Cantor (1845-1918). For most mathematical purposes we can think of a set intuitively, as
Cantor did, simply as collection of elements. For instance, if C is the set of all countries
that are currently in the United Nations, then the United States is an element of C, and if I
is the set of all integers from 1 to 100, then the number 57 is element of I. Developed the
theory on sets as an outgrowth of his studies on infinity
SETS
• A SET IS A COLLECTION OF ELEMENTS OR NUMBERS

OR OBJECTS, REPRESENTED WITHIN THE CURLY
BRACKETS {}.
SETS
• SETS are usually represented by uppercase letter like S. The

symbols R and N represent the set of real numbers and the
set of natural numbers, respectively. A lowercase letter near
the end of the alphabet like x , y, or z represents an element
of the set of real numbers. A lowercase letter near the
middle of the alphabet particularly form i to n may represent
an element of the set of integers.
• A set is a collection of any object. It is a mathematical expression in
which a name is given to some collection of objects. Elements or
members refer to the objects in a set. Otherwise, infinite set. A set
that has no members is called the empty set which is denoted using
{ } or
List method is a way of describing the set in which the members are
separated by commas and enclosed in braces like Set S = {4,8,12}. The
set has 3 elements which are 4, 8, and 12.
SET
• Represented by a capital letter

ELEMENT
• Each object in a set

• Member in a set
• Represented by a small letter
a
“ a is an element of set A “
3 { 1, 3, 5 }
NOTATION
If S is a set, the notation x ∈ S means that x is element of S. The notation x ∉ S means that x is not an
element of S. A set may be specified using the set roster notation by writing of all of its elements between
braces. For example, {1,2,3} denotes the set whose elements are 1,2, and 3. A variation of the notation is
sometimes used to describe a very large set, as when we write {1,2,3,,100} to refer to the set of all integers
from 1 to 100. A similar notation can also be describe an infinite set, as when we write {1,2,3,…} to refer
to the set of all positive integers. The symbol … is called an ellipsis and is read as “and so forth.”
• Throughout , we will use numbers to provide examples. Therefore, we
would like to standardize the symbols to denote various sets of numbers
as follows.
• N : The sets of all natural numbers (i.e. all positive integers)
• Natural numbers includes all the positive integers from 1 till infinity and
are also used for counting purpose excluding zero (0).
• Example: 1,2,3,4,5,6,7,…
• Z:The set of all integers
• The word integer originated from the Latin word “Integer” which means
whole or intact. Integers is a special set of numbers comprising zero,
positive numbers and negative numbers.
• Z* : The set of all nonzero integers
• E: The set of all even integers
• Q : The set of all rational numbers
• A rational number is a number that can be in the form p/q

where p and q are integers and q is not equal to zero.
• Q* : The set of all nonzero rational numbers
• Q+ : The set of all positive rational numbers
• I: The set of all irrational number
• The set of irrational numbers is represented by the letter I. Any real number that is not rational is irrational.
These are numbers that can be written as decimals, but not as fractions. They are non-repeating, non-
terminating decimals. Some examples of irrational numbers are:
• The most common irrational numbers are:

Pi (π) = 22/7 = 3.14159265358979…
Euler’s Number, e = 2.71828182845904…
Golden ratio, φ = 1.61803398874989….
Root, √ = √2, √3, √5, √7, √8, any number under root which cannot be simplified further.
• R: The set of all real numbers
• Real numbers can be defined as the union of both the rational and irrational numbers. They can be both
positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals and fractions
come under this category.
• R+ : The set of all positive real numbers
• C: The set of all complex numbers
A Complex Number is a combination of a

Real Number and an Imaginary Number
Imaginary Numbers when squared give a negative result.

Normally this doesn't happen, because: when we square a positive number we get a positive result, and
when we square a negative number we also get a positive result (because a negative times a negative
gives a positive), for example −2 × −2 = +4
Examples of Imaginary Numbers:
3i 1.04i −2.8i 3i/4 (√2)i 1998i
• C* : The set of all nonzero complex numbers
FINITE SET
•If the number of elements
in a set can be counted

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CHAPTER 2

• In Mathematics, an expression or mathematical expression is a

• Expressions are mathematical statements that have a minimum

• Mathematical sentence is an expression which is

• A SET IS A COLLECTION OF ELEMENTS OR NUMBERS

• SETS are usually represented by uppercase letter like S. The

• Represented by a capital letter

• Each object in a set

• N : The sets of all natural numbers (i.e. all positive integers)

• Z:The set of all integers

• E: The set of all even integers

• Q : The set of all rational numbers

• A rational number is a number that can be in the form p/q

• Q* : The set of all nonzero rational numbers

• Q+ : The set of all positive rational numbers

• I: The set of all irrational number

• The most common irrational numbers are:

• R+ : The set of all positive real numbers

• C: The set of all complex numbers

A Complex Number is a combination of a

Imaginary Numbers when squared give a negative result.

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