The story of Babel highlights

the importance of a
common language for
collaboration and progress.
How important is language
for engineers?
 Language is crucial for
engineers as it enables clear
communication of complex
ideas, collaboration across
diverse teams, and the
effective presentation of
technical solutions.
Mathematical
Language and
Symbol
Engr. Eldrick M. Fonollera
Professional Industrial Engineer
Many have troubles in understanding
mathematical ideas, not because ideas are
difficult, but because the ideas are being
presented in foreign language - the language
of mathematics.
The language of
mathematics is
difficult
But it can be
learned
Characteristics of Mathematical Language

1. PRECISE

Able to make very fine distinctions

1. CONCISE

Able to say things briefly

1. POWERFUL

Able to express complex thoughts with relative ease

Every language has its own vocabulary (words) and
rules for combining words into a complete thought
(sentences)

Barking is dog the

The barking dog is
The dog is barking

Same with Mathematics

 ENGLISH VS MATHEMATICS

NOUN = EXPRESSION
SENTENCE = MATHEMATICAL SENTENCE

An expression is the mathematical analogue of an English noun; it is

a correct arrangement of mathematical symbols used to represent a
mathematical object of interest.
 ENGLISH VS MATHEMATICS

NOUN = EXPRESSION
SENTENCE = MATHEMATICAL SENTENCE

An expression does not state a complete thought; it does not make

sense to ask if an expression is true or false.
Some types of expression are as follows:
● Numbers
○ 0, 1, 2, 3
● Sets
○ Set of natural numbers, ℕ = {1, 2, 3, ...}
○ Set of whole numbers, W = {0, 1, 2, 3, ...}
○ Set of integers, ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
● Functions
○ f(x)=3x+2
● Ordered Pairs
○ (1, 2)
● Matrices
○ [1, −2, 4]
● Vectors
○ velocity, momentum, force, electromagnetic fields and weight
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Complex Numbers is the most general set of numbers that cover all
numbers. All numbers are complex numbers.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Imaginary Numbers are numbers that can be expressed as a real number

multiplied by the imaginary unit 𝑖.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Real Numbers are set of all numbers found on the number line.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Rational Numbers are numbers that can be expressed as the ratio of two
integers.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Irrational Numbers are numbers that cannot be expressed as a ratio of

two integers. Unlike rational numbers, their decimal expansions are non-
terminating and non-repeating.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Fractions Integers
-⅔, ⅔ -2, 0, 1, 2

Fractions Numbers are used to represent parts of a whole or ratios

between quantities.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Fractions Integers
-⅔, ⅔ -2, 0, 1, 2

Integers are numbers that includes all whole numbers, both positive and
negative, as well as zero.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Fractions Integers
-⅔, ⅔ -2, 0, 1, 2

Negative Zero Natural Numbers

-2 0 1, 2

Negative Numbers are numbers that are less than zero. They are
represented with a minus sign (-) in front of the number and are used to
indicate values below zero.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Fractions Integers
-⅔, ⅔ -2, 0, 1, 2

Negative Zero Natural Numbers

-2 0 1, 2

Zero is an integer that represents a null value or no quantity. It is neither

positive nor negative.
 Complex Numbers
-2i, -2, -⅔, 0, 1, i, ⥌2, 2, ⅔ , 3i,

Real Numbers Imaginary Numbers

-2, -⅔, 0, 1, ⥌2, 2, ⅔ 3i, -2i, i

Rational Numbers Irrational Numbers

-2, -⅔, 0, 1, 2, ⅔ ⥌2

Fractions Integers
-⅔, ⅔ -2, 0, 1, 2

Negative Zero Natural Numbers

-2 0 1, 2

Natural Numbers are positive whole numbers used for counting and
ordering. They are typically denoted by the symbol 𝑁.
 Above are examples of expression. All look
different, but are all just different names for the
same number.
 ENGLISH VS MATHEMATICS

NOUN = EXPRESSION
SENTENCE = MATHEMATICAL SENTENCE

A mathematical sentence is the analogue of an English sentence; it is

a correct arrangement of mathematical symbols that states a
complete thought.
 ENGLISH VS MATHEMATICS

NOUN = EXPRESSION
SENTENCE = MATHEMATICAL SENTENCE

A mathematical sentence can be defined as true, false, sometimes

true, sometimes false
Conventions in
Mathematical
Language
Languages have conventions.

In the English language, it is conventional to use

capital letters for proper names. This
convention helps to distinguish between a
common name and a proper name.

Mathematical language also has its conventions

which help learners to distinguish between
different types of mathematical expressions.
Letters denotes numbers, constants and variables. It is
also used to label object such as line I, point P, function f,
set S, event E, and so on.

● Numbers are usually represented by lowercase

letters: a, b, m, x, y, z,....

● Sets are usually represented by uppercase letters: A,

B, C, S,...

● A set of real numbers usually uses lowercase from

the end of the alphabet particularly x, y, or z.

● A set of integers usually uses lowercase near the

middle of the alphabet particularly i, j, k, m, or n.
Four Basic
Concepts
Sets, Functions, Relations, Binary Operations
Sets
In 1879, Georg Cantor (1814 –
1918) introduced the use of word
set as a formal mathematical term.

A set is a collection of objects

called elements. We usually use
capital letters to denote sets.
Examples

If A is the set of all colors in the rainbow, then red

is an element of A.

If M is the set of all odd numbers from 1 to 50,

then 31 is an element of M.
If S is a set, the notation x ∈ S means that x is an
element of S.

A set may be specified using the set-roster

notation by writing all of its elements between
braces.
ROSTER NOTATION

Elements (or members) of a set are listed in a row inside the curly
brackets. Every two elements are separated by a comma symbol in
a roster notation if the set contains more than one element. The
roster form is also called the enumeration notation as the
enumeration is done one after one.

For example, if A is the set of the first 9 natural numbers, it can

be represented by: A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
ROSTER NOTATION

One of the limitations of roster notation is that we cannot

represent a large number of data in roster form.

For example, if we want to represent the first 100 or 200 natural

numbers in a set B.
B = {1, 3, 5, 7, ....., 199}
C = {a, b, c, d, ......., z}
If any set has an infinite number of elements like the set of all
the even positive integers
D = {2, 4, 6, 8, ......}
 SET BUILDER NOTATION

Represent the sets by using a condition instead of mentioning

the set of all elements.

The set of all 𝑥 such that 𝑥 is a natural number and 𝑥 is between 1 and 5
SET BUILDER NOTATION

Represent the sets by using a condition instead of mentioning

the set of all elements.

The set of all 𝑥 such that 𝑥 is an even number and 𝑥 is greater than 0
and less than 10.
SET BUILDER NOTATION

Represent the sets by using a condition instead of mentioning

the set of all elements.

The set of all 𝑥 such that 𝑥 is an odd number and less than 10.
Venn Diagram

A Venn diagram uses overlapping circles or other shapes to

illustrate the logical relationships between two or more sets of
items.
Universal Set
Whenever we use a set, it is easier to first
consider a larger set called a universal set
that contains all of the elements in all of
the sets that are being considered.

● A large rectangle is used to

represent the universal set and it is
usually denoted by the symbol E or
sometimes U.
● All the other sets are represented by
circles or closed figures within this
larger rectangle.
● Every set is the subset of the
universal set U.