BASIC MATHEMATICS.
Lecture Notes.
Lecture Notes
SECTION 1 – REAL NUMBERS
Learning Objectives:
Describe the set of real numbers, all its subsets and their relationships.
Identify and use the arithmetic properties of subsets of integers, rational, irrational and
real numbers, including closure properties for the four basic arithmetic operations
where applicable.
Lecture Notes 1 BASIC MATH
,1.1 The Real Number System
Numbers play a vital role in the day to day life as their applications and uses cannot be limited
to the numerical problem solving. There are different types of numbers such as Natural,
Integers, Rational, Irrational etc. In this section, we will review these types of numbers that
make up the real number system.
The Natural Numbers
The numbers 1, 2, 3, 4, 5, … are called the natural numbers (counting numbers). The group of all
natural numbers is denoted by N. The numbers 1, 3, 5, 7, … are odd natural numbers
whereas the numbers 2, 4, 6, 8, … are the even natural numbers.
Whole numbers
The collection of all natural numbers along with zero is known as the set of whole
numbers. They are 0, 1, 2, 3, …. It is denoted by W.
Integers
The set of all integers is denoted by Z. The set of integers include the natural
numbers, the number zero and the negatives of the natural numbers. Thus, they are
…, -3 , -2 , - 1, 0, 1, 2, 3, ….
Rational Numbers
p 0, are
The numbers which can be expressed in the form , where p and q are integers and q
q
3 1 6
called rational numbers. For eg, 22, , , , -3, 0, 8 etc. are rational numbers. Any
7 5 4 11
Lecture Notes 2 BASIC MATH
, terminating or non-terminating recurring decimal number is also a rational number. i.e,
Numbers such as 1.75, 2.333…, 0. 2134134134… are rational numbers. The group of rational
numbers is denoted by Q.
Operations on rational Numbers
Addition of Rational numbers
a
The sum of two rational numbers is always a rational number. To add two rational numbers
bc
and , we use the following rule:
d
a c ad bc
b d bd
Subtraction of Rational numbers
The difference of two rational numbers is again a rational number. In order to subtract one
rational number from another, we proceed in the same way as that of addition.
ac ad bc
bd bd
Multiplication of Rational numbers
The product of two rational numbers is another rational number whose numerator is the
product of the numerators and the denominator is the product of the denominators.
a c ac
b d bd
Division of Rational numbers
The ratio of two non-zero rational numbers is again a rational number. To divide a rational
number by another rational number, we multiply the first one by the reciprocal of the second.
Lecture Notes 3 BASIC MATH
Lecture Notes.
Lecture Notes
SECTION 1 – REAL NUMBERS
Learning Objectives:
Describe the set of real numbers, all its subsets and their relationships.
Identify and use the arithmetic properties of subsets of integers, rational, irrational and
real numbers, including closure properties for the four basic arithmetic operations
where applicable.
Lecture Notes 1 BASIC MATH
,1.1 The Real Number System
Numbers play a vital role in the day to day life as their applications and uses cannot be limited
to the numerical problem solving. There are different types of numbers such as Natural,
Integers, Rational, Irrational etc. In this section, we will review these types of numbers that
make up the real number system.
The Natural Numbers
The numbers 1, 2, 3, 4, 5, … are called the natural numbers (counting numbers). The group of all
natural numbers is denoted by N. The numbers 1, 3, 5, 7, … are odd natural numbers
whereas the numbers 2, 4, 6, 8, … are the even natural numbers.
Whole numbers
The collection of all natural numbers along with zero is known as the set of whole
numbers. They are 0, 1, 2, 3, …. It is denoted by W.
Integers
The set of all integers is denoted by Z. The set of integers include the natural
numbers, the number zero and the negatives of the natural numbers. Thus, they are
…, -3 , -2 , - 1, 0, 1, 2, 3, ….
Rational Numbers
p 0, are
The numbers which can be expressed in the form , where p and q are integers and q
q
3 1 6
called rational numbers. For eg, 22, , , , -3, 0, 8 etc. are rational numbers. Any
7 5 4 11
Lecture Notes 2 BASIC MATH
, terminating or non-terminating recurring decimal number is also a rational number. i.e,
Numbers such as 1.75, 2.333…, 0. 2134134134… are rational numbers. The group of rational
numbers is denoted by Q.
Operations on rational Numbers
Addition of Rational numbers
a
The sum of two rational numbers is always a rational number. To add two rational numbers
bc
and , we use the following rule:
d
a c ad bc
b d bd
Subtraction of Rational numbers
The difference of two rational numbers is again a rational number. In order to subtract one
rational number from another, we proceed in the same way as that of addition.
ac ad bc
bd bd
Multiplication of Rational numbers
The product of two rational numbers is another rational number whose numerator is the
product of the numerators and the denominator is the product of the denominators.
a c ac
b d bd
Division of Rational numbers
The ratio of two non-zero rational numbers is again a rational number. To divide a rational
number by another rational number, we multiply the first one by the reciprocal of the second.
Lecture Notes 3 BASIC MATH
