Natural Numbers: The counting numbers are called natural numbers.
Whole Numbers: All natural numbers together along with (zero) form the set W of all whole numbers.
Integers: All natural numbers, negative natural numbers and (zero) together form a set
of all Integers.
Properties of Addition of Integers
The sum of two integers is always an integer. If and
are two integers, then:
Associative Law: If , and
are integers, then:
Existence of Additive Identity: is the additive identity for integers. Therefore for any integer
we will have:
Existence of additive inverse: For each integer , there exists another integer
such that
Note: If be any integer, then
will be called the successor of
and the predecessor of
is
Properties of Subtraction
If and
are two integers, then
is also an integer.
For any integer , we have
. But
If are integers and
, then
Properties of Multiplication of Integers
Closure Property: The product of two integer is always an integer
Commutative Law: For any two integers and
we have:
Associative Law: For any three integers , and
we have:
Distributive Law of Multiplication over Addition: For any there integers , and
we have:
Existence of Multiplicative Identity: The integer is a multiplicative identity for Integers. So, for any integer a we have:
Property of Zero: For any integer we have:
Properties of Multiplication of Integers
If and
are integers, then
is not necessarily an integer.
If is an integer and
, then
If is an integer, then
If is a non-zero integer, then
but
is not defined.
If are integers, then
, unless
If , are integers and
then
, if
is positive
, if
is negative
Rational Numbers
The numbers that can be expressed in the form
, where
and
are integers and
are called rational numbers. Therefore, the set
of all rational numbers is given by
Note:
- Every integer is a rational number, since every integer
can be written as
- Every fraction is a rational number
- The square root of every perfect square is a rational number E.g.
- Every terminating decimal is a rational number E.g.,
- Every recurring decimal is a rational number E.g
Positive Rational Numbers: A rational number is said to be positive the numerator or denominator are either both positive and both negative. e.g.
Negative Rational Numbers: A rational number is said to be negative is any one of the numerator or denominator is negative (they are of opposite sign) e.g.
Equivalent Rational Numbers
If is a rational number and
is a non-zero integer, then we have
. We call these numbers
and
as equivalent rational numbers.
Standard form of a Rational Numbers
A rational number is said to be a standard rational number form if
and
are integers having no common divisors other than
and
is positive.
Note: Every rational number can also be represented on a number line.
Addition of Rational Numbers
In order to add rational numbers, we must first convert them into rational numbers with positive denominator.
When the rational numbers have the same denominator, then these numbers can be added as follows:
If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you add them.
Properties of Addition of Rational Numbers
Closure Property: The sum of two rational number is also a rational number. That is, if and
are rational numbers than
is also a rational number.
Commutative Law: If and
are rational numbers than
Associative Law: If are rational numbers, then
Existence of Additive Identity: The rational number 0 is the additive identity for any rational number. Therefore
Subtraction of Rational Numbers
(i) If and
are two rational numbers with the same denominator, then
(ii) If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you subtract them. i.e.
Properties of Multiplication
Closure Property: The product of two rational numbers is also a rational number. Therefore if and
are two rational numbers, then
is also a rational number.
Commutative Law: If and
are two rational numbers, then
Associative Law: If and
are rational numbers then
Existence of Multiplicative Identity: The rational number is a multiplicative identity for rational numbers. For any rational number
we have
Existence of Multiplicative Inverse: For every non zero rational number, there exists a multiplicative inverse. For a rational number the multiplicative inverse would be
such that
Distributive Law of Multiplication over Addition: For any three and
we have
Multiplicative Property of Zero: For any rational number that we have
Division of Rational Numbers
If and
are two rational numbers such that
. Then, we define
. When
is divided by
, then
is called the dividend,
is called the divisor and the result of the division is called the quotient.
Properties of Division of Rational Numbers
Closure Property: The division of two rational numbers is also a rational number. Therefore if
Properties of Division of Rational Numbers
Closure Property: The division of two rational numbers is also a rational number. Therefore if and
are two rational numbers, then
is also a rational number given
.
For any rational number , we have
For any non-zero rational number , we have
How to find a Rational Number between Two Given Rational Numbers?
Let there be two rational numbers and
. Then the rational number in between the two would be
.
If you want to find a large number of rational numbers between two given rational numbers, then first make the denominator equal by taking LCM. After that you can just insert values between the two numerators.
Irrational Numbers
A number that can neither be expressed as a terminating nor as a repeating decimals, is called an irrational number. These are non-terminating and non-repeating numbers. Examples
- Non-terminating and non-repeating numbers such as
or
- Square root of positive integers that are not perfect squares
- Cube root of numbers that are not perfect cubes
- Number
is irrational. It has a value that is non-terminating and non-repeating. We only approximate it to .
Properties of Irrational Numbers
Sum of two irrational numbers need not be an irrational number
- Take
and
as two irrational numbers. The sum is
which is a rational number.
Difference of two irrational numbers need not be an irrational number
- Take
and
as two irrational numbers. The difference is
which is a rational number
Product of two irrational numbers need not be an irrational number
- Take
and
as two irrational numbers. The product is
which is a rational number.
Quotient of two irrational numbers need not be an irrational number
- Take
and
as two irrational numbers. The quotient is
which is a rational number.
Commutative Law:
Associative Law
Distributive Law
The sum or difference of a rational and an irrational numbers is always irrational.
- Example
is irrational
- Example
is irrational
The product or quotient of a rational and an irrational numbers is always irrational
- Example
is irrational
- Example
is irrational
Real Numbers
All rational and irrational numbers forms the set of all real numbers.
Real number Line
- The real number line represents all real numbers.
- On a real number line, each point corresponds to a point and conversely, every point on the real number line corresponds to a real number.
- Between any two real numbers, there exist infinite real numbers.
Properties of Real Numbers
Closure property
- The sum of two real numbers is always a real number.
- The product of two real numbers is always a real number.
Commutative law
Associative law
Distributive Laws of multiplication over addition
Comparison of two irrational numbers
- If a and b are two irrational numbers such that
, then