Class 8 – Number System
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Introduction: A number which can be written in the form of , where and are integers and is called a rational number. In, , because division of an integer by zero is meaningless. Examples for rational numbers are Positive and Negative Rational Numbers: In a rational number , if both p and q are of the same sign i.e., both positive or both negative, then it is called as a positive rational number, If any one of p and q is of opposite sign, then rational number is called as a negative rational number. For example, etc., are positive rational numbers and etc., are negative rational numbers 
Rational Number 0 (i.e., etc.,) is neither positive nor negative. Some facts about rational numbers: 1. All natural numbers, whole numbers, integers and fractions are rational numbers. i) 24 can be written as here . ii) 12 can be written as here. In particular ‘zero’ is also a rational number. 2. Rational numbers do not have a unique representation in the form of (where p and q are integers and ). 

3. Arational number , is said to be in standard form if (i) q is positive and (ii) p, q do not have any common factors other than 1 (that is p, q are coprime).
Example: The standard form of the rational number is (G.C.Dof 72, 108 is 36).
4. Terminating decimals and nonterminating repeating decimals are also rational numbers.
Example: Convert the following rational numbers to standard form:
i) ii) iii) iv)
Solution:
i).
ii)In the rational number, the HCF of 16 and 20 is 4.
So, divide both numerator and denominator by 4.
which is in standard form.
iii) In denominatoris not positive. Multiply both denominator and numerator with (1)
HCF of 10 and 8 is 2,
So,
which is in standard form
iv) which is in standard form.
Equivalent Rational Numbers: Two rational numbers are said to be equivalent rational numbers if they have same standard form.
Example: and are equivalent rational numbers because and
i.e., they have the same standard form.
Similarly,
are equivalent rational numbers because standard form of each of them is .
There can be infinitely many equivalent rational numbers to a given rational number.
Example: Write in an equivalent rational number form so that.
i) Its numerator is 60
ii) Its denominator is 15
Solution:
i)
ii)
Operations on rational numbers: The operation of addition, subtraction, multiplication and division of rational numbers are similar to that of integers and fractions.
Addition:
a) Addition of rational numbers with same denominator: In general, to add two rational numbers and ,which are having same denominators.
The above rule can be extended to more than two rational numbers.
Example: Add and
Solution: We write
b) Addition of rational numbers with different denominators: It is always convenient to add two rational numbers with different denominators, by converting their denominators into LCM of two denominators.
In general, to add two (or more) rational numbers with different denominators, first convert them into equivalent rational numbers with common denominator equal to their LCM and then add as done before for adding rational numbers with same denominator.
Example: Add , and
Solution:
Subtraction of Rational Numbers:
a) Subtraction of rational numbers with same denominator: In general, if and are two rational numbers
Example: Subtract from
Solution:
b) Subtraction of rational numbers with different denominators: Here again, convert the given rational numbers into equivalent form with same denominator, preferably their LCM.
Example: Subtract from
Solution:
If and are two rational numbers such that then is called as additive inverse of and vice versa.
Multiplication of Rational Numbers: For two fractions
Product of fractions =
The above rule of multiplication of two fractions is same for rational numbers, and it can be extended to more than two rational numbers.
Example: Multiply 2, and
Solution: We have
Division of Rational Numbers: To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.
The same rule can be applied for dividing a rational number by a nonzero rational number.
Example: Divide
Solution:
Properties of rational numbers: If a, b, c are rational numbers:
1. Closure property:
Addition 
is rational 
Subtraction 
is rational 
Multiplication 
is rational 
Division 
need not be rational 
2. Commutative property:
Addition 
Commutative 

Subtraction 
Not commutative 

Multiplication 
Commutative 

Division 
Not commutative 
3. Associative property:
Addition 
Associative 

Subtraction 
Not associative 

Multiplication 
Associative 

Division 
Not associative 
4. Distributive property:
i) (Multiplication over addition)
ii) (Multiplication over subtraction)
5. Additive identity (zero property): ‘0’ is called the additive identity for the set of rational numbers.
6. Multiplicative identity (property of 1): is called multiplicative identity for the set of rational numbers.
7. Additive inverse (negative of a number): is called the additive inverse of ‘a’.
8. Multiplicative inverse (Reciprocal): is called the multiplicative inverse of ‘a’ or reciprocal of ‘a’.
To compare two rational numbers:
i) First make their denominators positive.
ii) If they are of opposite signs, positive rational number is always greater than the negative rational number.
iii)If both are positive, compare them as fractions.
iv) If both are negative, compare their corresponding fractions negative (ignoring the– sign). If one fraction is greater than the other, then the corresponding rational number is less than the other.
v) Every positive rational number is greater than 0 and every negative rational number is less than 0.
Example: Which is smaller of the given rational numbers?
(i) (ii)
Solution:
Here, denominators are same, compare the numerators 55 and42.
Clearly, 55 > 42.
So, Hence, is smaller.
Here, compare numerators 55 and 42, because denominators are the same.
–55 < –42
So,
Hence, is smaller.
Rational numbers between the rational numbers: Between two rational numbers there are infinitely many rational numbers.
Method 1: Let a and b be two rational numbers
[r1 is the rational number between a and b].
r2 is rational number between a and r1, then
r3 is the rational number between and b.In this manner infinite rational numbers can be found between two given distinct rational numbers.
Example: Find 3 rational numbers lying between and .
Solution: Let are any three rational numbers lying between and .
∴ 3 rational numbers between and , are
Method 2: Let and are two rational numbers.
Step 1: Make denominators equal in both rational numbers, (by multiplying the numerator and denominator of each rational number by LCM of denominators)
Step 2: To insert ‘n’ rational numbers between the two rational numbers (obtained in step 1, then multiply the numerator and denominator by such a number so that the difference between the numerators is at least ‘n’. Let us understand these steps through this example.
Example: Insert three rational numbers lying between and .
Solution: Make denominators equal by multiplying 2 and3. and . To insert 3 rational numbers, we multiply the numerators and denominators by such a number so that difference between the numerators is at least 3. Multiplying the numerators and denominators of both numbers by 4, we get and .
Hence the required 3 rational numbers are.
Applications of Rational Numbers in Problem solving:
Example: A person travels a distance of km towards west and then a distance of km towards east on the same road. Find the distance and direction of the person from the starting point.
Solution: Let the person start from a point, say, O towards west.
Distance travelled towards west = km
Distance travelled towards east = km
Position of the person is given by the expression
So, the person is at a distance of km from his original point in west direction.
Example: As a part of a km long marathon race, km wasto be covered by cycle, km by boat and rest on foot. How much is the distance travelled on foot?
Solution: Distance to be covered by cycle and boat = km
Distance to be covered on foot
Absolute value of a rational number:
For any rational number ‘x’, .
Example:
1. Every integer is a rational number.
2. Every fraction is a rational number.
3. Every terminating decimal is a rational number.
4. Every nonterminating repeating decimal is also a rational number.