Rational Numbers: Overview, Questions, Preparation

Numbers that can be written in the form of a ratio of two integers p and q where q≠0; that is if it can be written as p/q.

Examples:
1. -3/4
This is a rational number as it can be represented as (-3)/4 where p=-3 and q=4.

2. 0
This is a rational number as it is technically 0=0/1 and here q0

3. Even decimals can be expressed rationally. Taking 0.5, to remove the decimal point, multiply the numerator and denominator by 10
(0.5)x 10/10= 5/10 = 1/2  where p=1 and q=2.
Hence, 0.5 is a rational number.

4. 0/0
This is not a rational number as q=0 and will not be considered rational.

Positive and Negative Rational Numbers

Rational numbers with a negative sign either in the numerator or denominator are known as negative rational numbers. If they do not have a negative sign, then those are positive rational numbers.

Operations on Rational Numbers

Addition

The addition follows the same rules for fractions, where the denominators of two rational numbers must be the same.

Example:
1/2 + 1/3= (1/2)x 3+ (1/3)x 2 = 3/6 + 2/6= 5/6

Subtraction

Subtraction occurs when one negative rational number is added to a positive rational number or if there is a subtraction sign in the problem.

Example:
1/2 + (-1/3)= 1/2 - 1/3= (1/2)x 3 - (1/3)x 2= 3/6- 2/6= 1/6

Multiplication

Multiplication is done directly where numerators are multiplied, and denominators are also multiplied with each other.

Example:
1/2 x 1/3 = (1x1)/(2x3) = 1/6

Division

The division is done by multiplying the reciprocal of one number with the other.

Example:
1/2  1/3= 1/2 x 3/1 = 3/2

The number that can be represented in the form of p/q (where p and q are integer and q0) is called a rational number. All integers, finite decimals and repetitive decimals are rational numbers. 

Properties of Rational Number

Closure Property: This property states that the addition, subtraction and multiplication of two rational numbers is always a rational number. Closure property is applicable for the division of non-zero rational numbers only (as division by zero is undefined).

Commutative Property: Addition and multiplication of two rational numbers are always commutative while subtraction and division don’t follow the commutative property.

• Addition: (a+b)=(b+a) 
• Multiplication: (ab)=(ba) 

Associative Property: Addition and multiplication of rational numbers are associative.

• Addition: a+(b+c)=(a+b)+c 
• Multiplication: a(bc)=(ab)c|

Distributive Property: Multiplication of rational numbers is distributive over addition.

a(b+c)=(ab)+(ac)

Note: ‘a’ , ‘b’ and ‘c’ are rational numbers.

Identity Property: 0 is the additive identity, and 1 is the multiplicative identity of a rational number.

Inverse Property: If a is a rational number, then -a is its additive inverse and 1/a is its multiplicative inverse.

Apart from the above explained, there are other properties that have been explained below.

1. If x/y is a rational number and m is a non-zero integer then 

x/y=(xm)/(ym) and x/y=(xm)/(ym)

2. If a/b and c/d are two rational numbers

then, a/b=c/d (a*d)=(b*c)

5/7=15/21 (5*21)=(7*15)

3. For any rational number n, only one of the following conditions will hold true.
i) n>0            ii) n=0        iii) n<0

4. For any two rational numbers ‘a’ and ‘b’, exactly one condition will hold true.
i) a>b            ii) a=b        iii)a

5. For three rational number ‘a’, ‘b’ and ‘c’ if a>b and b>c then a>c

The properties of rational numbers have been explained in the Class 8 textbook of mathematics under the chapter Rational Numbers. This topic is essential in higher classes too. 

Students are made familiar with the concept of rational numbers in class 7. Class 8 syllabus also covers the chapter, but it is dealt with in detail Class 9 onwards. This chapter carries about 5-8 marks in the final examination. Understanding rational numbers pave the way to higher-level problems in Maths and Physics and have far-reaching applications.

1. Solve 5/63- (- 6/21)

Solution:

Multiplied by - becomes + and LCM of 63 and 21 is 63.

5/63 + 6/21
= 5/63 + (6/21)x3 
= 5/63 + 18/63 
= 23/63

2. -2(1/3)+ 4(3/5)

Solution: 
Convert mixed fractions to improper fractions. -2(1/3) will become -7/3 and 4(3/5) will be 23/5

-7/3 + 23/5= (-7/3)x5 + (23/5)x 3
= -35/15 + 69/15
= (69-35)/15 
= 34/15

3. Find the value of -(1/8)÷ 3/4

Solution: 
Take reciprocal of the second term
-1/8 x 4/3
= -4/24
= -1/6

4. Using the appropriate property, find (-2/3)x(3/5)+(5/2)-(3/5)x(1/6)

Solution.

.Step 1: Regrouping:(-2/3)x(3/5)-(3/5) x (1/6)+(5/2)
Step 2: Using distributive property:  3/5( -2/3- 1/6) +5/2
= 3/5 x (-5/6)+5/2=2

5. Name the property under multiplication used in each of the following.
i) -4/5 x 1=1 x -4/5 = -4/5    ii) -19/29 x  -29/19= 1
Solution: 

i) Commutative property ( ∵ a x b=b x a)

ii) Multiplicative Inverse Property ( ∵ a x 1/a=1)

6. Multiply 6/13 by the reciprocal of -7/16.
Solution: 
Reciprocal of -7/16 =16/-7= -16/7
∴ 6/13  x -16/7 = -96/91

7. Write the additive inverse of -5/9
Solution: 

Additive inverse of -5/9= -(-5/9)=5/9

8. Write the multiplicative inverse of -13/19
Solution: 
Write the multiplicative inverse of -13/19=19/-13= -19/13

Q: What are the rational numbers?

Q: What is the total number of rational numbers?

Q: How many rational numbers are there between 6 and 7 in the number line?

Q: Is 0 a positive or negative rational number?

Q: When is a rational number said to be in standard form?

Q: Is zero a rational number?

Q: What is the standard form of rational number?

Q: Is every rational number a fraction?

Q: Which rational number is equal to its negative?

Q: Is every rational number a natural number?

Q: Why ℼ is an irrational number?

Q: What is the difference between rational and non-rational numbers?