Rational numbers

1.Rational Numbers

Rational Numbers

A number is called Rational if it can be expressed in the form p/q where p and q are integers (q > 0). It includes all natural, whole number and integers.

Example: 1/2, 4/3, 5/7,1 etc. Rational Numbers

Natural Numbers

All the positive integers from 1, 2, 3,……, ∞.

Whole Numbers

All the natural numbers including zero are called Whole Numbers.

Integers

All negative and positive numbers including zero are called Integers.

Properties of Rational Numbers

1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a. Whole Numbers

If p and q are two whole numbers then

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + q will also be the whole number.	p – q will not always be a whole number.	pq will also be the whole number.	p ÷ q will not always be a whole number.
Example	6 + 0 = 6	8 – 10 = – 2	3 × 5 = 15	3 ÷ 5 = 3/5
Closed or Not	Closed	Not closed	Closed	Not closed

b. Integers If p and q are two integers then

Operation	Addition	Subtraction	Multiplication	Division
Integers	p+q will also be an integer.	p-q will also be an integer.	pq will also be an integer.	p ÷ q will not always be an integer.
Example	- 3 + 2 = – 1	5 – 7 = – 2	- 5 × 8 = – 40	- 5 ÷ 7 = – 5/7
Closed or not	Closed	Closed	Closed	Not closed

c. Rational Numbers

If p and q are two rational numbers then

Operation	Addition	Subtraction	Multiplication	Division
Rational Numbers	p + q will also be a rational number.	p – q will also be a rational number.	pq will also be a rational number.	p ÷ q will not always be a rational number
Example				p ÷ 0 = not defined
Closed or Not	Closed	Closed	Closed	Not closed

2. Commutative Property

This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.

a. Whole Numbers

If p and q are two whole numbers then

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + q = q + p	p – q ≠ q – p	p × q = q × p	p ÷ q ≠ q ÷ p
Example	3 + 2 = 2 + 3	8 –10 ≠ 10 – 8 – 2 ≠ 2	3 × 5 = 5 × 3	3 ÷ 5 ≠ 5 ÷ 3
Commutative	yes	No	yes	No

3. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a. Whole Numbers

If p, q and r are three whole numbers then

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + (q + r) = (p + q) + r	p – (q – r) = (p – q) – r	p × (q × r) = (p × q) × r	p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r
Example	3 + (2 + 5) = (3 + 2) + 5	8 – (10 – 2) ≠ (8 -10) – 2	3 × (5 × 2) = (3 × 5) × 2	10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1
Associative	yes	No	yes	No

b. Integers

If p, q and r are three integers then

Operation	Integers	Example	Associative
Addition	p + (q + r) = (p + q) + r	(– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5)	Yes
Subtraction	p – (q – r) = (p – q) – r	5 – (7 – 3) ≠ (5 – 7) – 3	No
Multiplication	p × (q × r) = (p × q) × r	(– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5)	Yes
Division	p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r	[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)]	No

c. Rational Numbers

If p, q and r are three rational numbers then

Operation	Integers	Example	Associative
Addition	p + (q + r) = (p + q) + r		yes
Subtraction	p – (q – r) = (p – q) – r		No
Multiplication	p × (q × r) = (p × q) × r		yes
Division	p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r		No

The Role of Zero in Numbers (Additive Identity)

Zero is the additive identity for whole numbers, integers and rational numbers.

	Identity		Example
Whole number	a + 0 = 0 + a = a	Addition of zero to whole number	2 + 0 = 0 + 2 = 2
Integer	b + 0 = 0 + b = b	Addition of zero to an integer	False
Rational number	c + 0 = 0 + c = c	Addition of zero to a rational number	2/5 + 0 = 0 + 2/5 = 2/5