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Introduction

Introduction of Rational numbers

A number r is called a rational number, if it can be written in the form p⁄q, where p and q are integers and q≠0. i.e. 0.3796, 4.1333...,4⁄5 etc.

Properties of rational numbers

(1) Closure
(2) Commutativity
(3) Associativity
(4) Additive identity
(5) Multiplicative identity
(6) Additive inverse
(7) Multiplicative inverse or Reciprocal
(8) Distributivity

Closure property consists numbers

Closure property consists three numbers following below :

(i) Whole numbers : In whole numbers there have four operation consists for closure property :-

a. Addition
In addition opertaion, 0 + 3 = 3, 4 + 5 = 9, is a whole number, so we can say that whole numbers are closed in under addition. i.e., a + b
b. Subtraction
In Subtraction opertaion, 4 - 3 = 1 is a whole number, but 4 - 5 = -1, is not a whole number, so we can say that whole numbers are not closed in under Subtraction. i.e., a - b
c. Multiplication
In multiplication opertaion, 3 x 2 = 6, is a whole number, so we can say that whole numbers are closed in under multiplication. i.e., a x b
d. Division
In division opertaion, 4 ÷ 3 = 4/3, is not a whole number, so we can say that whole numbers are not closed in under division. i.e., a ÷ b

(ii) Rational numbers : In rational numbers there have four operation consists for closure property :-

a. Addition
In addition opertaion, 1/2 + 3/5 = (5 + 6)/10 = 11/10, 4/5 + (-5/4) = (16 - 25)/20 = -(9/20), are a rational numbers, so we can say that rational numbers are closed in under addition.
b. Subtraction
In Subtraction opertaion, 4/3 - 3/2 = (8 - 9)/6 = -(1/6) is a rational number, so we can say that rational numbers are closed in under Subtraction.
c. Multiplication
In multiplication opertaion, 3/2 x 2/5 = 6/10, is a rational number, so we can say that rational numbers are closed in under multiplication.
d. Division
In division opertaion, 4/3 ÷ 0 = ∞, is not a rational number, so we can say that rational numbers are not closed in under division.


(iii) Integer numbers : In integer numbers there have four operation consists for closure property :-

a. Addition
In addition opertaion, 4 + (-5) = -1, is an integer number, so we can say that integer numbers are closed in under addition.
b. Subtraction
In Subtraction opertaion, -4 - 3 = -7 is an integer number, so we can say that integer numbers are closed in under Subtraction.
c. Multiplication
In multiplication opertaion, (-3) x 2 = -6, is an integer number, so we can say that integer numbers are closed in under multiplication.
d. Division
In division opertaion, 4 ÷ 5 = 4/5, is not an integer number, so we can say that integer numbers are not closed in under division.

Commutativity property consists numbers

Commutativity property consists three numbers following below :

(i) Whole numbers : In whole numbers there have four operation consists for Commutativity property :-

a. Addition
In addition opertaion, 4 + 3 = 3 + 4, is a whole number, so we can say that whole numbers are commutative in under addition. i.e., a + b = b + a.
b. Subtraction
In Subtraction opertaion, 4 - 3 ≠ 3 - 4, is not a whole number, so we can say that whole numbers are not commutative in under Subtraction. i.e., a - b ≠ b - a
c. Multiplication
In multiplication opertaion, 3 x 2 = 2 x 3, is a whole number, so we can say that whole numbers are commutative in under multiplication. i.e., a x b = b x a
d. Division
In division opertaion, 4 ÷ 3 ≠ 3 ÷ 4, is not a whole number, so we can say that whole numbers are not commutative in under division. i.e., a ÷ b ≠ b ÷ a

(ii) Rational numbers : In rational numbers there have four operation consists for Commutativity property :-

a. Addition
In addition opertaion, 1/2 + 3/5 = 3/5 + 1/2, are a rational numbers, so we can say that rational numbers are commutative in under addition.
b. Subtraction
In Subtraction opertaion, 4/3 - 3/2 ≠ 3/2 - 4/3 is not a rational number, so we can say that rational numbers are not commutative in under Subtraction.
c. Multiplication
In multiplication opertaion, 3/2 x 2/5 = 2/5 x 3/2, is a rational number, so we can say that rational numbers are commutative in under multiplication.
d. Division
In division opertaion, 4/3 ÷ 0 ≠ 0 ÷ 4/3 , is not a rational number, so we can say that rational numbers are not commutative in under division.


(iii) Integer numbers : In integer numbers there have four operation consists for commutativity property :-

a. Addition
In addition opertaion, 4 + (-5) = (-5) + 4, is an integer number, so we can say that integer numbers are commutative in under addition.
b. Subtraction
In Subtraction opertaion, (-4) - 3 ≠ 3 - (-4) is not an integer number, so we can say that integer numbers are not commutative in under Subtraction.
c. Multiplication
In multiplication opertaion, (-3) x 2 = 2 x (-3), is an integer number, so we can say that integer numbers are commutative in under multiplication.
d. Division
In division opertaion, 4 ÷ 2 ≠ 2 ÷ 4, is not an integer number, so we can say that integer numbers are not commutative in under division.

Associativity property consists numbers

Associativity property consists three numbers following below :

(i) Whole numbers : In whole numbers there have four operation consists for Associativity property :-

a. Addition
In addition opertaion, (4 + 3) + 2 = 4 + (3 + 2), is a whole number, so we can say that whole numbers are associative in under addition. i.e., (a + b) + c = a + (b + c).
b. Subtraction
In Subtraction opertaion, (4 - 3) - 1 ≠ 4 - (3 - 1), is not a whole number, so we can say that whole numbers are not associative in under Subtraction. i.e., (a - b) - c ≠ a - (b - c)
c. Multiplication
In multiplication opertaion, (3 x 2) x 4 = 3 x (2 x 4), is a whole number, so we can say that whole numbers are associative in under multiplication. i.e., (a x b) x c = a x (b x c)
d. Division
In division opertaion, (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2), is not a whole number, so we can say that whole numbers are not associative in under division. i.e., (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

(ii) Rational numbers : In rational numbers there have four operation consists for Associativity property :-

a. Addition
In addition opertaion, (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4), are a rational numbers, so we can say that rational numbers are associative in under addition.
b. Subtraction
In Subtraction opertaion, (1/2 - 1/3) - 1/4 ≠ 1/2 - (1/3 - 1/4) is not a rational number, so we can say that rational numbers are not associative in under Subtraction.
c. Multiplication
In multiplication opertaion, (1/2 x 1/3) x 1/4 = 1/2 x (1/3 x 1/4), is a rational number, so we can say that rational numbers are associative in under multiplication.
d. Division
In division opertaion, (1/2 ÷ 1/3) ÷ 1/4 ≠ 1/2 ÷ (1/3 ÷ 1/4), is not a rational number, so we can say that rational numbers are not associative in under division.

(iii) Integer numbers : In integer numbers there have four operation consists for Associativity property :-

a. Addition
In addition opertaion, [(–2) + 3)] + (– 4) = (–2) + [3 + (– 4)], is an integer number, so we can say that integer numbers are associative in under addition.
b. Subtraction
In Subtraction opertaion,(5 – 7) – 3 ≠ 5 – (7 – 3) is not an integer number, so we can say that integer numbers are not associative in under Subtraction.
c. Multiplication
In multiplication opertaion, [(– 4) × (– 8)] × (–5) = (– 4) × [(– 8) × (–5)], is an integer number, so we can say that integer numbers are associative in under multiplication.
d. Division
In division opertaion, [(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)], is not an integer number, so we can say that integer numbers are not associative in under division.

The role of zero (0)

Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
In general,
a + 0 = 0 + a = a, where a is a whole number. e.g., 2 + 0 = 0 + 2 = 2.
b + 0 = 0 + b = b, where b is an integer. e.g., (-2) + 0 = 0 + (-2) = -2.
c + 0 = 0 + c = c, where c is a rational number. e.g., 1/2 + 0 = 0 + 1/2 = 1/2.

The role of one (1)

1 is the multiplicative identity for rational numbers.
In general,
a x 1 = 1 x a = a, where a is a whole number. e.g., 2 x 1 = 1 x 2 = 2.
b x 1 = 1 x b = b, where b is an integer. e.g., (-2) x 1 = 1 x (-2) = -2.
c x 1 = 1 x c = c, where c is a rational number. e.g., 1/2 x 1 = 1 x 1/2 = 1/2.

Negative of a number (additive inverse)

Here, -(a/b) is the additive inverse of a/b and a/b is the additive inverse of -(a/b) or vice-versa.
In general, for an integer a, we have, a + (–a) = (–a) + a = 0; so, a is the negative of –a and –a is the negative of a. e.g., 2 + (-2) = (-2) + 2 = 0.
In general, for a rational number a/b, we have a/b + (-a/b) = (-a/b) + a/b = 0. e.g., 1/2 + (-1/2) = (-1/2) + 1/2 = 0

Reciprocal (multiplicative inverse)

p/q is the multiplicative inverse or reciprocal of another rational numbers a/b if a/b x p/q = 1. And zero (0) has no any reciprocal values.
In general,
e.g., 4/5 x 5/4 = 1 is a reciprocal values.

Distributivity of multiplication

For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac.
For addition, e.g., 1/2 x (1/3 + 1/4) = (1/2 x 1/3) + (1/2 x 1/4) = 1/6 + 1/8 = 7/24.
For Subtraction, e.g., 1/2 x (1/3 - 1/4) = (1/2 x 1/3) - (1/2 x 1/4) = 1/6 - 1/8 = 1/24.

Rational Numbers between Two Rational Numbers

Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.
If a and b are two rational numbers, then (a + b)/2 is a rational number between a and b such that a < (a + b)/2 < b.

Representation of number line

We have learnt to represent natural numbers, whole numbers, integers and rational numbers on a number line

Natural number

The line extends indefinitely only to the right side of 1.
number line

Whole number

The line extends indefinitely to the right, but from 0. There are no numbers to the left of 0.
number line

Integer number

The line extends indefinitely on both sides. Do you see any numbers between –1, 0; 0, 1 etc.
number line

Rational number

The line extends indefinitely on both sides. But you can now see numbers between –1, 0; 0, 1 etc.
number line

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