There, we covered the basic operations such as Addition, Subtraction, Multiplication and Division and their properties such as Closure, Commutativity, Associativity, Distributivity etc.

Need for Extension of Set of Integers

We have seen that closure property is not true for Division in the set of Integers, with examples 8 ÷ 3 = ? ∉ Z; 36 ÷ -5 = ? ∉ Z;

i.e. when an integer is divided by another integer which is not a factor of it, we do not get an integer.

In other words, the quotient of two integers need not always be an integer.

This shows the need to extend the set of all integers to include such quotients.

We write 8 ÷ 3 in the form 8⁄3 and 36 ÷ -5 in the form -36⁄5.

We can say that these are ratios of two integers.

Now, we give formal definition to such quotients using the symbolism given above.

Definition

A number of the form a⁄b, where a and b are integers, b≠0 is called a Rational Number.

The Rational Number a⁄b where b = 1, ( i.e. the Rational Number a⁄1) is same as the integer a.

Thus,

The Rational Numbers include all the Integers.

We denote the set of Rational Numbers by Q.

Since every rational number is the quotient of two integers, it is quite reasonable to denote the set by Q, the first letter in the word quotient.

You might have noticed by now, that every Fraction is a rationmal number.

Comparison of the elements of the set Q

From the knowledge of
Comparing Fractions, we can compare positive elements of Q and know which is bigger or which is smaller.

For example, (i) (2⁄3) > (1⁄3) ; (ii) (7⁄4) > (7⁄5) ; (iii) 5⁄6 < 11⁄13.

In addition to that knowledge, the following points will help in deciding which is bigger or smaller.

0 is less than every positive rational number.

0 is greater than every negative rational number.

Every negative rational number is less than every positive rational number.

The greater is the rational number, the lesser is its negative.

What is seen to be true for in these examples, is true for any three elements of Q. So, we can say :

For any three Rational Numbers a⁄b, c⁄d and e⁄f, a⁄b + (c⁄d + e⁄f) = (a⁄b + c⁄d) + e⁄f This is called the Associative Property of Addition in the set Q.

What is seen to be true for these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, a⁄b + 0 = 0 + a⁄b = a⁄b 0 is called the Additive Identity in the set Q.

Additive Inverse :

We have, 1⁄9 + (-1⁄9) = 0, 2⁄3 + (-2⁄3) = 0, 3⁄2 + (-3⁄2) = 0 and so on.

What is seen to be true for these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, we have a⁄b + (-a⁄b) = 0. We call a⁄b and -a⁄b as the additive inverse of each other.

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1⁄3 x 7⁄8 = 7⁄24 ∈ Q; -7⁄3 x 2⁄9 = -14⁄27 ∈ Q; -1⁄8 x -4⁄3 = 1⁄6 ∈ Q;

What is seen to be true for these examples, is true for any two elements of Q. So, we can say :

For any two rational numbers a⁄b and c⁄d, [(a⁄b) x (c⁄d)] is also a rational number. This is called the Closure Property of Multiplication in the set Q.

Commutative Property of Multiplication :

Look at the following Examples.

1⁄10 x 1⁄8 = 1⁄8 x 1⁄10 = 1⁄80; -2⁄5 x 1⁄4 = 1⁄4 x -2⁄5 = -1⁄10; -11⁄9 x -5⁄6 = -5⁄6 x -11⁄9 = 55⁄54

What is seen to be true for in these examples, is true for any two elements of Q. So, we can say :

For any two rational numbers a⁄b and c⁄d, a⁄b x c⁄d = c⁄d x a⁄b This is called the Commutative Property of Multiplication in the set Q.

Associative Property of Multiplication :

Look at the following Examples.

(4⁄5 x -2⁄7) x 3⁄2 = 4⁄5 x (-2⁄7 x 3⁄2) = -12⁄35; (-1⁄6 x -9⁄2) x 2⁄3 = -1⁄6 x (-9⁄2 x 2⁄3) = 1⁄2

What is seen to be true for in these examples, is true for any three elements of Q. So, we can say :

For any three Rational Numbers a⁄b, c⁄d and e⁄f, a⁄b x (c⁄d x e⁄f) = (a⁄b x c⁄d) x e⁄f This is called the Associative Property of Multiplication in the set Q.

Multiplicative Identity :

Look at the following Examples.

4⁄7 x 1 = 1 x 4⁄7 = 4⁄7 ; -98⁄75 x 1 = 1 x -98⁄75 = -98⁄75

What is seen to be true for these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, a⁄b x 1 = 1 x a⁄b = a⁄b 1 is called the Multiplicative Identity in the set Q.

Zero Property of Multiplication :

Look at the following Examples.

-23⁄4 x 0 = 0 x -23⁄4 = 0 ; 56⁄47 x 0 = 0 x 56⁄47 = 0

What is seen to be true for these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, a⁄b x 0 = 0 x a⁄b = 0. This is called the Zero Propertyof Multiplication in the set Q.

Also,

For any two elements of the set Q, a⁄b and c⁄d, If a⁄b x c⁄d = 0, then either a = 0 or c = 0 or a = c = 0

Distributive Property of Multiplication over Addition :

Look at the following Example.

1⁄5 x (-3⁄4 + 2⁄3) = 1⁄5 x -3⁄4 + 1⁄5 x 2⁄3 = -3⁄20 + 2⁄15 = -1⁄60 same as 1⁄5 x -1⁄12 = -1⁄60;

-2 x (-7 + 8) = -2 x -7 + (-2) x 8 = 14 - 16 = -2 same as -2 x 1;

What is seen to be true for this example, is true for any three elements of Q. So, we can say :

For any three Rational Numbers a⁄b, c⁄d and e⁄f, a⁄b x (c⁄d + e⁄f) = (a⁄b x c⁄d) + (a⁄b x e⁄f) This is called the Distributive Property of Multiplication over Addition.

Properties of Division in the set Q

Closure Property is true for Division in the set Q :

What is seen to be true for in these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, a⁄b ÷ 1 = a⁄b

Division by zero is not defined :

-1⁄9 ÷ 0 = ? Is it possible to assign any whole number to this quotient ? Suppose, if possible, -1⁄9 ÷ 0 = a⁄b where a⁄b is a rational number. Since division is the inverse process of multiplication, we have a⁄b x 0 = -1⁄9. But, any whole number x 0 = 0 [see zero property of multiplication above.] So a⁄b x 0 = 0 and not -1⁄9. Thus there is no element in Q which when multiplied by zero gives a non zero value.

Therefore -1⁄9 ÷ 0 can not be equal to any integer.

Thus division by zero is not defined.

Division by zero is not defined.

Division of zero with any number gives zero :

Look at the following Examples.

0 ÷ (-3⁄4) = 0 ; 0 ÷ (56⁄17) = 0

What is seen to be true for in these examples, is true for any element of Q. So, we can say :

For every rational number a⁄b, 0 ÷ a⁄b = 0

Denseness in the set Q

Between any two rational numbers (however close they may be) there is an infinite set of rational numbers.

This important property is called property of density. In other words, the set Q is dense.

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