RATIONAL NUMBERS - DEFINITION, BASIC OPERATIONS, PROPERTIES SUCH AS CLOSURE ETC.

Please study  Integers before Rational Numbers, if you have not already done so.

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

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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.

As (7⁄4) > (7⁄5), we have (-7⁄4) < (-7⁄5);

Operations in the set Q

For the four basic operations
in the set Q, go to

Adding Fractions

Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Properties of Addition in the set Q

Closure Property of Addition :

Observe the following Examples :

(+2⁄3) + (+1⁄6) = (+5⁄6) ∈ Q;
(-1⁄8) + (-1⁄7) = -(1⁄8 + 1⁄7) = -15⁄56 ∈ Q;
(+1⁄2) + (-1⁄8) = +(+1⁄2 - 1⁄8) = +3⁄8 = 3⁄8 ∈ Q;
(- 13⁄6) + (+9⁄7) = - (13⁄6 - 9⁄7) = -37⁄42 ∈ 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 elements of Q a⁄b and c⁄d,
(a⁄b + c⁄d) is also a Rational Number.
This is called the Closure Property
of Addition in the set Q.

Commutative Property of Addition :

Look at the following Examples.

(+1⁄3) + (+1⁄6) = (+1⁄6) + (+1⁄3)
(-3⁄8) + (-4⁄7) = (-4⁄7) + (-3⁄8)
(+7⁄12) + (-1⁄9) = (-1⁄9) + (+7⁄12)
(- 1⁄136) + (+5⁄96) = (+5⁄96) + (- 1⁄136)

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 + c⁄d = c⁄d + a⁄b
This is called the Commutative Property
of Addition in the set Q.

Associative Property of Addition :

Look at the following Examples.

(1⁄5 + 2⁄7) + (-1⁄8) = 1⁄5 + {2⁄7 + (-1⁄8)} = 61⁄280;
(-3⁄2 + 1⁄6) + 1⁄5 = -3⁄2 + (1⁄6 + 1⁄5) = -34⁄30 = -17⁄15

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.

Additive Identity :

Look at the following Examples.

1⁄9 + 0 = 0 + 1⁄9 = 1⁄9 ; -13⁄45 + 0 = 0 + (-13⁄45) = -13⁄45

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.

Properties of Subtraction in the set Q

Closure Property is true for Subtraction in the set Q :

Observe the following Examples :

1⁄3 - 1⁄2 = -1⁄6 ∉ Q;
2⁄3 - 1⁄4 = 5⁄12 ∉ 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 - c⁄d) is also a rational number.
This is called the Closure Property of Subtraction
in the set Q.

Subtraction is not Commutative :

Observe the following Examples :

2⁄7 - 1⁄4 ≠ 1⁄4 - 2⁄7;
7⁄8 - 6⁄7 ≠ 6⁄7 - 7⁄8

So, we can say Subtraction is not Commutative.

Subtraction is not Commutative in the set Q.

Subtraction is not Associative :

Observe the following Examples :

(1⁄8 - 1⁄5) - 1⁄2 ≠ 1⁄8 - (1⁄5 - 1⁄2);
(7⁄8 - 6⁄7) - 1⁄2 ≠ 7⁄8 - (6⁄7 - 1⁄2)

So, we can say Subtraction is not Associative.

Subtraction is not Associative in the set Q.

Properties of Multiplication in the set Q

Closure Property of Multiplication :

Observe the following Examples :

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 Property of 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 :

Observe the following Examples :

1⁄3 ÷ 7⁄8 = 1⁄3 x 8⁄7 = 8⁄21 ∈ Q; -7⁄3 ÷ 2⁄9 = -7⁄3 x 9⁄2 = -21⁄2 ∈ Q; -1⁄8 ÷ -4⁄3 = -1⁄8 x -3⁄4 = 3⁄32 ∈ 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) ÷ (c⁄d)] is also a rational number.
This is called the Closure Property
of Division in the set Q.

Division is not Commutative :

Observe the following Examples :

1⁄8 ÷ 3⁄4 ≠ 3⁄4 ÷ 1⁄8

11⁄4 ÷ -1⁄9 ≠ -1⁄9 ÷ 11⁄4

So, we can say Division is not Commutative.

Division is not Commutative in the set Q.

Division is not Associative :

Observe the following Examples :

(1⁄8 ÷ 1⁄4) ÷ 1⁄2 ≠ 1⁄8 ÷ (1⁄4 ÷ 1⁄2);
(5⁄4 ÷ -1⁄6) ÷ 1⁄3 ≠ 5⁄4 ÷ (-1⁄6 ÷ 1⁄3)

So, we can say Division is not Associative.

Division is not Associative in rational numbers.

Division with 1 of any number gives that number :

Look at the following Examples.

-1⁄9 ÷ 1 = 1⁄9 ; 97⁄76 ÷ 1 = 97⁄76

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.

Representation as terminating or
non-terminating but repeating decimals.

Every rational number when expressed in
decimal form is expressible either in
terminating or in repeating decimals.

Also,

every terminating or repeating decimal
is a rational number.

For study of terminating and repeating decimals, go to

Repeating Decimal.