Rational Numbers

1.1 INTRODUCTION

The word ‘rational’ is derived from the word ‘ratio’.

A rational number is any number that can be named in the form ab where a and b are integers and b 0.

Thus, each of the numbers 56,611,139,617 is a rational number.

1. Every natural number is a rational number but a rational number need not be a natural number.

We can write 1=11, 2=21, 3=31 and so on.

This shown that every natural number n can be written as n1 which is a rational number.

But none of the rational numbers like 56,38,13 , etc., is a natural number.

2. Zero is a rational number.

We can write 0 in anyone of the forms 01,01,02,02,03,03 and so on. Thus, 0 can be expressed as pq , where p = 0 and q is any non-zero integer. Hence, 0 is a rational number.

3. Every integer is a rational number but a rational number need not be an integer.

We know that

1=11, 2=21, 3=31, 1=11, 2=21, 3=31

and so on.

In general, any integer n can be written as n=n1 , which is a rational number.

But rational numbers like 57,78,116 are not integers.

4. Every fraction is a rational number but a rational number need not be a fraction.

Let p/q by any fraction. Then p and q are natural numbers. Since every natural number is an integer, therefore, p and q are integers.

Thus, the fraction p/q is the quotient of two integers such that q 0 .

Hence, p/q is a rational number.

5. There are infinitely many rational numbers between two rational numbers.

Numerator and Denominator
Let pq (q0) be a rational number. It has two terms. One is p above the line ‘____’ and the other is q below the line.
p is called the numerator of the rational number and q is called the denominator.

Positive and Negative Rational Numbers
1. A rational number is said to be positive if its numerator and denominator are either both positive or both negative.
For example, 57,2918,619,2783 are all positive rational numbers.
2. A rational number is said to be negative if its numerator and denominator are such that one of them is a positive integer and the other is a negative integer.
For example, Each of the numbers 79,2859,3711,56217 is a negative rational numbers.
Note:
1. Every negative integer is a negative rational number.
Example., –1, –2, –3 and so on, which may be expressed as 1=11,2=21,3=31. are all negative rational numbers.
2. The rational number 0 is neither positive nor negative.
Equivalent rational numbers
If pq is a rational number and m is a non-zero integer, then pq=p×mq×m

Reducing to a simpler form
If pq is a rational number and m is a common divisor of p and q then pq=p÷mq÷m.

1.2 RATIONAL NUMBERS IN STANDARD FORM
A rational number is said to be in standard form if its denominator is positive and it is in the lowest terms.
To express a given rational number in standard form, proceed as under:
Step1. Make the denominator of the given rational number positive.
Step2. Divide both the numerator and the denominator by their HCF.
Example 3: Find which of the following rational number is in the standard form
i 912   ii 34   iii 24   iv 48
Solution
A rational number is said to be in standard form if its denominator is positive and it is in the lowest terms.
i 912=34     (ii) 34 is in standard form     (iii) 24=12     (iv) 48=12

1.3 PROPERTIES OF RATIONAL NUMBER
If a, b, c are rational numbers, then
1. Closure
Addition: a + b is rational.
Subtraction: a – b is rational.
Multiplication: a × b is rational.
Division: a ÷ b need not be rational.
2. Commutativity
Addition: a + b = b + a
Subtraction: a + b ≠ b + a
Multiplication: a × b = b × a
Division: a ÷ b ≠ b ÷ a
3. Associativity
Addition: a + (b + c) = (a + b) + c
Subtraction: a – (b – c) ≠ (a – b) – c
Multiplication: a × (b × c) = (a × b) × c
Division: a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
Addition on rational numbers satisfies the closure property, the commutative law and the associative law.
Zero is the identity element for the addition of rational numbers.
Every rational number pq has its additive inverse qp.
Multiplication on rational numbers satisfies the closure property, the commutative law, the associative law and the distributive law over addition.
One is the multiplicative identity for rational numbers.
Every nonzero rational number pq has its multiplicative inverse qp.
Zero does not have its multiplicative inverse.

4. Order properties of rational numbers
Property 1
For each rational number x, one and only one of the following is true.
(i) x > 0 (ii) x = 0 (iii) x < 0
Property 2
For any two rational numbers x and y, one and only one of the following is true.
(i) x > y (ii) x = y (iii) x < y
Property 3
If x, y, z be any three rational numbers such that x > y and y > z; then x > z.

5. The role of zero (0)
Look at the following.
2 + 0 = 0 + 2 = 2
(Addition of 0 to a whole number)
– 5 + 0 =… +… = – 5
(Addition of 0 to an integer)
27+=027=27
(Addition of 0 to a rational number)
You have done such additions earlier also. Do a few more such additions.
What do you observe? You will find that when you add 0 to a whole number, the sum is again that whole number. This happens for integers and rational numbers also.
In general, a + 0 = 0 + a = a, where a is a whole number
b + 0 = 0 + b = b, where b is an integer
c + 0 = 0 + c = c, where c is a rational number
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.

6. The role of 1
We have, 5 × 1 = 5 = 1 × 5     (Multiplication of 1 with a whole number)
27×1=×=27
38×=1×38=38
What do you find?
You will find that when you multiply any rational number with 1, you get back that rational number as the product. Check this for a few more rational numbers. You will find that, a × 1 = 1 × a = a for any rational number a.

7. Additive Inverse (Negative of a number)
While studying integers you have come across negatives of integers. What is the negative of 1? It is – 1 because 1 + (– 1) = (–1) + 1 = 0
So, what will be the negative of (–1)? It will be 1.
Also, 2 + (–2) = (–2) + 2 = 0, so we say 2 is the negative or additive inverse of –2 and 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.
For the rational number 23, we have,
23+23=2+(2)3=0
Also, 23+23=0
Similarly, 89+=+89=0+117=117+=0
In general, for a rational number ab,
we have, ab+ab=ab+ab=0
We say that ab is the additive inverse of ab and ab is the additive inverse of ab.

8. Multiplicative Inverse (Reciprocal)
By which rational number would you multiply 821, to get the product 1? Obviously by 218, since
821×218=1 .
Similarly, 57 must be multiplied by 75 so as to get the product 1.
We say that 218 is the reciprocal of 821 and 75 is the reciprocal of 57.
Can you say what is the reciprocal of 0 (zero)?
Is there a rational number which when multiplied by 0 gives 1? Thus, zero has no reciprocal.
We say that a rational number cd is called the reciprocal or multiplicative inverse of another rational number
ab if ab×cd=1 .

9. Distributivity
a × (b +c) = ab + ac
a × (b – c) = a × b – a ×c

1.4 RATIONAL NUMBERS ON NUMBER LINE
You have learnt how to represent integers on a number line.
If you draw any line as shown below, take a point O on it which you may call the zero point, set off equal distances on both sides of O on the line, then these distances will be considered as of unit length.
If we name the points on the right as A, B, C, D, E, … and the corresponding points on the left as A’,B’,C’,D’, E’ ,…
Then A, B, C, D, E, …. will represent the points 1, 2, 3, 4, 5, …. and A’,B’,C’,D’, E’ , …. Will represent the points –1, –2, –3, –4, –5….

In the same manner as done, we can represent rational numbers on a number line and obtain a rational number line.
(i) If we bisect OA, we get the point P which represent the rational number 12. Similarly, if we bisect OA’ , we get the point P’ which represents the rational number 12

(ii) If we divide the lengths OA and OA’ into three equal parts and label the points of division as P,Q,P’,Q’ as shown,

then P, Q will represent the rational numbers 13 and  23 respectively. Likewise, points P’,Q’ will represent the rational numbers 13 and 23 respectively.

1.5 OPERATIONS ON RATIONAL NUMBERS
You know how to add, subtract, multiply and divide integers as well as fractions. Let us now study these basic
operations on rational numbers.
Addition and Subtraction
Let us add two rational numbers with same denominators, say 73 and 53. We find 73+53
On the number line, we have:

The distance between two consecutive points is 13. So adding 53 to 73 will mean, moving to the left of 73, making 5 jumps. Where do we reach? We reach at 23. So, 73+53=23. Similarly, subtraction can be done.
Multiplication
Let us multiply the rational number 35 by 2, i.e., we find 35×2.
On the number line, it will mean two jumps of 35 to the left.

Where do we reach? We reach at 65. Let us find it as we did in fractions.
35×2=3×25=65
Division
We have studied reciprocals of a fraction earlier. What is the reciprocal of 27? It will be 72. We extend this idea of reciprocals to rational numbers also.

The reciprocal of 27 will be 72 i.e., 72; that of 35 would be 53.
Dividing a rational number 49 by another rational number 57 as, 49÷57=49×75=2845.

1.6 COMPARISON OF RATIONAL NUMBERS
There are two ways to compare Rational numbers:
(i) Number Line (ii) Arithmetical Process
Number Line: One can compare rational numbers by using a number line easily. Arrange the rational numbers on the number line in ascending order from left to right.
Example: The Rational numbers having 4 as denominator can be represented on the number line as follows. If the two rational numbers represented on the number line, the number on the left is smaller than the number to its right,
Thus

34<74,34<74 and 74<54,34>54.

We observe from the above example that:
• A positive rational number is always greater than a negative rational number.
• Zero is greater than each one of the negative rational numbers and less than each one of the positive rational numbers.
Arithmetical Process
(i) Express each rational number with a positive denominator.
(ii) Find L.C.M of the positive denominators.
(iii) Express each of the given rational numbers with L.C.M as the common denominator.

Thus, 25,43

25=25×33,43=43×55

615,2015

615<2015

(iv) The number having greater numerator is greater.
(v) A positive rational number is always greater than a negative rational number
(vi) Zero is greater than each one of the negative rational numbers and less than each one of the positive rational numbers.

1.7 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS
Can you tell the natural numbers between 1 and 5? They are 2, 3 and 4.
How many natural numbers are there between 7 and 9? There is one and it is 8.
How many natural numbers are there between 10 and 11? Obviously none.
List the integers that lie between –5 and 4. They are – 4, – 3, –2, –1, 0, 1, 2, 3.
How many integers are there between –1 and 1?
How many integers are there between –9 and –10?
You will find a definite number of natural numbers (integers) between two natural numbers (integers).
Example: How many rational numbers are there between 310 and 710?
Solution
You may have thought that they are only 410,510 and 610.
But you can also write 310 as 30100 and 710 as 70100.
Now the numbers, 31100,32100,33100,.68100,69100, are all between 310 and 710.
The number of these rational numbers is 39.
Also 310 can be expressed as 300010000 and 710 as 700010000.
Now, we see that the rational numbers

300110000,300210000,,699810000,699910000 are between 310 and 710. These are 3999 numbers in all.
In this way, we can go on inserting more and more rational numbers between 310 and 710. So unlike natural numbers and integers, the number of rational numbers between two rational numbers is not definite.

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