Class 8- Maths Rational numbers on a number line

Rational numbers on a number line

Rational numbers representation on number line: Example: We represent on the number line as under -

The point P represents the rational number  

The point P’ represents the rational number 

Similarly, we represent 3.5 and – 3.5.

All rational numbers can be represented on a number line just the way we represent integers on number line.

We already know that rational numbers are in between two successive integers i.e. between two successive positive or negative integers.

In other words, Rational number = ‘Integer + fraction’ i.e. a ‘positive or negative whole number’ plus a fraction makes a rational number.

Rational numbers can be represented on a number line by

1.    identifying the ‘sign’ of the rational number

2.    placing the ‘integer part of the rational number’ on the number line in a way that

a.    the placement is on the left of the origin (‘0’) if the sign is negative

b.    the placement is on the right of the origin if the sign is positive

3.    placing the fraction (which is less than one) ahead of the integer 

4.    The point of fraction (ahead of the integer part) represents the rational number on the number line

Comparison of rational numbers

Here are some facts about rational numbers which help in comparing them:

(i) Every positive rational number is greater than 0.

(ii) Every negative rational number is less than 0.

(iii) Every positive rational number is greater than every negative rational number.

(iv) Every rational number represented by a point on the number line is greater than every rational number represented by points on its left.

(v) Every rational number represented by a point on the number line is less than every rational number represented by points on its right.

Specifically, to compare any two rational numbers, we use the following steps:

As the saying goes, you cannot compare apples and oranges; every comparison requires something in common between the compared things.

In the case of rational numbers, between the denominators and the numerators of the numbers to be compared, comparison is easily possible if the denominators are ‘common’ or same and if they are not we try to make them same by finding their LCM.  

Specifically, the steps in comparing two rational numbers are -

1: Rewrite the given rational numbers such that their denominators are positive.

If a number has a negative denominator than make it positive by shifting the negative sign to the numerator. For e.g.

If two numbers are; (2 / - 7) and (3 / - 5)

Then we may write them as (-2 / 7) and (- 3 /5)

Step 2: Find the LCM of the positive denominators of the rational numbers obtained in Step 1.

From the above example, Now let’s take the LCM of 7 and 5 i.e. 35.

Step 3: Express each rational number in terms of the LCM (obtained in step 2) as the common denominator.

Again from the above example,

To get the common denominator (i.e. 35 in this example), we multiply numerator and denominators of -2/7 and -3/5 from 5 and 7 respectively.

Therefore, we get

-10/ 35 and -21/35

Step 4: Compare the numerator of rational numbers obtained in step 3. The number having the greater numerator is the greater rational number.  

Now by comparing the both numerators (-10 and -21), we can say that -10 is greater than – 21 (due to negative sign).

Insertion of Rational Numbers between two given Rational numbers

The mean (average) of any two rational numbers will always be rational and will be between them.

(1/14 + 3/10)/2

= (5/70 + 21/70)/2

= (26/70)/2

= 13/70

However, you must know that there are an infinite number of rational numbers between any two given rational numbers.