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Numbers

Look at this number line. How many integers are there between (i) -5 and 4 and (ii) -4 and -3?

 


(i) There are 8 integers.
-4, -3, -2, -1, 0, 1, 2, 3

(ii) -4 and -3 are successive integers, so there will be no integers between them.

The number of integers that lie between two integers is such that it depends on the given integers.

To find the number of rational numbers between two rational numbers:

Method 1:-
Consider the rational numbers . Let us try to find the number of rational numbers between them.

 


Consider the rational numbers . Definitely these numbers lie between .

Now let us consider the next set of rational numbers . These numbers also lie between .
 

is yet another set of rational numbers which lie between .

Like this we can find infinite number of rational numbers that lie between

Thus there are infinite number of rational numbers between any two rational numbers.
Method 2 :-
Consider the rational numbers . Find the average for these two rational numbers which is  
-1 lies in the middle of and it is also a rational number. 

Similarly the average of -1 and is and the average of - 1 and is . Proceeding this way we find that there are infinite number of rational numbers between .

Thus there are infinite number of rational numbers between any two rational numbers.





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