Question

There are countless rational numbers between 5/6 and 8/9. True or False? Why?

Answer 1

true because
Example: How many rational numbers are there between 1/5 and 4/5?
Solution: You may say there are only two – 2/5, 3/5
But if you rewrite 1/5 as 5/25 and 4/5 as 20/25, then you would have
6/25, 7/25, 8/25, 9/25, 10/25………..10/25.

In all, there are 14 rational numbers and all of them lie between 1/5 and 4/5.
Now if we write 1/5 as 100/500 and 4/5 as 400 500, we would have a new set of rational numbers
occurring between 1/5 and 4/5.
These are 101/500, 102/500, 103/500, 104/500, 105/500,…………..397/500, 398/500, 399/500.
And, you can go on endlessly finding new sets of rational numbers between 1/5 and 4/5/
This method of finding rational numbers can be applied to any two randomly selected rational numbers.
Therefore, it follows that there are countless rational numbers between any two rational numbers.  

Answer 2

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