Question

How many rational numbers are there between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ ?

Answer 1

Hint: Rational numbers are the numbers which can be written in the form of $ \dfrac{p}{q} $ , where $ p $ and $ q $ are integers and $ q \ne 0 $ . An integer is also a rational number. A rational number added to a rational number always yields a rational number. To find a rational number between two rational numbers we can multiply integers in numerator or denominator to get another rational number. We can use these properties to find other rational numbers between two rational numbers.

Complete step by step solution:
We are given to find how many rational numbers exist between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ .
We can observe that both the numbers given are rational numbers as both can be written in the form of $ \dfrac{p}{q} $ , where $ p $ and $ q $ are integers and $ q \ne 0 $ .
First we try to make the denominator the same for both the numbers. Multiplying a same rational number in the numerator and denominator does not change the value of the rational number.
Thus we can write,
 $ - \dfrac{1}{{10000}} = - \dfrac{{1 \times 2}}{{10000 \times 2}} = - \dfrac{2}{{20000}} $
First we can find the average of these two given numbers as,
 $ \dfrac{{\left( { - \dfrac{2}{{20000}}} \right) + \left( { - \dfrac{1}{{20000}}} \right)}}{2} = - \dfrac{3}{{20000 \times 2}} = - \dfrac{3}{{40000}} $
This is a rational number lying between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ .
Similarly we can find an average of $ - \dfrac{3}{{40000}} $ and $ - \dfrac{1}{{10000}} $ which will lie between them. And an average of $ - \dfrac{3}{{40000}} $ and $ - \dfrac{1}{{20000}} $ which will lie between them.
We can keep doing this exercise many times to get as many rational numbers as we want. All these numbers will lie between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ .
Actually we can get an infinite number of rational numbers between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ .
Hence, there exists an infinite number of rational numbers between $ - \dfrac{1}{{10000}} $ and $ - \dfrac{1}{{20000}} $ .

Note: To find the rational numbers between two given rational numbers we used the method to find the average which always lies between the given numbers. We can use any other method to find rational numbers like increments in the numerator to reach one number from the other when denominators are the same.