Question

Give one example for the fact that a number which is rational but not an integer.

Answer 1

Hint: First, before proceeding for this, we must know the condition that the rational number is always an integer is not necessary but an integer can be written as a rational number always. Then, rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ and an integer is the number that has an integer value and always comes in total range of number from $ -\infty $ to $ \infty $ .Then, the example considered by us as $ \dfrac{3}{4} $ is not an integer but still a rational number.

Complete step-by-step answer:
In this question, we are supposed to find a number which is rational but not an integer.
So, before proceeding for this, we must know the condition that the rational number is always an integer is not necessary but an integer can be written as a rational number always.
Here, we are said to give an example of the rational number which is not an integer.
But, before that, we all should know what is a rational number.
So, the rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ .
However, an integer is the number that has an integer value and always comes in the total range of numbers from $ -\infty $ to $ \infty $ .
Now, to take an example of any rational number which is not an integer is:
 $ \dfrac{3}{4} $
Here, the example considered by us as $ \dfrac{3}{4} $ is not an integer but still a rational number.
Now, to prove that it is not an integer, find the decimal value of the number considered as:
 $ \dfrac{3}{4}=0.75 $
So, it gives the value 0.75 and by the definition of integers, it is not an integer value.
But when we go for the definition of rational number, $ \dfrac{3}{4} $ is of form $ \dfrac{p}{q} $ and also its denominator is not zero which states that it is a valid rational number.
Similarly, we can take many numbers which are rational numbers but not integers.
Hence, it is proved that $ \dfrac{3}{4} $ is a rational number but not an integer.

Note: Now, to solve these types of questions, it was not necessary to take this fixed example as $ \dfrac{3}{4} $ to prove that a rational number is not necessarily an integer. So, in mathematics of the numbers, we have a number of rational numbers like $ \dfrac{1}{2},\dfrac{6}{8},\dfrac{1}{5} $ and much more that are not integers. So we can take any rational number whose fraction doesn’t give an integer value to prove this statement.