State whether the following statement is true or false. Give reasons for your answer. Every rational number is a whole number [a] True. [b] False.
ANSWER
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Hint: In order to prove that a statement is incorrect, we have to come up with a counterexample, and in order to prove it correct, we have to come up with a formal proof. Recall the definitions of a rational number. Try finding a counterexample in the above case, i.e. find a rational number which is not a whole number.
Complete step-by-step answer: Rational Numbers: Number which can be expressed in the form of $\dfrac{p}{q}$ where p and q are integers and $q\ne 0$ are called rational numbers. Whole numbers: The numbers 0,1,2,…, are called whole numbers. Consider the number $\dfrac{3}{2}$. Since 2 and 3 are integers and 2 is non-zero, we have $\dfrac{3}{2}$ is a rational number. But $\dfrac{3}{2}$ is not a whole number. Hence there exists a rational number which is not a whole number. Hence the claim that every rational number is a whole number is incorrect. Hence option [b] is correct.
Note: [1] The method of proof, as done above, is called proof by counterexample. [2] Every rational number is not a whole number, but every whole number is a rational number. This is because every whole number is an integer and every integer n can be expressed in the form $\dfrac{n}{1}$ and since n and 1 both are integers and 1 is non-zero, n is a rational number.