Question

What is the difference between real numbers and rational numbers?

Answer 1

Hint: First we will define terms real numbers and rational numbers one by one with the help of some examples. Now, we will check if all the rational numbers are real or not and similarly if all the real numbers can be included in the set of rational numbers or not.

Complete step by step answer:
Here we have been asked the differences between real numbers and rational numbers but first we need to know about these terms. Let us check their definitions one by one to detect the differences.
(1) Rational numbers: A number of the form $\dfrac{p}{q}$ where p and q are integers and q is not equal to 0 is called a rational number. For example: $\dfrac{1}{2},\dfrac{5}{11},\dfrac{9}{18}$ etc. All the whole numbers are also rational numbers as they can be written in the form $\dfrac{p}{q}$ where q will be taken as 1. For example: we can write 5 as $\dfrac{5}{1}$.
(2) Real numbers: Real numbers are the sets of all the rational numbers as well as irrational numbers. We have seen the definition of rational numbers above so let us now know about irrational numbers. Irrational numbers are the numbers which are non – repeating and non – terminating. It is different from rational numbers which are generally terminating or non – terminating but repeating. For example: $\sqrt{2},\sqrt{3},\sqrt{5}$ etc. are irrational numbers. We cannot write these numbers in the form $\dfrac{p}{q}$.
From the above definitions we can say that real numbers include both rational and irrational numbers that means we can say that all the rational numbers are real numbers while all the real numbers are not rational. This is the basic difference between the two.

Note: You must remember the definitions of all the basic terms used in the number theory like: natural number, whole number, fractions, rational numbers, irrational numbers, real numbers and at last complex numbers. Note that all the numbers are subsets of real numbers except complex numbers. Complex numbers contain both real and imaginary numbers hence all the real numbers are the subsets of complex numbers.