Question

Are rational and irrational numbers real numbers?

Answer 1

Hint: In the above question, we are given two sets of numbers. The first is the set of rational numbers and the other is the set of irrational numbers. We have to determine if the both given sets of numbers fall in the category of real numbers i.e. are also a subset of the set of real numbers or not.

Complete step by step answer:
Given that, two sets of numbers are rational numbers and irrational numbers.
We have to determine if both of the sets are also subsets of real numbers.
First let us define the given two sets of numbers.
1. Rational numbers: A number which can be written in the form of \[\dfrac{p}{q}\] , where \[p\] and \[q\] are integers and \[q \ne 0\] , is known as a rational number. It contains all the integers and the fractional numbers and also radicals who have complete square factors and some other similar numbers which can be written in their simpler forms so that they become a rational number. Ex- \[\dfrac{{ - 2}}{7}, - 9,\dfrac{1}{2},\sqrt {36} ,5,\dfrac{{22}}{7}\] etc.
2. Irrational numbers: A number which can not be written in the form of \[\dfrac{p}{q}\] , where \[p\] and \[q\] are integers and \[q \ne 0\] , is known as an irrational number. Generally it contains the radical numbers most of the times and all the non-terminating non-repeating decimal numbers and some special numbers like pie and the Euler’s number. Ex- \[\sqrt 3 ,\dfrac{1}{{\sqrt 2 }},e,\pi ,\dfrac{{\sqrt 5 }}{2},5.121122111222...\] etc.
Whereas integers are just positive and negative of the whole numbers as \[... - 3, - 2, - 1,0,1,2,3,...\] etc.
On the other hand, real numbers contain all the numbers that are real, i.e. all of those numbers that are used in the real world, that means they have their physical significance. The set of real numbers, as a result, contains all of the numbers mentioned above in the sets of rational and irrational numbers.
Thus, all rational and irrational numbers are real numbers.

Note:
In mathematics, there are different sets of numbers defined as per their qualities.
They all can be understood at once by making a table as shown below.

Sets of numbers	Elements
Complex numbers	Real and non-real/imaginary numbers $a+ib$
Imaginary numbers	Numbers having ONLY imaginary part \[ib,b \ne 0\]
Real numbers	Rational and irrational numbers
Irrational numbers	Radicals, non-terminating non-repeating decimals
Rational numbers	Integers, fractions \[p/q,q \ne 0\]
Integers	Negative and positive of Whole numbers
Whole numbers	Natural numbers and zero
Natural numbers	\[1,2,3,4,5,...\]