Question

State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is in the form of$\sqrt{m}$ , where m is a natural number.
(iii) Every real number is an irrational number.

Answer 1

Hint: There are 3 statements given in the question. Take each statement one by one and try to prove it or disprove it. By that you can say the result, true or false. Use all definitions of irrational numbers, real numbers.

Complete step-by-step answer:
Real number: Numbers which value a continuous quantity that can represent a distance along a line.
Irrational number: Numbers which cannot be written in the $\dfrac{p}{q}$ form are called irrational numbers.
Rational numbers: Numbers which can be written in the $\dfrac{p}{q}$ form are called rational numbers.
In the above 2 definitions p, q are said to be integers.
Integers: Numbers which can be written without any fractional component are called integers.
Natural Numbers: All the positive integers are called natural numbers.
Whole Numbers: All the natural numbers with 0 as an extra element are called as whole numbers.

(i) True
Real numbers are numbers we can think about. So, all irrational numbers are real numbers.
For example \[\sqrt{2},\sqrt{5},1+\sqrt{7}\] are real numbers.

(ii) False
A number line can have negative numbers. Since no negative can be the square root of a natural number, thus every point on the number cannot be of the form $\sqrt{m}$ , where m is the natural number.
For example we cannot write -2.3 as the square root of a natural number.

(iii) False
All the numbers are real number but all real numbers cannot be said as particular type
Ex: 2, 3, 4……….are not irrational but they are real.

Note: The method we follow is if the statement is true try to prove it. If it is false try to give a counterexample. Use the definition of irrational and real numbers carefully. By definition of natural numbers we can say they are examples for disproving of statement 3.