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NCERT

Class 9

Mathematics

Solutions

State whether the following statements are true or false. Justify your answer.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form, where m is a natural number.

(iii) Every real number is an irrational number.

(i) This statement is true because irrational and rational numbers together form real numbers. Therefore, every irrational number is a real number.

(ii) This statement is false, because all real numbers can be represented on the number line. Here m is a natural number which show that only the points √1 , √2 , √3, ... lie on the number line. Whereas the reality is that there is a wide gap between the points representing any two consecutive numbers; For example √2 = 1.414 and √3 = 1.732, then the next number between 1.414 and 1.732 also get a place on the number line. Apart from this, negative number lie on the number line. No negative number can be the square roof of any natural number. Hence, every point on the number line cannot be represented by √m , where m is a natural number.

(iii) This statement is false, because the set of real numbers is formed by the collection of rational and irrational numbers. So, every irrational number can be a real number but every real number need not be irrational.

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