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 of the form \(\sqrt m,\) where m is a natural number.

(iii) Every real number is an irrational number.

Asked by Vishal kumar | 1 year ago |  248

1 Answer

Solution :-

(i) Consider the irrational numbers and the real numbers separately.

  • The irrational numbers are the numbers that cannot be converted in the form \(\frac{p}{q}\), where p and q are integers and q \(\ne\) 0. (Eg: \( \sqrt{2},3\pi,..011011011...)\)
  • The real number is the collection of rational numbers and irrational numbers.
  • Therefore, we conclude that, every irrational number is a real number.

(ii) Consider a number line. on a number line, we can represent negative as well as positive numbers.

  • Positive numbers are represented in the form of \( \sqrt 1, \sqrt{1.1},\sqrt{1.2}...\)
  • But we cannot get a negative number after taking square root of any number. (Eg: \( \sqrt{-5}=5i\) is a complex number (which you will study in higher classes))

Therefore, we conclude that every number point on the number line is not of the form \( \sqrt m,\) where m is a natural number.

(iii) Consider the irrational numbers and the real numbers separately.

  • Irrational numbers are the numbers that cannot be converted in the form \(\frac{p}{q},\) where p and q are integers and q \( \ne 0\).
  • A real number is the collection of rational numbers (Eg: \( \frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},......)\) and irrational numbers (Eg: \( \sqrt 2, 3\pi,.011011011...)\)

So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.

Therefore, every real number is not an irrational number.

Answered by Shivani Kumari | 1 year ago

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