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 of the form \[\sqrt m \] , where \[m\] is a natural number.
iii. Every real number is an irrational number.

Answer 1

Hint: The set of real numbers is all those numbers that can be shown on a number line. This includes natural, whole numbers, and integers. It also includes rational numbers, which are numbers that can be written as a ratio of two integers, and irrational numbers, which cannot be written as the ratio of two integers.

Complete step-by-step answer:
i. An irrational number is a real number that cannot be expressed as a ratio of integers.
For example, √ 2 is an irrational number. Real numbers are simply the combination of rational and irrational numbers, in the number system. For example-23, -12, 6.99, 5/2, are real numbers. Answer is True. As per definition irrational numbers are also a real number. Hence the answer is True
ii. A natural number is an integer greater than 0. Natural numbers begin at 1 and go up to infinity: i.e. 1, 2, 3, 4, 5, etc. Natural numbers are also called "counting numbers" because they are used for counting. For example, 1, 2, 3…….
Since there are negative numbers also whose square cannot be taken out. On the number line there are positive numbers and negative numbers. For example if we have \[ - 3\] then if we put it in the form of \[\sqrt m \] , then its value will come \[\sqrt { - 9} \] which is not possible as in square root of negative does not comes in case of real numbers. Hence the answer is False.

iii. Real numbers are simply the combination of rational and irrational numbers, in the number system. All the arithmetic operations can be performed on these numbers and they can be represented in the number line. For example-23, -12, 6.99, 5/2, \[\pi \] are real numbers. As per definition real numbers have both rational and irrational numbers hence the answer is False.


Note: - It is always advisable to remember the concepts and definitions of various terms like real numbers, natural numbers and irrational numbers as it helps in solving these types of questions easily.