Question

State whether the following statement is true or false.
All real numbers are irrational.
A. TRUE
B. FALSE

Answer 1

Hint: Using the definitions of real numbers and irrational numbers we check if all real numbers are irrational or not.

Complete step-by-step answer:
We know Real numbers are the numbers that can be represented on the real line, which consists of positive numbers, negative numbers, rational numbers as well as irrational numbers. So, we can say Real numbers is a set of numbers that contains rational numbers as well irrational numbers.
Rational numbers are the numbers that can be written in the form \[\dfrac{p}{q}\] where \[q\] is non-zero and both the numerator and denominator are integers, i.e. the decimal representation of a rational number is either terminating or recurring. Examples: \[\dfrac{1}{2},\dfrac{0}{5},\sqrt 4 = \pm 2,7\]etc.
An irrational number is a number that cannot be represented in the form \[\dfrac{p}{q}\], and the value of an irrational number has the decimal representation as non-recurring, non-repeating and non-terminating. Also, any number that is not a rational number is called an irrational number. Examples: \[\sqrt 2 ,\sqrt 3 ,\pi \]etc.
The statement given in the question states that all real numbers are irrational, this will mean that rational numbers are not real numbers which contradicts the definition of real numbers.
So, the statement given in the question is FALSE.
So, option B is correct.

Note: Students should avoid wrong assumptions that every number with square root is irrational as many numbers which are perfect squares can be solved by cancelling the square root by square power making the number a rational number.
Students should know that converse of this statement is true because all rational numbers and all irrational numbers together form real numbers, so it is okay to say that all rational numbers are real numbers.