Question

Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Answer 1

Hint: Here we will first take one example of the positive integer whose square root value will give a rational number. Rational number is defined as a number that can be expressed as a ratio of two integers or can be expressed as a fraction. Irrational number is defined as a number that cannot be expressed as the ratio of two integers.

Complete step by step solution:
We know that irrational numbers cannot be expressed as fractions. Also integers are the numbers which can be both positive and negative. The examples of positive integers are 1,2,3 and so on.
The square roots of all positive integers are not irrational because we know that the square roots of all positive integers include both rational and irrational numbers.
Now, we will take one example of a positive integer whose square root will give a rational number.
Let’s take a positive integer 4. Taking square root of 4, we get
\[\sqrt 4 = 2\]
We know 2 is a rational number.
Therefore, the square of all positive integers are not irrational.

Note: Here in this question, we require the knowledge of rational and irrational numbers. We need to keep in mind that the rational includes only those decimal numbers which are recurring in nature whereas irrational includes only those decimal numbers which are non-recurring in nature or non-terminating in nature. The numerator and denominator of a rational number can be whole numbers, integers, etc. Every integer can be a rational number however not every rational number can be an integer.