Question

Find two rational numbers between \[\sqrt 2 \] and \[\sqrt 3 \].

Answer 1

Hint: Here we need to find two rational numbers between \[\sqrt 2 \] and \[\sqrt 3 \]. First, we will write the value of \[\sqrt 2 \] in the decimal form and then the value of \[\sqrt 3 \] in the decimal form. Then we will choose two random numbers between them which would satisfy the condition of a rational number. From there, we will get our final result.

Complete step-by-step answer:
Here we need to find the value of two rational numbers between \[\sqrt 2 \] and \[\sqrt 3 \].
We will write the value of \[\sqrt 2 \] in the decimal form and the value of \[\sqrt 3 \] in the decimal form i.e.
We know that
\[\sqrt 2 = 1.414...\] and \[\sqrt 3 = 1.732...\]
Now, we will find two rational numbers between \[\sqrt 2 \] and \[\sqrt 3 \].
We have to take two numbers between \[1.414\] and \[1.732\]. We can choose any number between them but the number should be a rational number.
We know 1.5 and 1.6 are rational numbers as these can be expressed in the form of \[\dfrac{p}{q}\], where \[q\] cannot be zero i.e.
1.5 can be written as \[\dfrac{3}{2}\] and 1.6 can be written as \[\dfrac{8}{5}\] , which are satisfying the conditions of a number to be a rational number.
Therefore, two rational numbers between \[\sqrt 2 \] and \[\sqrt 3 \] are 1.5 and 1.6.

Note: Rational number is defined as a number that can be expressed in the form \[\dfrac{p}{q}\], where \[q\] cannot be equal to zero. We need to keep in mind that the sum that we obtain after adding two rational numbers will also be a rational number. The addition of two rational numbers satisfies the commutative property of addition but the difference or subtraction of two rational numbers is not commutative.