Question

Is 5 a rational number between \[\sqrt 2 \] and \[\sqrt 3 \].
A.True
B.False

Answer 1

Hint: Firstly note the definition of rational number and irrational number. Rational number is a number that can be expressed in the form of \[\dfrac{p}{q}\], where \[p\] and \[q\] are integers. More importantly \[q \ne 0\]. If \[q\] becomes zero we will get infinity. Irrational means no ratio. While in case of irrational numbers it cannot be expressed as a simple fraction. For example, \[\sqrt 2 \] and\[\sqrt 3 \]. Irrational numbers are not a finite number. Rational numbers are finite.

Complete step-by-step answer:
Since, we know that 5 is integer as well as a natural number.
But, in this case we have, the value of \[\sqrt 2 = 1.4142135......\] and
\[\sqrt 3 = 1.73205089.......\]
Since, 5 is greater than \[\sqrt 2 \] and \[\sqrt 3 \]. Meaning that 5 does not lie in-between \[1.4142135......\] and \[1.73205089......\]
Therefore the result is: 5 is not a rational number between \[\sqrt 2 \] and \[\sqrt 3 \].
You don’t need to solve this at all, because we know that \[\sqrt 2 \] and \[\sqrt 3 \] are not rational.
Both are irrational. Because \[\sqrt 2 \]and\[\sqrt 3 \] cannot be expressed as \[\dfrac{p}{q}\] form. We can say that 5 is not a rational number between \[\sqrt 2 \] and \[\sqrt 3 \], hence it is $false$.
So, the correct answer is “Option B”.

Note: We can see that by the definition of rational and irrational, 5 is neither rational nor irrational. Infect 5 is a pure natural number. The rational that exists in-between \[\sqrt 2 \] and \[\sqrt 3 \] is \[1.5\] which can be expressed as \[\dfrac{p}{q}\] form. That is \[1.5 = \dfrac{{15}}{{10}} = \dfrac{3}{2}\]. You can also find the rational in-between \[\sqrt 2 \] and \[\sqrt 3 \] is \[\dfrac{8}{5}\]. Further note that \[q\] may be equal to 1. Also every integer is a rational number.