Question

Find two rational and two irrational number between \[\sqrt 2 \] and \[\sqrt 3 \]

Answer 1

Hint: As we know that \[\sqrt 2 \] and \[\sqrt {\text{3}} \] are irrational numbers so their approximate values are \[1.414\] and \[{\text{1}}{\text{.732}}\]. Now , we need to calculate rational an irrational number between \[1.414\] and \[{\text{1}}{\text{.732}}\]

Complete step by step answer:

Given Irrational Numbers \[\sqrt 2 \] and \[\sqrt {\text{3}} \]
Firstly find its rational and so we need to consider rational points for that,
Calculating rational numbers between \[1.4\]and \[1.7\].
So, they are \[ \Rightarrow \dfrac{{1.4 + 1.7}}{2} = 1.55\] and can also be integers as \[1.5,1.6...\]
And now calculating the irrational terms,
\[ \Rightarrow \dfrac{{\sqrt 2 + \sqrt 3 }}{2} = 1.572\]
So the numbers between \[\sqrt 2 \] and \[\sqrt {\text{3}} \] which are non-terminating and cannot be expressed in \[\dfrac{{\text{p}}}{q}\] form, so it can be \[1.665\overline 7 ,1.543\overline 9 ,....\]
Hence, \[1.5,1.6\] are 2 rational numbers and \[1.665\overline 7 ,1.543\overline 9 \] are irrational terms between \[\sqrt 2 \] and \[\sqrt {\text{3}} \].

Note: An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means, not Rational number.
A number that can be made by dividing two integers (an integer is a number with no fractional part). The word comes from "ratio".
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.