Question

Which of the following is irrational?
A.\[\dfrac{\sqrt{4}}{\sqrt{9}}=\dfrac{2}{3}\]
B.\[\dfrac{\sqrt{12}}{\sqrt{3}}=\dfrac{2\sqrt{3}}{\sqrt{3}}=2\]
C.\[\sqrt{7}=2.64575131106\]
D.\[\sqrt{81}=9\]

Answer 1

Hint: In this question we will check all the options that are irrational or not. To check that the number is irrational we need to check that they are not written \[\dfrac{p}{q}\] form where \[p\] and \[q\] both are integers and the value of \[q\] must be nonzero and then check which option is correct in the given options.

Complete step-by-step answer:
We are familiar with whole numbers, natural numbers and integers.
Whole numbers are those numbers that start from \[0\] hence we can say that \[0\] and natural numbers are called whole numbers \[0,1,2,3,4,.........\]
Natural numbers are counting numbers \[1,2,3,4,.........\] and there are finite natural numbers and the smallest natural number is \[1\] .
Integers are the whole numbers from the set of negative, non \[-\] negative, positive and zero numbers.
Rational numbers are those numbers which are written in \[\dfrac{p}{q}\] form where \[p\] and \[q\] both are integers and the value of \[q\] must be nonzero.
As a result, a rational number is one that can be expressed as a simple fraction, or as a ratio.
Around \[300\] BC, the Pythagoras in Greece, followers of the famed mathematician and philosopher Pythagoras, were the first to discover non-rational numbers. These numbers are called irrational numbers because they cannot be expressed as a ratio of integers.
Now in the question we will check each of the options to find which number is irrational.
\[\dfrac{\sqrt{4}}{\sqrt{9}}=\dfrac{2}{3}\]
Here \[\dfrac{2}{3}\] is written in \[\dfrac{p}{q}\] form and the value of \[q=3\] hence \[q\ne 0\]
Thus \[\dfrac{2}{3}\] is a rational number, therefore option \[(1)\] is incorrect.
\[\dfrac{\sqrt{12}}{\sqrt{3}}=\dfrac{2\sqrt{3}}{\sqrt{3}}=2\]
Here we can write \[2\] as \[\dfrac{2}{1}\] which is in \[\dfrac{p}{q}\] form and the value of \[q=1\] hence \[q\ne 0\]
Hence \[2\]is a rational number, hence option \[(2)\] is incorrect.
\[\sqrt{7}=2.64575131106\]
Here \[2.64575131106\] is an irrational number because it cannot be written in the form of a ratio of integers.
Hence \[2.64575131106\] is an irrational number therefore option \[(3)\] is correct.
Option \[(4)\] is also incorrect as \[\sqrt{81}=9\]
As here we can write \[9\] as \[\dfrac{9}{1}\] which is in \[\dfrac{p}{q}\] form and the value of \[q=1\] hence \[q\ne 0\]
So, the correct answer is “Option C”.

Note: Students you know that in \[1870s\] there were two German mathematician, Cantor and Dedekind, showed that: Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number. Two rational numbers \[\dfrac{a}{b}=\dfrac{c}{d}\] are equal if \[ad=bc\].