Questions on identifying rational & irrational numbers

In this post we will solve questions related to identification of rational and irrational numbers.

All the questions are Grade 9 standard.


Question 01
Identify if the given numbers are rational or irrational

\mathtt{( a) \ \sqrt{9}}\\\ \\ \mathtt{( b) \ \sqrt{5}}\\\ \\ \mathtt{( c) \ \sqrt{21}}\\\ \\ \mathtt{( d) \ 2+\sqrt{6}}\\\ \\ \mathtt{( e) \ \sqrt{16} +\sqrt{4}}

Solution
\mathtt{( a) \ \sqrt{9}}\\\ \\ \mathtt{\sqrt{9} =\ 3}

The number is a perfect square, hence it’s a rational number.

\mathtt{( b) \ \sqrt{5}}

It’s an irrational number since 5 is not a perfect square.

\mathtt{( c) \ \sqrt{21}}

Number 21 is not perfect square so the given number can’t be reduce out of square root.

Hence, it’s an irrational number.

\mathtt{( d) \ 2+\sqrt{6}}

2 ⟹ rational number
\mathtt{\sqrt{6}} ⟹ irrational number

Addition of rational and irrational number results in irrational number.

Hence, the given number is irrational number.

\mathtt{( e) \ \sqrt{16} +\sqrt{4}}

\mathtt{\sqrt{16} \ =\ 4}\\\ \\ \mathtt{\sqrt{4} \ =\ 2}

⟹ 4 + 2
⟹ 6

Hence, the given number is rational number.

Question 02
Check if the following expression results in rational or irrational number.

\mathtt{( a) \ \sqrt{3} +\sqrt{5}}\\\ \\ \mathtt{( b) \ \left(\sqrt{3} +6\right)^{2}}\\\ \\ \mathtt{( c) \ \left( 5+\sqrt{5}\right)\left( 5-\sqrt{5}\right)}\\\ \\ \mathtt{( d) \ 10-\sqrt{7}}\\\ \\ \mathtt{( e) \ \left(\sqrt{2} +\sqrt{3}\right)^{2}}

Solution
\mathtt{( a) \ \sqrt{3} +\sqrt{5}}

\mathtt{\sqrt{3}} = irrational number
\mathtt{\sqrt{5}} = irrational number

Addition of irrational numbers results in irrational number.

\mathtt{( b) \ \left(\sqrt{3} +6\right)^{2}}

Using the square of sum formula;

\mathtt{( a+b)^{2} =a^{2} +b^{2} +2ab}

Putting the values;

\mathtt{\Longrightarrow \left(\sqrt{3}\right)^{2} +6^{2} +2.\left(\sqrt{3}\right)( 6)}\\\ \\ \mathtt{\Longrightarrow \ 3+\ 36\ +18\sqrt{3}}\\\ \\ \mathtt{\Longrightarrow \ 39+18\sqrt{3}}

39 ⟹ rational number
\mathtt{18\sqrt{3}} ⟹ irrational number

Addition of rational number with irrational number results in irrational number. Hence, the given number is irrational number.

\mathtt{( c) \ \left( 5+\sqrt{5}\right)\left( 5-\sqrt{5}\right)}

Solving the expression;

\mathtt{\Longrightarrow \ 5^{2} -5\sqrt{5} +5\sqrt{5} -5}\\\ \\ \mathtt{\Longrightarrow \ 20}

Hence, the above expression results in rational number.

\mathtt{( d) \ 10-\sqrt{7}}

10 = rational number
\mathtt{\sqrt{7}} = irrational number

Subtraction/Addition of rational with irrational number results in irrational number.

Hence, the above expression is irrational number.

\mathtt{( e) \ \left(\sqrt{2} +\sqrt{3}\right)^{2}}

Again using square of sum formula;

\mathtt{\Longrightarrow \left(\sqrt{2}\right)^{2} +\left(\sqrt{3}\right)^{2} +2.\left(\sqrt{2}\right)\left(\sqrt{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ 2+\ 3\ +2\sqrt{6}}\\\ \\ \mathtt{\Longrightarrow \ 5+2\sqrt{6}}\\ \\


The above expression is irrational number.

Question 03
Check of the below numbers are rational or irrational numbers.

\mathtt{( a) \ \sqrt{144}}\\\ \\ \mathtt{( b) \ \sqrt{1.69}}\\\ \\ \mathtt{( c) \ \sqrt{\frac{121}{81}}}\\\ \\ \mathtt{( d) \ -\sqrt{36}}\\\ \\ \mathtt{( e) \ \sqrt{15}}

Solution

\mathtt{( a) \ \sqrt{144}}\\\ \\ \mathtt{\sqrt{144} \ =\ 12}

It’s a rational number

\mathtt{( b) \ \sqrt{1.69}}\\\ \\ \mathtt{\sqrt{1.69} =\sqrt{\frac{169}{100}}}\\\ \\ \mathtt{\sqrt{\frac{169}{100}} \ =\ \frac{13}{10}}

Hence, it’s a rational number.

\mathtt{( c) \ \sqrt{\frac{121}{81}}}\\\ \\ \mathtt{\sqrt{\frac{121}{81}} \ =\ \frac{11}{9}}

Hence, it’s a rational number.

\mathtt{( d) \ -\sqrt{36}}\\\ \\ \mathtt{-\sqrt{36} \ =\ -6}

It’s a rational number.

\mathtt{( e) \ \sqrt{15}}

15 is not a perfect square.
Hence, the given number is irrational number.

Question 04
Identify if the below decimal number is rational or irrational.

\mathtt{( a) \ 1.23\overline{45}}\\\ \\ \mathtt{( b) \ 0.369369369\ .\ \ .\ \ .}\\\ \\ \mathtt{( c) \ 11.279033679812\ .\ \ .\ \ .\ .}\\\ \\ \mathtt{( d) \ 3.663666366663\ .\ \ .\ .\ }\\\ \\ \mathtt{( e) \ \pi }

Solution

\mathtt{( a) \ 1.23\overline{45}}

It’s a non terminating repeating decimal, hence it’s a rational number.

\mathtt{( b) \ 0.369369369\ .\ \ .\ \ .}

Again it’s a non terminating repeating decimal as number 369 after decimal point is repeated again & again.

Hence, it’s a rational number.

\mathtt{( c) \ 11.279033679812\ .\ \ .\ \ .}

It’s a non terminating, non repeating decimal.

Hence, it’s an irrational number.

\mathtt{( d) \ 3.663666366663\ .\ \ .\ .\ }

It’s non terminating and non repeating decimal.
Hence, it’s an irrational number.

\mathtt{( e) \ \pi }

Pi is an irrational number.

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