# Ex.1.3 Q3 Real Numbers Solution - NCERT Maths Class 10

## Question

Prove that the following are irrationals:

(i) \(\begin{align}\frac{1}{{\sqrt 2 }}\end{align}\) (ii) \(\begin{align}7\sqrt 5 \end{align}\) (iii) \(\begin{align}6 +\sqrt 2 \end{align}\)

## Text Solution

**What is unknown?**

(i) \(\begin{align}\frac{1}{{\sqrt 2 }}\end{align}\) (ii) \(\begin{align}7\sqrt 5 \end{align}\) (iii) \(\begin{align}6 +\sqrt 2 \end{align}\) are Irrationals

**Steps:**

(i) \( \begin{align} \frac{1}{{\sqrt 2 }}\end{align} \)

Let us assume, to the contrary \( \begin{align} \frac{1}{{\sqrt 2 }}\end{align} \) that is a rational number.

\[ \begin{align} \frac{1}{{\sqrt 2 }} = \frac{p}{q}\end{align} \]

Let *\(p\)* and *\(q\)* have common factors, so by cancelling them we will get \( \begin{align} \frac{a}{b},\end{align} \) where *\(a\)* and *\(b\)* are co-primes.

\[\begin{align}\frac{1}{{\sqrt 2 }}& = \frac{a}{b} \\\sqrt 2 {\text{a}} &= {\text{b}}\\\sqrt 2 &= \frac{b}{a}\end{align} \]

(*where \(a\) and \(b\) are co - primes and have no common factor otherthan* \(1\))

Since, *\(b\)* and \(a \) are integers, \( \begin{align} \frac{b}{a}\end{align} \) is rational number and so, \( \begin{align} \sqrt 2 \end{align} \) is rational.

But we know that \( \begin{align} \sqrt 2 \end{align} \) is irrational this contradicts the fact that \( \begin{align} \sqrt 2 \end{align} \) is rational.

So, our assumption was wrong. Therefore, \( \begin{align} \frac{1}{{\sqrt 2 }}\end{align} \) is a rational number.

(ii) \( \begin{align} 7\sqrt 5 \end{align} \)

Let us assume, to the contrary that \( \begin{align} 7\sqrt 5 \end{align} \) is a rational number.

\[ \begin{align} 7\sqrt 5 = \frac{p}{q}\end{align} \]

Let *\(p\)* and *\(q\)* have common factors, so by cancelling them we will get \( \begin{align} \frac{a}{b},\end{align} \) where *\(a\)* and *\(b\)* are co-primes.

\[\begin{align} 7\sqrt 5 &= \frac{a}{b} \\7\sqrt 5 {\text{b}} &= {\text{a}}\\\sqrt 5 &= \frac{a}{{7b}}\end{align} \]

(*where \(a\) and \(b\) are co - primes and have no common factor other than *\(1\))

Since, \(a,\; 7\) and *\(b\)* are integers. So, \( \begin{align} \frac{a}{{7b}}\end{align} \) is rational number and so, \( \begin{align} \sqrt 5 \end{align} \) is rational. But this contradict the fact that \( \begin{align} \sqrt 5 \end{align} \)

So, our assumption was wrong. Therefore, \( \begin{align} 7\sqrt 5 \end{align} \) is a rational number.

(iìi) \( \begin{align} 6{\text{ }} + \sqrt 2 \end{align} \)

Let us assume, to the contrary that \( \begin{align} 6{\text{ }} + \sqrt 2 \end{align} \) is a rational number.

\[\begin{align} 6{\text{ }} + \sqrt 2 = \frac{p}{q}\end{align}\]

Let *\(p\)* and *\(q\)* have common factors, so by cancelling them we will get \( \begin{align} \frac{a}{b},\end{align} \) where *\(a\)* and *\(b\)* are co-primes.

\[\begin{align} 6 + \sqrt 2 &= \frac{a}{b}\\\sqrt 2 &= \frac{a}{b} - 6 \end{align} \]

*(where \(a\) and \(b\) are co - primes and have no common factor other than \(1\))*

Since, \(a, \,b\) and \(6 \) are integers. So, \( \begin{align} \frac{a}{b} - 6\end{align} \) is rational number and so, \( \begin{align} \sqrt 2 \end{align} \) is also a rational number.

But this contradicts the fact that \( \begin{align} \sqrt 2 \end{align} \) is irrational. So, our assumption was wrong.

Therefore, \( \begin{align} 6{\text{ }} + \sqrt 2 \end{align} \) is a rational number.