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1. Show that the following numbers are irrational.(i)

    \[\mathbf{1}/\surd\mathbf{2}\]

(ii)

    \[\mathbf{7}\surd\mathbf{5}\]

CBSE | Class 10 | Maths | Maths | RD Sharma | Real Numbers
1. Show that the following numbers are irrational.(i)

    \[\mathbf{1}/\surd\mathbf{2}\]

(ii)

    \[\mathbf{7}\surd\mathbf{5}\]

Solution:

Consider

    \[1/\surd 2\]

is a rational number

Let us assume

    \[1/\surd 2\]

= r where r is a rational number

On further calculation we get

    \[1/r\text{ }=\text{ }\surd 2\]

Since r is a rational number

    \[,\text{ }1/r\text{ }=\text{ }\surd 2\]

is also a rational number

But we know that

    \[\surd 2\]

is an irrational number

So our supposition is wrong.

Hence

    \[,\text{ }1/\surd 2\]

is an irrational number.

Solution:

Let’s assume on the contrary that

    \[7\surd 5\]

is a rational number. Then, there exist positive integers a and b such that

    \[7\surd 5\]

= a/b where, a and b, are co-primes

⇒

    \[\surd 5\text{ }=\text{ }a/7b\]

⇒

    \[\surd 5\]

 is rational [∵

    \[7\]

, a and b are integers ∴

    \[a/7b\]

is a rational number]

This contradicts the fact that

    \[\surd 5\]

is irrational. So, our assumption is incorrect.

Hence,

    \[7\surd 5\]

is an irrational number.

i

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