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7. Prove that

    \[\mathbf{2}\surd \mathbf{3}\text{ }-\text{ }\mathbf{1}~\]

is an irrational number.
CBSE | Class 10 | Maths | RD Sharma | Real Numbers
7. Prove that

    \[\mathbf{2}\surd \mathbf{3}\text{ }-\text{ }\mathbf{1}~\]

is an irrational number.

Solution:

Let’s assume on the contrary that

    \[2\surd 3\text{ }\text{ }1\]

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

    \[2\surd 3\text{ }\text{ }1\text{ }=\text{ }a/b\]

⇒

    \[2\surd 3\text{ }=\text{ }a/b\text{ }+\text{ }1\]

⇒

    \[\surd 3\text{ }=\text{ }\left( a\text{ }+\text{ }b \right)/\left( 2b \right)\]

⇒

    \[\surd 3\]

is rational [∵

    \[2\]

, a and b are integers ∴

    \[\left( a\text{ }+\text{ }b \right)/2b\]

is a rational number]

This contradicts the fact that

    \[\surd 3\]

is irrational. So, our assumption is incorrect.

Hence,

    \[2\surd 3\text{ }\text{ }1\]

 is an irrational number.

i

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