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8. Prove that

    \[\mathbf{2}\text{ }-\text{ }\mathbf{3}\surd \mathbf{5}\]

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

    \[\mathbf{2}\text{ }-\text{ }\mathbf{3}\surd \mathbf{5}\]

is an irrational number.

Solution:

Let’s assume on the contrary that

    \[2\text{ }\text{ }3\surd 5\]

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

    \[2\text{ }\text{ }3\surd 5\text{ }=\text{ }a/b\]

⇒

    \[3\surd 5\text{ }=\text{ }2\text{ }\text{ }a/b\]

⇒

    \[\surd 5\text{ }=\text{ }\left( 2b\text{ }\text{ }a \right)/\left( 3b \right)\]

⇒

    \[\surd 5\]

is rational [∵

    \[3\]

, a and b are integers ∴

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

is a rational number]

This contradicts the fact that

    \[\surd 5\]

is irrational. So, our assumption is incorrect.

Hence,

    \[2\text{ }\text{ }3\surd 5\]

is an irrational number.

i

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