Prove that $2-3\sqrt{5}$ is an irrational number.


Given: Number $2-3\sqrt{5}$
To do: To prove that the given number is an irrational number.
Solution:
Let us assume that $2-3\sqrt{5} =x$ and$\ x$ is a rational number.

$\therefore \ 2-x=3\sqrt{5} ,\ 2-x$ would also be a rational number.

$\therefore \frac{\ ( 2-x)}{3} =\frac{( 3\surd 5)}{3} =\sqrt{5}$ 
 
If x is a rational number then $2-x$ is also a rational number and $\frac{2-x}{3}$  also should be a rational number.

But here we find that $\therefore \frac{\ ( 2-x)}{3} =5\ $ and $\sqrt{5}$ can never be rational number. 

Our assumption was wrong x to be a rational number.

Thus it has been proved that $2-3\sqrt{5}$ is an irrational number.

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Updated on: 10-Oct-2022

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