Prove That 2 − 3√5 Is an Irrational Number

Video Explanation

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Solution

Question: Prove that 2 − 3√5 is an irrational number.

Proof:

Let us assume that 2 − 3√5 is a rational number.

Since 2 is a rational number, subtracting 2 from both sides, we get:

−3√5 = (2 − 3√5) − 2

This implies that −3√5 is a rational number.

Dividing both sides by −3 (which is a non-zero rational number), we get:

√5 is a rational number.

But this is a contradiction because √5 is an irrational number.

∴ Our assumption is wrong.

Hence, 2 − 3√5 is an irrational number.

Final Answer

∴ 2 − 3√5 is an irrational number.

Conclusion

Thus, by the method of contradiction , we have proved that 2 − 3√5 is irrational.