Prove That (2 + √3)/5 Is an Irrational Number

Video Explanation

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Solution

Question: Given that √3 is irrational, prove that (2 + √3)/5 is an irrational number.

Proof:

Let us assume that (2 + √3)/5 is a rational number.

Since 5 is a non-zero rational number, multiplying both sides by 5, we get:

2 + √3 is a rational number.

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

√3 is a rational number.

But this contradicts the given fact that √3 is irrational.

∴ Our assumption is wrong.

Hence, (2 + √3)/5 is an irrational number.

Final Answer

∴ (2 + √3)/5 is an irrational number.

Conclusion

Thus, using the method of contradiction and the given fact that √3 is irrational, we have proved that (2 + √3)/5 is irrational.