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Prove that 3 + 2√5 is irrational.
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24/05/2021 11:49 am
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Prove that 3 + 2√5 is irrational.
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24/05/2021 11:51 am
Let us assume 3 + 2√5 is rational.
Then we can find coprime x and y (y ≠ 0) such that 3 + 2√5 = x/y
Rearranging, we get,
\(2 \sqrt{5} = \frac{x}{y}3\)
\(\sqrt{5} = \frac{1}{2}(\frac{x}{y}  3)\)
Since, x and y are integers, thus,
\(\frac{1}{2}(\frac{x}{y}  3)\)
is a rational number.
Therefore, √5 is also a rational number. But this contradicts the fact that √5 is irrational.
So, we conclude that 3 + 2√5 is irrational.
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