Show that (5 - 2√3) is irrational

1.	Show that (5 - 2√3) is irrational
Answer» Let x = 5 - 2√3 be a rational number. x = 5 - 2√3 ⇒ x² = (5 - 2√3)² ⇒ x² = 5² + (2√3)² – 2(5) (2√3) ⇒ x² = 25 + 12 – 20√3 ⇒ x² – 37 = – 20√3 ⇒ \(\frac{37-x^2}{20}\) = √3 Since x is a rational number, x² is also a rational number. ⇒ 37 - x² is a rational number ⇒ \(\frac{37-x^2}{20}\) is a rational number ⇒√3 is a rational number But √3 is an irrational number, which is a contradiction. Hence, our assumption is wrong. Thus, (5 - 2√3) is an irrational number.

Answer»

Let x = 5 - 2√3 be a rational number.

x = 5 - 2√3

⇒ x² = (5 - 2√3)²

⇒ x² = 5² + (2√3)² – 2(5) (2√3)

⇒ x² = 25 + 12 – 20√3

⇒ x² – 37 = – 20√3

⇒ \(\frac{37-x^2}{20}\) = √3

Since x is a rational number, x² is also a rational number.

⇒ 37 - x² is a rational number

⇒ \(\frac{37-x^2}{20}\) is a rational number

⇒√3 is a rational number But √3 is an irrational number, which is a contradiction.

Hence, our assumption is wrong.

Thus, (5 - 2√3) is an irrational number.

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