Example 10 - Class 10

Transcript

Example 10 Show that 5 - √3 is irrational. We have to prove 5 - √3 is irrational Let us assume the opposite, i.e., 5 - √3 is rational Hence, 5 - √3 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, 5 - √3 = 𝑎/𝑏 −√3 = 𝑎/𝑏 - 5 −√3 = (𝑎 −5𝑏)/𝑏 − √3 = (𝑎 − 5𝑏)/𝑏 √3 = −((𝑎 − 5𝑏)/𝑏) √3 = (5𝑏 − 𝑎 )/𝑏 Here, (−𝑎 +5𝑏)/𝑏 is a rational number But √3 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence, 5 - √3 is irrational Hence proved.