Example 6 - Chapter 1 Class 10 Real Numbers

Transcript

Example 6 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, (5𝑏 − 𝑎)/𝑏 is a rational number But √3 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Therefore, 5 - √𝟑 is irrational Hence proved.