Example 12

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Example 12(Method 1) Check whether 7 √5, 7/√5, √2 + 21, 𝜋 – 2 are irrational numbers or not. 7 √𝟓 7 is rational √5 = 2.236…. which is non- terminating non-repeating is an irrational number We know that Rational × Irrational = Irrational Since, 7 √5 = 7 × √5 it is irrational 𝟕/√𝟓 7 is rational √5 = 2.236…. which is non- terminating non-repeating is an irrational number We know that 𝑅𝑎𝑡𝑖𝑜𝑛𝑎𝑙/𝐼𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = Irrational Thus, 7/√5 is irrational √𝟐 + 21 21 is rational √2 " = 1.4142…" which is non- terminating non-repeating is an irrational number We know that Irrational + Rational = Irrational Thus, √2 + 21 is an irrational number 𝝅 – 2 2 is rational 𝜋 = 3.1415…. which is non- terminating non-repeating is an irrational number We know that Irrational – Rational = Irrational Thus, 𝜋 - 2 is an irrational number Example 12 (Method 2) Check whether 7 √5, 7/√5, √2 + 21, 𝜋 – 2 are irrational numbers or not. 7 √𝟓 √5 = 2.236…. 7 √5 = 7 × (2.346) = 15.652….., It is a non-terminating and non-repeating decimal expansion. So,7 √5 is an irrational number 𝟕/√𝟓 7/√5 = (7√5)/(√5 √5) = (7√5)/5 = 7/5 × (2.346) = 3.1304…. It is a non-terminating and non-repeating decimal expansion. So, 7/√5 is an irrational number √𝟐 + 21 √2 = 1.4142… √2 + 21 = 1.4142 + 21 = 22.4142……, It is a non-terminating and non-repeating decimal expansion. So, √2 + 21 is an irrational number 𝝅 – 2 𝜋 = 3.1415…. 𝜋 – 2 = 3.1415…. – 2 = 1.1415…… It is a non-terminating and non-repeating decimal expansion. So, 𝜋 - 2 is an irrational number