Ex 3.5, 3

Transcript

Ex 3.5, 3 (i) Classify the following numbers as rational or irrational: (i) √81 Simplifying our number √𝟖𝟏 = √(9^2 ) = 9 And, we can express 9 as 9 = 𝟗/𝟏 Thus, we can express √81 as 𝑝/𝑞 So, √81 is a rational number Ex 3.5, 3 (ii) Classify the following numbers as rational or irrational: (ii) √12 Simplifying our number √𝟏𝟐 = √(4 × 3) = √(2^2 × 3) = √(2^2 ) ×√3 = 2√𝟑 Since √𝟑 is irrational, 𝟐 × √𝟑 will also be irrational As Rational × Irrational = Irrational Thus, √12 is irrational Ex 3.5, 3 (iii) Classify the following numbers as rational or irrational: (iii) 0.33333 … Here, 0.33333… = 0.3 ̅ So, the decimal expansion is non-terminating, repeating And, non-terminating repeating decimal expansions are rational numbers Thus, 0.33333 … is a rational number Ex 3.5, 3 (iv) Classify the following numbers as rational or irrational: (iv) 0.123451234512345 … Here, 0.123451234512345… = 0.(12345) ̅ So, the decimal expansion is non-terminating, repeating And, non-terminating repeating decimal expansions are rational numbers Thus, 0.123451234512345… is a rational number Ex 3.5, 3 (v) Classify the following numbers as rational or irrational: (v) 1.01001000100001… (Notice the pattern: Is it repeating a single block?) Our number is 1.01001000100001… Here, the decimal expansion looks repeating, but it isn’t repeating Thus, our decimal expansion is non-terminating, non-repeating And, non-terminating non-repeating decimal expansions are irrational numbers Thus, 1.01001000100001… is irrational Ex 3.5, 3 (vi) Classify the following numbers as rational or irrational: (vi) 23.560185612239874790120 Here, 23.560185612239874790120 Since the decimal expansion ends, it is a terminating decimal expansion And, terminating decimal expansions are rational numbers Thus, 23.560185612239874790120 is a rational number