Classify the following numbers as rational or irrational: (i) (ii) (iii) (iv) (v) (Notice the pattern: Is it repeating a single block?) (vi) Find the explicit fractions in case they are rational.

Question

KnowledgeBoat · Accepted Answer

(i) ⇒ Since 9 can be written as with q ≠ 0, is rational. (ii) ⇒ Since is irrational, is also irrational. So, is irrational. (iii) 0.33333… This is a repeating decimal with the digit 3 repeating. Let x = 0.3333… ⇒ 10x = 3.3333… ⇒ 10x - x = 3.3333… - 0.3333… = 3 ⇒ 9x = 3 ⇒ x = . Since 0.33333… can be written as with q ≠ 0, So, 0.33333… is rational and equals . (iv) 0.123451234512345… This is a repeating decimal with the block "12345" repeating. Let x = 0.12345 12345… ⇒ 100000x = 12345.12345… ⇒ 100000x - x = 12345.12345… - 0.12345… = 12345 ⇒ 99999x = 12345 ⇒ x = . Since 0.123451234512345… can be written as with q ≠ 0, So, 0.123451234512345… is rational and equals . (v) 1.01001000100001… The pattern increases the number of zeros each time (one 0, then two 0's, then three 0's, etc.). This means there is no single repeating block. So, 1.01001000100001… is non-terminating and non-repeating. Hence, it is irrational. (vi) 23.560185612239874790120 This is a terminating decimal with 21 decimal places. It can be written as : ⇒ . Since 23.560185612239874790120 can be written as with q ≠ 0, So, 23.560185612239874790120 is rational.

Mathematics

Classify the following numbers as rational or irrational: (i) (ii) (iii) (iv) (v) (Notice the pattern: Is it repeating a single block?) (vi) Find the explicit fractions in case they are rational.

Whole Numbers

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