### Real Number System Class 9th Mathematics Term 1 Tamilnadu Board Solution

##### Question 1.A number having non-terminating and recurring decimal expansion is –A. an integerB. a rational numberC. an irrational numberD. a whole numberAnswer:By definition, a rational number is one which follows either of the following 2 conditions:1. It can be expressed in form where p and q are integers (q≠0).2. Its decimal expansion is either terminating or non-terminating recurring.Ex. – 0.35 (terminating decimal expansion) → Rational number0.7777…. = (non-terminating but recurring i.e. the number does not terminate but its digits follow a pattern.) → Rational numberπ = 3.14…… (Non-terminating non recurring) → Irrational numberHence, the answer.Question 2.If a number has a non-terminating non-recurring decimal expansion then the number is –A. a rational numberB. a natural numberC. an irrational numberD. an integer.Answer:As explained in the above question, for a number to be rational its decimal expansion has to be either terminating or non-terminating and recurring.For example –π = 3.14…… (Non-terminating non recurring) → Irrational numberHence, the answer.Question 3.Decimal form of isA. −0.75B. −0.50C. −0.25D. −0.125Answer:To convert into its equivalent decimal expansion, we can either proceed by long division method or multiply the denominator and (hence) numerator by a number such that the denominator turns out to be 100.Now, let 4×a = 100Solving, we get a = 25.Hence multiplying numerator and denominator by 25.∴ (-3/4) × (25/25) = (-75/100) = -0.75 (Placing decimal point before 2 digits).Hence, the answer.Question 4.The form of isA. B. C. D. Answer:Let x = 0. = 0.3333... → (eq)1Since there is only one repeating digit after decimal point, multiply equation 1 by 10.10x = 3.3333… = 3 + 0.3333… = 3 + x → (eq)2(eq)2 –(eq)110x-x = 3.3333… - 0.3333… = 3∴ 9x = 3∴ x = 1/3Hence, the answer.Question 5.Which one of the following is not true?A. Every natural number is a rational numberB. Every real number is a rational numberC. Every whole number is a rational numberD. Every integer is a rational number.Answer:Real numbers is the set which includes both rational and irrational numbers.∴ Rational numbers and Irrational numbers are 2 independent subsets of Real numbers. Hence every real number is not a rational number (since real numbers also consists of irrational numbers.)Option (A) is correct because every natural number can be expressed in form (q≠0 i.e a rational number).Option (C) is correct because every whole number can be expressed in form (q≠0 i.e a rational number).Option (D) is correct because every integer can be expressed in form (q≠0 i.e a rational number).Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real numbersReal numbers ⊃ (Rational numbers + Irrational numbers).Hence, only option B is incorrect.Question 6.Which one of the following has a terminating decimal expansion?A. B. C. D. Answer:If a rational number (q≠0) can be expressed in the form , where p ∈ Z, and m,n ∈ W, then the rational number will have a terminating decimal expansion.Otherwise, the rational number will have a non-terminating and recurring decimalexpansion.Now, 32 = 25 × 50 Hence, it has terminating decimal expansion.Hence, option A is correct.Option B is incorrect ∵ 9 = 32 and it cannot be expressed in the form 2m × 5n.Option C is incorrect ∵ 15 = 3 × 5 and it cannot be expressed in the form 2m × 5n.Option D is incorrect ∵ 12 = 3 × 22 and it cannot be expressed in the form 2m × 5n.Hence, only option A is correct.Question 7.Which one of the following is an irrational number?A. πB. C. D. Answer:Option A is correct since π = 3.141592… i.e. the decimal expansion is non-terminating and non-recurring.Option B is incorrect because = ±3 and it is an integer (and hence a rational number, Integers ⊂ Rational Numbers).Option C is incorrect as it is expressed in the form (p = 1 and q = 4) and is a rational number (by definition, q≠0).Option D is incorrect as it is expressed in the form (p = 1 and q = 5) and is a rational number (by definition, q≠0).Hence, option A is correct.Question 8.Which of the following are irrational numbers?i. ii. iii. iv. A. (ii), (iii) and (iv)B. (i), (ii) and (iv)C. (i), (ii) and (iii)D. (i), (iii) and (iv)Answer:i.√(2 + √3 ) is irrational ∵ √3 is irrational and 2 is rational and sum of a rational and irrational number is always irrational. Hence 2 + √3 is irrational. Hence, its square root is also irrational.ii.√(4 + √25) is rational ∵ √25( = 5) is rational and 4 is also rational.Hence, √(4 + 5) = √9 = 3 is rational.iii. ∛(5 + √7) is irrational ∵ √7 is irrational and 5 is rational and sum of a rational and irrational number is always irrational. Hence 5 + √7 is irrational. Hence, its cube root is also irrational.iv.√(8-∛8 ) is irrational. Here both ( = 2) and 8 are rational. Hence8 – 81/3 = 8 – 2 = 6 is rational. But √6 is irrational.Hence, the answer.

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