Question 1. Write the following in decimal form and say what kind of decimal expansion each has: (i) 36/100 (ii) 1/11 (iii) 4⅛ (iv) 3/13 (v) 2/11 (vi) 329/400
Solution.
(i) 36/100 = 0.36 — terminating decimal expansion.
(ii) 1/11 = 0.090909… = 0.09 — non-terminating, repeating decimal expansion.
(iii) 4⅛ = 33/8 = 4.125 — terminating decimal expansion.
(iv) 3/13 = 0.23076923… = 0.230769 — non-terminating, repeating decimal expansion.
(v) 2/11 = 0.1818… = 0.18 — non-terminating, repeating decimal expansion.
(vi) 329/400 = 0.8225 — terminating decimal.
In these examples, rational numbers expressed in decimal form terminate without leaving a remainder — such decimals are called terminating decimals (also called finite decimal forms).Ans.
Question 2. You know that 1/7 = 0.142857. Can you predict the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 without doing long division? If so, how?
Solution. Yes. All of these have repeating decimals that are permutations (cyclic shifts) of the digits 1, 4, 2, 8, 5, 7.
2/7 = 0.285714 3/7 = 0.428571 4/7 = 0.571428
5/7 = 0.714285 6/7 = 0.857142Ans.
Question 3. Express the following in the form p/q, where p and q are integers and q ≠ 0: (i) 0.6 (ii) 0.47 (iii) 0.001
Solution.
(i) Let x = 0.6 = 0.666… Multiplying by 10: 10x = 6.666… Subtracting: 9x = 6 ⇒ x = 6/9 = 2/3.Ans.
(ii) Let x = 0.47. Multiplying by 10: 10x = 4.7 = 4 + 7/9 = 43/9 ⇒ x = 43/90.Ans.
(iii) Let x = 0.001 = 0.001001001… Multiplying by 1000: 1000x = 001.001001… Subtracting: 999x = 1 ⇒ x = 1/999.Ans.
Question 4. Express 0.99999… in the form p/q. Are you surprised by your answer? Discuss why the answer makes sense.
Solution. Let x = 0.9999… Multiplying by 10: 10x = 9.9999… Subtracting: 9x = 9 ⇒ x = 1.
Hence, 0.99999… = 1. Since 0.99999… goes on forever, there is no gap between 1 and 0.99999…, so they are equal.Ans.
Question 5. What is the maximum number of digits in the repeating block of the decimal expansion of 1/17? Perform the division to check your answer.
Solution. Performing the long division:
1/17 = 0.0588235294117647
Hence, the maximum number of digits in the repeating block is 16.Ans.
Question 6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q have no common factors other than 1 and have terminating decimal expansions. What property must q satisfy?
Solution. Examples: 1/2 = 0.5, 1/4 = 0.25, 7/8 = 0.875, 37/25 = 1.48, 8/125 = 0.064, 17/20 = 0.85, 31/16 = 1.9375.
In each case, the denominator can be multiplied by a suitable integer to make it a power of 10. This is only possible when the denominator’s only prime factors are 2 or 5.
Property: If the denominator of a rational number in standard form has no prime factors other than 2 or 5, then — and only then — it can be represented as a terminating decimal.Ans.
Question 7. Write three numbers whose decimal expansions are non-terminating and non-recurring.
Solution. √2, √3, and √5 — or equivalently: 0.1001000100001…, 0.202002000200002…, and 0.003000300003….Ans.
Question 8. Find 3 different irrational numbers between the rational numbers 5/7 and 9/11.
Solution. 5/7 = 0.714… and 9/11 = 0.818…
Three irrational numbers between them:
0.72072007200072000072…
0.76076007600076000076…
0.79079007900079000079…Ans.
Question 9. Classify the following numbers as rational or irrational: (i) √23 (ii) √225 (iii) 0.3796 (iv) 7.478478… (v) 1.101001000100001…
Solution.
(i) √23 is irrational — 23 is not a perfect square.
(ii) √225 = √(3×3×5×5) = 3×5 = 15, a rational number.
(iii) 0.3796 is a terminating decimal — rational.
(iv) 7.478478… is non-terminating but repeating — rational.
(v) 1.101001000100001… is non-terminating and non-repeating — irrational.Ans