1. Write the following in decimal form and say what kind of decimal expansion each has: 4. Express 0.99999… in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Let x = 0.99999... ...(1) Multiply both sides by 10, we have [∵ There is only one repeating digit.] 10 x x = 10 x (0.99999…) or 10x = 9.9999 ...(2) Subtracting (1) from (2), we get 10x – x = (9.9999…) – (0.9999…) or 9x = 9 or x = 9/9 = 1 Thus, 0.9999… = 1 As 0.9999… goes on forever, there is no gap between 1 and 0.9999 Hence both are equal. 5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of (1/17)? Perform the division to check your answer. Since, the number of entries in the repeating block of digits is less than the divisor. In 1/17, the divisor is 17. ∴ The maximum number of digits in the repeating block is 16. To perform the long division, we have. The remainder 1 is the same digit from which we started the division. Thus, there are 16 digits in the repeating block in the decimal expansion of 1/17. Hence, our answer is verified. 6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy? Let us look at decimal expansion of the following terminating rational numbers: We observe that the prime factorization of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both. Note: If the denominator of a rational number (in its standard form) has prime factors either 2 or 5 or both, then and only then it can be represented as a terminating decimal. 7. Write three numbers whose decimal expansions are non-terminating non-recurring. 8. Find three different irrational numbers between the rational numbers (5/7) and (9 ). To express decimal expansion of 5/7 and 9/11, we have: (i) 0.750750075000750… (ii) 0.767076700767000767… (iii) 0.78080078008000780… 9. Classify the following numbers as rational or irrational: <!--[if !supportLineBreakNewLine]--> <!--[endif]-->