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Representing Rational Numbers Binary numbering 16 1. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 8 0 4 0 2 1 1 1 1 32 16 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 8 4 2 1 Representing Rational Numbers 2. Find the value of each of the binary expressions (a) 10101012 (b) 11011012 (c) 11.011012 3. Find the binary and ternary representations of (a) 1995 using both the remainder method and the subtraction method. (b) 1996 using both the remainder method and the subtraction method. (c) 0.25 (d) 0.2. (e) 1/3. 4. Perform the indicated arithmetic (a) 11012 × 101112 (b) 10012 × .12 (c) 100002 × 1.001012 (d) 11012 + 101112 (e) 10012 + 1011.12 (f) 10111002 − 1001012 5. Find the base -2 representation of each of the integers from 1 up to 30. 6. Find the factorial representation of each of the integers from 1 up to 30. 7. Find a Fibonacci representation of each of the integers from 1 up to 30. 8. An increasing sequence 1, 3, 4, 9, 10, 12, 13, . . . consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Fine the 100th term of the sequence (where 1 is the first term, 3 is the second, etc.) 2 Representing Rational Numbers 9. Each column in the table provides the representation of the numbers from 1 to 12 for a certain system of enumeration. Of course, the entries in column A are decimal representations. Study the pattern, and determine the method of representation. Then replace the ?’s with the appropriate representations. A B C 1 0 1 2 1 10 3 10 100 4 2 101 100 1000 5 11 1001 6 7 1000 1010 3 10000 8 20 10001 9 10 101 10010 11 10000 10100 12 10101 12 ? ? 13 ? ? 14 ? ? 15 16 ? ? ? ? 17 ? ? 18 D 1 10 11 20 21 100 101 110 111 120 121 200 ? ? ? ? ? ? E 1 2 10 11 12 20 21 22 100 101 102 110 ? ? ? ? ? ? F 1 110 111 100 101 11010 11011 11000 11001 11110 11111 11100 ? ? ? ? ? ? 10. The floor of a real number x, denoted bxc is the largest integer that is not bigger than x. The fractional part of a real number x, denoted hxi, is defined by hxi = x − bxc. For example, 17 23 = b17 23 c + h17 23 i = 17 + 23 and −2 13 = b−2 13 c + h−2 13 i = −3 + 23 . Solve simultaneously: x + byc + hzi = 1.1 hxi + y + bzc = 2.2 bxc + hyi + z = 3.3 3