Chapter 2 6. (a) (b) (c) (d) (e) (f) (g) (h) 1110 = 1 × 23 + 1 × 22 + 1 × 21 = 8 + 4 + 2 = 14 1010 = 1 × 23 + 1 × 21 = 8 + 2 = 10 11100 = 1 × 24 + 1 × 23 + 1 × 22 = 16 + 8 + 4 = 28 10000 = 1 × 24 = 16 10101 = 1 × 24 + 1 × 22 + 1 × 20 = 16 + 4 + 1 = 21 11101 = 1 × 24 + 1 × 23 + 1 × 22 + 1 × 20 = 16 + 8 + 4 + 1 = 29 10111 = 1 × 24 + 1 × 22 + 1 × 21 + 1 × 20 = 16 + 4 + 2 + 1 = 23 11111 = 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 = 16 + 8 + 4 + 2 + 1 = 31 7. (a) 110011.11 = 1 × 25 + 1 × 24 + 1 × 21 + 1 × 20 + 1 × 2−1 + 1 × 2−2 = 32 + 16 + 2 + 1 + 0.5 + 0.25 = 51.75 101010.01 = 1 × 25 + 1 × 23 + 1 × 21 + 1 × 2−2 = 32 + 8 + 2 + 0.25 = 42.25 1000001.111 = 1 × 26 + 1 × 20 + 1 × 2−1 + 1 × 2−2 + 1 × 2−3 = 64 + 1 + 0.5 + 0.25 + 0.125 = 65.875 1111000.101 = 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 2−1 + 1 × 2−3 = 64 + 32 + 16 + 8 + 0.5 + 0.125 = 120.625 1011100.10101 = 1 × 26 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 2−1 + 1 × 2−3 + 1 × 2−5 = 64 + 16 + 8 + 4 + 0.5 + 0.125 + 0.03125 = 92.65625 1110001.0001 = 1 × 26 + 1 × 25 + 1 × 24 + 1 × 20 + 1 × 2−4 = 64 + 32 + 16 + 1 + 0.0625 = 113.0625 1011010.1010 = 1 × 26 + 1 × 24 + 1 × 23 + 1 × 21 + 1 × 2−1 + 1 × 2−3 = 64 + 16 + 8 + 2 + 0.5 + 0.125 = 90.625 1111111.11111 = 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 + 1 × 2−1 + 1 × 2−2 + 1 × 2−3 + 1 × 2−4 + 1 × 2−5 = 64 + 32 + 16 + 8 + 4 + 2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 = 127.96875 (b) (c) (d) (e) (f) (g) (h) 8. (a) (c) (e) (g) (i) 22 − 1 = 3 24 − 1 = 15 26 − 1 = 63 28 − 1 = 255 210 − 1 = 1023 9. (a) (b) (c) (d) (e) (f) (g) (h) (24 − 1) < 17 < (25 − 1); 5 bits (25 − 1) < 35 < (26 − 1); 6 bits (25 − 1) < 49 < (26 − 1); 6 bits (26 − 1) < 68 < (27 − 1); 7 bits (26 − 1) < 81 < (27 − 1); 7 bits (26 − 1) < 114 < (27 − 1); 7 bits (27 − 1) < 132 < (28 − 1); 8 bits (27 − 1) < 205 < (28 − 1); 8 bits (b) (d) (f) (h) (j) 23 − 1 = 7 25 − 1 = 31 27 − 1 = 127 29 − 1 = 511 211 − 1 = 2047 8