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## Anis Den Ghorbal

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󰁓󰁯󰁬󰁵󰁴󰁩󰁯󰁮󰁳 󰁯󰁦 󰁴󰁨󰁥 󰁥󰁸󰁥󰁲󰁣󰁩󰁳󰁥󰁳 󰁯󰁦 󰀺    Chapter 1 Number Systems and Error 1.  Convert the following binary numbers to decimal form(1010) 2 ,  (100101) 2 ,  ( . 1100011) 2 Solution: (1010) 2  = (0 × 2 0 ) + (1 × 2 1 ) + (0 × 2 2 ) + (1 × 2 3 )= 0 + 2 + 0 + 8 = (10) 10 (100101) 2  = (1 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 )= (0 × 2 4 ) + (1 × 2 5 )= 1 + 0 + 4 + 0 + 0 + 32 = (37) 10 ( . 1100011) 2  = (1 × 2 − 7 ) + (1 × 2 − 6 ) + (0 × 2 − 5 ) + (0 × 2 − 4 )= (0 × 2 − 3 ) + (1 × 2 − 2 ) + (1 × 2 − 1 )= 1 / 2 7 + 1 / 2 6 + 0 + 0 + 0 + 1 / 2 2 + 1 / 2 = (99 / 128) 10 2.  Convert the following binary numbers to decimal form.(101101) 2 ,  (10110) 2 ,  (100111) 2 ,  (10000001) 2 Solution: (101101) 2  = (1 × 2 0 ) + (0 × 2 1 ) + (1 × 2 2 ) + (1 × 2 3 )= (0 × 2 4 ) + (1 × 2 5 )= 1 + 0 + 4 + 8 + 0 + 32 = (45) 10 (101111) 2  = (1 × 2 0 ) + (1 × 2 1 ) + (1 × 2 2 ) + (1 × 2 3 )= (0 × 2 4 ) + (1 × 2 5 )= 1 + 2 + 4 + 8 + 0 + 32 = (47) 10 (100111) 2  = (1 × 2 0 ) + (1 × 2 1 ) + (1 × 2 2 ) + (0 × 2 3 )= (0 × 2 4 ) + (1 × 2 5 )= 1 + 2 + 4 + 0 + 0 + 32 = (39) 10 (10000001) 2  = (1 × 2 0 ) + (0 × 2 1 ) + (0 × 2 2 ) + (0 × 2 3 )= (0 × 2 4 ) + (0 × 2 5 ) + (0 × 2 6 ) + (1 × 2 7 )= 1 + 0 + 0 + 0 + 0 + 0 + 0 + 128 = (129) 10 1  3.  Find the ﬁrst ﬁve binary digits of (0 . 1) 10 . Obtain values for the absolute andrelative errors in yours results. Solution:  Apply the conversion procedure as follows:0 . 1 × 20 . 2 + integer part 0 ( d − 1 ) × 20 . 4 + integer part 0 ( d − 2 ) × 20 . 8 + integer part 0 ( d − 3 ) × 20 . 6 + integer part 1 ( d − 4 ) × 20 . 2 + integer part 1 ( d − 5 )Thus(0 . 1) 10  = ( . 00011) 2 4.  Convert the following: (a)  decimal numbers to binary numbers form.165 ,  3433 ,  111 ,  2345 ,  278 . 5 ,  347 . 45 (b)  decimal numbers to hexadecimal decimal numbers.1025 ,  278 . 5 ,  14 . 09375 ,  1445 ,  347 . 45 (c)  hexadecimal numbers to both decimal and binary.1 F.C, FFF. 118 ,  1 A 4 .C,  1023 ,  11 . 1 Solution: (a) 165 = 2 × 82 + 1 , b 0  = 182 = 2 × 41 + 0 , b 1  = 041 = 2 × 20 + 1 , b 2  = 120 = 2 × 10 + 0 , b 3  = 010 = 2 × 5 + 0 , b 4  = 05 = 2 × 2 + 1 , b 5  = 12 = 2 × 1 + 0 , b 6  = 01 = 2 × 0 + 1 , b 7  = 12  Thus the binary representation for 165 is165 = ( b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 ) 2  = (10100101) 2 111 = 2 × 55 + 1 , b 0  = 155 = 2 × 27 + 1 , b 1  = 127 = 2 × 13 + 1 , b 2  = 113 = 2 × 6 + 1 , b 3  = 16 = 2 × 3 + 0 , b 4  = 03 = 2 × 1 + 1 , b 5  = 11 = 2 × 0 + 1 , b 6  = 1Thus the binary representation for 111 is111 = ( b 6 b 5 b 4 b 3 b 2 b 1 b 0 ) 2  = (1001111) 2 278 = 2 × 139 + 0 , b 0  = 0139 = 2 × 69 + 1 , b 1  = 169 = 2 × 34 + 1 , b 2  = 134 = 2 × 17 + 0 , b 3  = 017 = 2 × 8 + 1 , b 4  = 18 = 2 × 4 + 0 , b 5  = 04 = 2 × 2 + 0 , b 6  = 02 = 2 × 1 + 0 , b 7  = 01 = 2 × 0 + 1 , b 8  = 1Also0 . 5 × 20 . 0 + integer part 1 ( d − 1 )Thus the binary representation for 278 . 5 is278 . 5 = (100010110 . 1) 2 (c) (1 F.C  ) 16  =  C  (16) − 1 + F  (16) 0 + 1(16) 1 = 12 / 16 + 15 + 16 = (31 . 75) 10 Thus the decimal and binary representation for (1 F.C  ) 16  is(1 F.C  ) 16  = (31 . 75) 10  = (11111 . 11) 2 (1 A 4 .C  ) 16  =  C  (16) − 1 + 4(16) 0 +  A (16) 1 + 1(16) 2 = 12 / 16 + 4 + 160 + 256 = (420 . 75) 10 Thus the decimal and binary representation for (1 A 4 .C  ) 16  is(1 A 4 .C  ) 16  = (420 . 75) 10  = (110100100 . 11) 2 3  (11 . 1) 16  = 1(16) − 1 + 1(16) 0 + 1(16) 1 = 1 / 16 + 1 + 16 = (17 . 0625) 10 Thus the decimal and binary representation for (11 . 1) 16  is(11 . 1) 16  = (17 . 0625) 10  = (10001 . 0001) 2 5.  What is the absolute error in approximating 1 / 3 by 0 . 3333 ? What is the corre-sponding relative error ? Solution:  Let  x  = 1 / 3 = 0 . 33333 and ˆ x  = 0 . 3333, then the absolute error is Abs. Error  = | 0 . 33333 − 0 . 3333 | = 3 × 10 − 5 and the relative error is Rel. Error  = 3 × 10 − 5 0 . 33333 = 9 . 0001 × 10 − 5 Since 9 . 0001 × 10 − 5 <  12 × 10 − 4 , therefore, ˆ x  approximates  x  to 4 signiﬁcant digits. 6.  Evaluate the absolute error in each of the following calculations and hence givethe answer to a suitable degree of accuracy.( a ) 9 . 01 + 9 . 96 ,  ( b ) 4 . 65 − 3 . 429 ,  ( c ) 0 . 7425 × 0 . 7199 ,  ( d ) 0 . 7078 ÷ 0 . 87 Solution: (a)  Let  x 1  = 9 . 01 ,x 2  = 9 . 96, and  S   =  x 1  + x 2  = 18 . 97. Also, let  e 1  and e 2  be the errors in  x 1  and  x 2  respectively. Then the respective absolute errors areas follows: | e 1 |≤  12  × 10 − 2 ,  | e 2 |≤  12  × 10 − 2 and Abs. Error ≤| e 1 | + | e 2 | = 1 × 10 − 2 The relative error is Rel. Error  =  Abs. ErrorS   = (1 × 10 − 2 )18 . 97 = 5 . 2715 × 10 − 4 Thus, the result lies in the range  S  ± Abs. Error 18 . 97 ± 1 × 10 − 2 ,  or 18 . 96 ≤ S   ≤ 18 . 98The answer may be rounded meaningfully to 18 . 9 which is correct to 3 signiﬁcantdigits (sd)(1 decimal place (dp)). (c)  Let  x 1  = 0 . 7425 ,x 2  = 0 . 7199, and  P   =  x 1 x 2  = 0 . 5345. Let  e 1  and  e 2  be theerrors in  x 1  and  x 2  respectively and | e 1 |≤  12  × 10 − 4 ,  | e 2 |≤  12  × 10 − 4 4
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