2000. Convert the following Binary numbers (1 bit sign, 15 bit value) to Decimal, Octal and Hex

          a.       0,110010111011111      

          b.       0,001101110111011

          c.       1,010100111001100          (One’s Complement)

          d.       1,111111111111010          (Two’s Complement)

2.       a.       Decimal:

          0,110010111011111₂ = 2¹⁴        + 2¹³      + 2¹⁰      + 2⁸    + 2⁷    + 2⁶   + 2⁴ + 2³ + 2² + 2¹ + 2⁰

                                             = 16,384 + 8,192 + 1,024 + 256  + 128 + 64 + 16 + 8 + 4   + 2  + 1

                                             = + 26,079₁₀

          Octal:

                                                                            Chk:       6 * 8⁴ + 2 * 8³  + 7 * 8²  + 3 * 8¹      + 7 * 8⁰

+           6           2          7          3          7 =  6 * 4,096   + 2 * 512 + 7 * 64 + 3 * 8 + 7 * 1

= 24,576          + 1,024        + 448         + 24           + 7

          0,         110        010     111      011     111₂ = 26,079₁₀

          Hexadecimal:

                                                                    Chk:  6 * 16³ + 5 * 16² + D = 13 * 16¹   + F = 15 * 16⁰

+           6            5             D         F = 6 *  4,096 + 5 * 256 + 13 * 16 + 15 * 1

=           24,576    + 1,280    + 208      + 15

          0,        110      0101      1101     1111₂ =   26,079₁₀

2.       b.       Decimal:

          0,001101110111011   = + 2¹² + 2¹¹ + 2⁹ + 2⁸ + 2⁷ + 2⁵ + 2⁴ + 2³ + 2¹ + 2⁰

                                             = + 4,096 + 2,048 + 512 + 256 + 128 + 32 + 16 + 8 + 2   + 1

                                             = + 7,099₁₀

          Octal:

+        1                    5                6          7    3 Chk:   1 * 8⁴ + 5 * 8³ + 6 * 8² + 7 * 8¹ + 3 * 8⁰

= 1 * 4,096 + 5 * 512 +  6 * 64 + 7 * 8 + 3 * 1

          0,     001        101      110      111      011           =  4,096 +  2,560      + 384 + 56 + 3

= + 7,099₁₀

          Hexadecimal:

+        1                       B         B                  B Chk:              = 1 * 16³ + B=11* 16² + D=13 * 16¹ + B=11 * 16⁰

= 1 * 4,096   + 11 * 256 + 11* 16 + 11 * 1

          0,     001     1011    1011    1011                        = 4,096 + 2,816    + 176+ 11

                                                                                                                = +7,099

2.       c.       Decimal:

          Since negative, we must first complement:      1,010100111001100 ® 0,101011000110011

          101011000110011 = 2¹⁴ + 2¹² + 2¹⁰ + 2⁹ + 2⁵ + 2⁴ + 2¹ + 2⁰

                                         = 16,384 + 4,096 + 1,024 + 512 + 32 + 16 + 2 + 1

                                         = -22,067

          Octal:

5       3       0         6       3 Chk:    = 5 * 8⁴           + 3 * 8³       + 6 * 8¹   + 3

            = 5 * 4,096     + 3 * 512    + 6 * 8    + 3

          101    011   000   110    011                                     = 20,480          + 1,536        + 48         + 3

= -22,067

          Hexadecimal:

Chk:    = 1 * 16³         + 6 * 16²     + 3 * 16¹ + 3

5           6            3           3 = 5 *  4,096    + 6 *  256   + 3 * 16    + 3

= 20,480          + 1,536        + 48           + 3

          101       0110     0011    0011                                   = -22,067

2.d.    Decimal:

Since the binary number:       1,111111111111010  has the value ‘1’ in the left-most position, the number is negative and we must complement the right 15-bits AND add ‘1’ to the complemented result:

         000000000000101

+                         1

         000000000000110₂ =  - 6₁₀ = - 6₈ = - 6₁₆

          Are we sure??  As a check, let’s store the value –6₁₀.

          6 / 2  =  3           6 % 2  =  0            =  110₂  = 0000000000000110₂  on 16-bits

          3 / 2  =  1           3 % 2  =  1

          1 / 2  =  0           1 % 2  =  1

         Since the value is negative (and stored as a two’s complement), we must complement and add 1 to the complement:

                   0000000000000110              The Value +6

                   1111111111111001              One’s Complement

+                     1                  Add 1

                   1111111111111010              Two’s Complement