ENEE 244 (01**). Spring 2006 

Homework 1 solutions

1.  Convert the following binary numbers to decimal: (a) 1101; (b) 10111001.

    (a) 1101 = 1.23 + 1.22 + 0.21 + 1.20 = 8 + 4 + 0 + 1 = (13)10.

    (b) 10111001 = 1 + 8 + 16 + 32 + 128 (written in backwards order) = 185.

 

2.  Convert the following decimal numbers to binary: (a) 23; (b) 498.

    (a) By repeated division by radix 2:

          23
          11    1
            5    1
            2    1
            1    0
            0    1

            Answer from above, bottom to top: (10111)2 .

    (b) By repeated division, as above (steps not shown) = (111110010)2.

 

3. Convert the following numbers to decimal: (a) (34)7; (b) (76)8.

    (a) (34)7= 3.71 + 4.70 = 21 + 4 = 25.

    (b) (76)8 = 7.81 + 6.80 = 56 + 6 = 62.

 

4. Convert (a) (47)10 to radix 8; (b) (63)10 to base 16.

    (a) By repeated division by radix 8:

            47
              5    7
              0    5

              Answer from above, bottom to top: (57)8.

    (b) By repeated division by radix 16:

            63
              3    F
              0    3

              Answer from above, bottom to top: (3F)16     /* Also written as 0x3F */

 

5. Convert (a) (11010110111)2 to hexadecimal; (b) (4532)8 to binary.

    (a) Grouping into groups of 4 bits from right: 0x6B7.

    (b) Converting each digit separately: (100 101 011 010)2.

 

6. Convert the following decimal numbers to binary: (a) 0.75; (b) 5.3.

    (a) 0.75 X 2 = 1 + 0.5
          0.50 X 2 = 1 + 0.0

            Answer from above, top to bottom: (0.11)2.

    (b) 5 is 101.

            0.3 X 2 = 0 + 0.6
            0.6 X 2 = 1 + 0.2
            0.2 X 2 = 0 + 0.4
            0.4 X 2 = 0 + 0.8
            0.8 X 2 = 1 + 0.6    /* We stop here since 0.6 repeats */

            Answer: (101.01001)2 /* Last four digits of fraction repeat forever */
 

7. Add the binary numbers (a) 0101 and 1001; (b) 01101.01 and 1001011.101.

    (a)    01101
          +10  01
             11 10

    (b)       101111101.01
         +10  0  1  0  11.101
            10 1  1  0  00.111

 

8. Do the following subtractions: (a) 1100 - 1001; (b) 101101 - 11010.

    (a)   Answer: 0011.  (See book for workings for similar problems).

    (b)   Answer: 010011.
           

9. Multiply: 10100101.

        10100
            101
        10100
      00000
    10100
    1100100  /* Answer obtained by adding preceding three rows */

 

10.  Find the diminished radix complement of (a) (76)8; (b) (10010)2.

    (a) 7's complement = 77-76 = (01)8.

    (b) 1's complement = (01101)2.  /* Obtained by switching all the bits */

 

11.  Find the radix complement of (445)10.

             9's complement = 999-445 = 554.
     
10's  complement = 554 + 1 = 555.

 

12.  Represent the following binary numbers in an 8-bit signed-magnitude scheme: (a) +1001; (b) -1011.

    (a) 00001001.

    (b) 10001011.    /* First bit is sign bit */

 

13.  A 6-bit computer uses the signed 2's complement representation.  It adds 10011 + 11101.  State the answer.  How is the overflow detected?   /* Hint: In such a representation, negative numbers are stored as 2's complement of their absolute value. Positive numbers are represented just like in the sign-magnitude scheme. */

       10111010111
     + 0  1  1  1  01
        1  1  0  0  00

        Since carry into sign bit (1) ≠ carry out of sign bit (0) Overflow is detected.
 

14.  An 8-bit computer uses the signed 2's complement representation.  Show how this computer performs (a) 111010 - 1100; (b) 101 - 10110.  

     (a) 111010 - 1100 = 111010 + (-1100)
          -1100 = 2's complement of 00001100 = 11110011+ 1= 11110100
           Expression = 00111010 + 11110100 = 100101110.
           Discarding carry out of sign bit (left most digit) = 00101110.  /*Answer */
           In doing this calculation, carry into sign bit (1) = carry out of sign bit (1)
No overflow; answer correct.

      (b) 00000101 + (-00010110)
            -00010110 = 2's complement of 00010110 = 11101001 + 1 = 11101010.
            Expression = 00000101 + 11101010 = 11101111. /* Nothing to discard, so this is answer */
            In doing this calculation, carry into sign bit (0) = carry out of sign bit (0)
No overflow; answer correct.

15.  A 7-bit computer uses the signed 2's complement representation. What is the range of integers that can be represented in this scheme?

        Range is [-(26) to +26- 1] = [-64 to +63].