The Number System

Question 7 - 20

7. What is the ​ largest positive number​ one can represent in
    a ​ 12-bit 2's complement​ code? Write your result in binary
    and decimal?

Ans: 

    Largest 12-bit binary positive = 0111 1111 1111 
    In Decimal = (1+2+4+8+16+32+64+128+256+512+1024+0) = 2047 = 2^11 -1 


8. What are the 8-bit patterns used to represent each of the
    characters in the string "​ CODE/THS 2020 " ? (Only represent
    the characters between the quotation marks.)
    **Note: There is space between ​ THS ​ and ​ 2020 ​ .

Ans: 
    Using the ASCII Table 

    CODE/THS 2020
        Dec         Binary 
    C = 67          0100 0011
    O = 79          0100 1111
    D = 68          0100 0100
    E = 69          0100 0101
    / = 47          0010 1111
    T = 84          0101 0100
    H = 72          0100 1000
    S = 83          0101 0011
    SPACE = 32      0010 0000
    2 = 50          0011 0010
    0 = 48          0011 0000
    2 = 50          0011 0010
    0 = 48          0011 0000    

    Decimal = 67 79 68 69 47 84 72 83 32 50 48 50 48 
    Binary = 01000011-01001111-01000100-01000101-00101111-01010100-01001000-01010011-00100000-00110010-00110000-00110010-00110000

    9. Perform the following additions on​ 12-bit numbers,
        generating an ​ 12-bit result.​ ​ Negative numbers​ are
        represented using two’s complement. For each addition,
        clearly indicate if unsigned and/or ​ signed overflow
        occurred or not.

        ● 1111 1111 0101
          0000 1100 1100
        ● 1101 0101 1010
          1111 1110 1010
        ● 1111 1111 0100
          1101 0111 1111
        ● 1111 1111 0011
          0011 1001 1001


      Ans. 

          ● 
           11111 1111 1       (Carry)
            1111 1111 0101
            0000 1100 1100
          -----------------
          1 0000 1100 0001
         = Signed Overflow is  1

          ● 
        1 1111 1111  1        (Carry)  
          1101 0101 1010
          1111 1110 1010
        -----------------
        1 1101 0100 0100
        = Signed Overflow is  1

          ● 
          1 1111 1111 1        (Carry)
            1111 1111 0100
            1101 0111 1111
          -----------------
          1 1101 0111 0011
        = Signed Overflow is  1

          ● 

            1111 1111 0011
            0011 1001 1001
          -------------------
          1 0011 1000 1100
        = Signed Overflow is  1

    10. (1101)​2​ × (101)​2

    Ans: 
            1101                    1101 = 1+0+4+8 = 13
             101 *                   101 = 1+0+4 =    5   
           -------                                  ----                                
          111   (carry)                             65
            1101
           0000| +
          1101|| +
         ---------
        01000001  = 1+64 = (65)10
        
    
    11. (1001)​2​ ÷ (101)​2

    Ans:  
                    1    (Borrow)        101 = 1+0+4 = 5
            101 )  1001 ( 1             1001 = 1+0+0+8=9 
                    101                                9/5 = 
                   ----
                   0100
                   


        (1001)​2​ ÷ (101)​2 = (1)2 

    12. What is the biggest binary number you can write with 5 bits?

    Ans: 11111 = 1+2+4+8+16 = 2^5 -1 = 31

    13. What is the biggest binary number you can write with
        n bits?
    Ans: 2^n -1 the max no 

    14. Which fractions recur infinitely in binary and which terminate?
    
    Ans: if demoninator of fraction in lowest form is power of 2 then it terminates, else not.

    15. In  hex, 2BFC + 54AF ?
  
    Ans : 2BFC = (11260)10 = (10101111111100)2
          54AF = (21679)10 = (101010010101111)2
			   
             111 1111 1111 1  (Carry) 
		         010 1011 1111 1100
           + 101 0100 1010 1111
          ----------------------
            1000 0000 1010 1011 

            = 32768+128+32+8+2+1 = (32939)10


          16  32939 
              2058  11  = B
              128   10  = A
              8     0   

              = (80AB)16
        2BFC + 54AF = 80AB


    16. If a number has ​ k ​ digits in hex, how many digits (bits) does it have in ​ binary​ ?
		
      		
    Ans : k/4

    17. Convert the binary number ​ 1101101111110101 to hex ?

    Ans : 
            1101 1011 1111 0101 
      
            = 32768+16384+4096+2048+512+256+128+64+32+16+4+1
            = (56309)10

            16  56309 
                3519  5
                219   15
                13    11

        1101 1011 1111 0101  = (DBF5)16

     18. Convert the hex number ​ ABC7 to binary?

      Ans. 
      
           ABC7 
           = 10*16^3 + 11*16^2 + 12*16 + 7*1 
           = (43975)10

           2  43975 
              21987 1
              10993 1
              5496  1
              2748  0
              1374  0
              687   0
              343   1
              171   1
              85    1
              42    1
              21    0
              10    1
              5     0
              2     1
              1     0

             (ABC7)16 = (1010 1011 1100 0111)2
                 

      19. In hex, ​ AC74 − B3F?

      Ans: 

            AC74 = (10 × 16³) + (12 × 16²) + (7 × 16¹) + (4 × 16⁰) 
                 = (44148)10
                 = (1010 1100 0111 0100)2

            B3F = (11 × 16²) + (3 × 16¹) + (15 × 16⁰)
                = (2879)10
                = (0000 1011 0011 1111)2

                      011  000 1011 (Borrow)
                1010 1100 0111 0100
              - 0000 1011 0011 1111
            -------------------------
                1010 0001 0011 0101

                = (41269)10 
                = (A135)16
      
      20. Convert the following binary fractions to ordinary fractions
          
          ● 0.1001
          ● 1.0011
          ● 1.1111


      Ans: 
          A. 0.1001
            = 0*2^0 + 1*2^-1 + 0 + 0 + 1*2^-4
            = (0.5625)10

          B. 1.0011
            = 1 + 1*2^-3 + 1*2^-4
            = (1.1875)10

          C. 1.1111  
            = 1 + 1*2^-1 + 1*2^-2 + 1*2^-3 + 1*2^-4
            = (1.9375)10