Assignment 4

Solutions to Problems on Information Representation

1. Convert each of the following binary numbers into decimal (base 10).
   1. 1000000

      Note that the rightmost bit represents 2⁰, the bit to the left of that represents 2¹, the third bit from the right represents 2², and so on. In this case, there is only a single 1 at the 2⁶ positions. Note that 2⁶ = 64, which is the decimal value of this bit pattern.

   2. 1101010

      The 1's in this bit pattern are at positions (starting from the right) 2¹, 2³, 2⁵, and 2⁶. Thus, the decimal value of the binary number is 2⁶+2⁵+2³+2¹ = 64 + 32 + 8 + 2 = 106.

   3. 10111011

       27 +  25 + 24 + 23 + 21 + 20 = 
      128 + 32 + 16 +  8 +  2 +  1 = 
      187.
      

   4. 101011001

       28 +  26 + 24 + 23 + 20 = 
      256 + 64 + 16 + 8 +  1 = 
      345.
      

2. Give the binary representation for each of the following decimal numbers.
   1. 23

      Note that 23 is the sum of the following powers of 2: 23 = 16 + 4 + 2 + 1, or equivalently: 23 = 2⁴ + 2² + 2¹ + 2⁰. So, the binary representation of 23 is the bit pattern: 1 0 1 1 1.

   2. 47

      47 = 32 + 8 + 4 + 3 + 2 + 1. So, binary representation is: 1 0 1 1 1 1.

   3. 64

      64 = 2⁶. Binary representation: 1 0 0 0 0 0 0.

   4. 109

      109 = 64 + 32 + 8 + 4 + 1. Binary representation: 1 1 0 1 1 0 1.

3. For the each of the following if the number is in binary, give its Hexadecimal representation; and if the number is in Hex, give its representation in binary.
   1. 1011011010101101

      To convert to Hex, starting from the right divide bit pattern into groups of 4 bits (if the leftmost set of bits is less than 4, then add some leading 0's to make it 4 bits). Then convert each group of 4 bits into its binary equivalent.

      1011 0110 1010 1101
        B    6    A    D
      

   2. 1011110111

      0010 1111 0111
        2    F    7
      

   3. D37A

        D    3    7    A
      1101 0011 0111 1010
      

   4. 6BC0E2

        6    B    C    0    E    2
      0110 1011 1100 0000 1110 0010
      

4. GIF images can have up to 256 colors (note that 256 is 2⁸, so 8 bits are required to represent all of the 256 different colors). Each pixel in the image can be any of the available colors. Suppose that we have a GIF image with a size of 320x240 pixels. How many bytes is required to store this image?

   The total number of pixels in the image is 320 x 240 = 76800. Each pixel requires 8 bits (or one byte). Thus, to store the image, we need 76800 bytes which is 75K bytes.