Fractional Numbers

• Remember in a positional number system with base b.
  • $d_1d_2d_3.d_4d_5d_6 = d_1 \times b^2 + d_2 \times b^1 + d_3 \times b^0 + d_4 \times b^{-1} + d_5 \times b^{-2} + d_6 \times b^{-3}$
• We will need to convert between fractions in different bases.
• Convert .1875 to binary
  • A brute force method is to continue to subtract negative powers of two until we reach 0 or the accuracy we want.

    $2^{-1} = 1/2$ or .5
       This is larger than .1875 so record .0
    
    $2^{-2} = 1/4$ or  .25
       This is larger than .1875 so record .00
    
    $2^{-3} = 1/8$ or  .125
       This is smaller than or equal to  .1875 so record .001
       Subtract  .1875
               - .125
                 ------
                 .0625
    
    $2^{-4} = 1/16$ or  .0625
       This is smaller than or equal to  .0625 so record .0011
       Subtract  .0625
               - .0625
               -------
                     0
    
    $.1875 = .0011_2$ 

• A second method is to multiply by 2 and record any 1s

  .1875 x 2 = .375   record 0
  .375 x 2 = .75     record 0
  .75 x 2  = 1.5     record 1
  .5 x 2 = 1.0       record 1 stop, read down
  
  $.1875 = .0011_2$
      

• We can process other bases the same way.
  • To decimal:
    • Just expand the digit * weight
  • Convert $.1452_7$ to decimal
    • $1 \times 7^{-1} + 4 \times 7^{-2} + 5 \times 7^{-3} + 2\times 7^{-4} = .2399$
  • From decimal
    • Multiply by the base
    • The integer part becomes the digit
  • Convert .6012 to base 15

    • .6012 * 15 = 9.018    so record .9
      .018 * 15 =  0.27     so record .90
      .27 * 15 =  4.05      so record .904
      .05 * 15 = 0.75       SO record .9040