Converting to Binary

Date: 05/27/2001 at 05:31:17
From: maya karipel
Subject: Binary numbers

Dear sir,
 
I would like to know how to convert the number .78125 to its binary 
equivalent.
 
Thanking you,
 
Yours faithfully,
Maya A.Karipel

Date: 06/05/2001 at 15:06:30
From: Doctor Twe
Subject: Re: Binary numbers

Hi Maya - thanks for writing to Dr. Math.

To convert a whole number into binary you can repeatedly divide by 2, 
keeping the remainders. See for example,

   Converting Binary to and From Decimal
   http://mathforum.org/dr.math/problems/shroff.6.16.99.html   

To convert a fraction into binary, we can repeatedly multiply by 2, 
keeping the integer part. After each multiplication, the integer part 
becomes the next binary digit (left to right), and the fractional part 
gets multiplied by 2 again. We can continue until the fractional part 
is zero, or until we have as many digits as we desire. For example, to 
convert 0.59375 (base 10) into binary we multiply:

                                                  .1 0 0 1 1
                                                  / / / / /
     .59375 * 2 = 1.1875   keep the 1 -----------+ / / / /
                    \__/                          / / / /
        /------------/                           / / / /
       v                                        / / / /
     .1875  * 2 = 0.375    keep the 0 ---------+ / / /
                    \_/                         / / /
        /-----------/                          / / /
       v                                      / / /
     .375   * 2 = 0.75     keep the 0 -------+ / /
                    \/                        / /
        /-----------/                        / /
       v                                    / /
     .75    * 2 = 1.5      keep the 1 -----+ /
                    \/                      /
       /------------/                      /
      v                                   /
     .5     * 2 = 1.0      keep the 1 ---+

Since the fractional part of the last multiplication is 0, we can stop 
and we have an exact answer. (Any additional digits would simply be 
0.) So our answer is 0.59375 (base 10) = 0.010011 (base 2).

We can check this by converting the binary back to decimal. The 
fractional place values in binary are:

        1     1     1     1     1
     . ---   ---   ---   ---   --- ...
       2^1   2^2   2^3   2^4   2^5
or:
       1   1   1    1    1
     . -   -   -   --   -- ...
       2   4   8   16   32
or:
     . (.5)   (.25)   (.125)   (.0625)   (.03125) ...

So 0.10011 (base 2) is:

       1 * .5     = .5
     + 0 * .25    = .0
     + 0 * .125   = .0
     + 1 * .0625  = .0625
     + 1 * .03125 = .03125
                   --------
                    .59375 (base 10)

So it double-checks. Notice that most fractions that terminate in a 
decimal will be repeating fractions in binary. For example, let's 
convert the fraction 0.6 (base 10) to binary:

                                   .1 0 0 1 1 0 0 1 ...
                                   / / / / / / / /
     .6 * 2 = 1.2   keep the 1 ---+ / / / / / / /
     .2 * 2 = 0.4   keep the 0 ----+ / / / / / /
     .4 * 2 = 0.8   keep the 0 -----+ / / / / /
     .8 * 2 = 1.6   keep the 1 ------+ / / / /
     .6 * 2 = 1.2   keep the 1 -------+ / / /
     .2 * 2 = 0.4   keep the 0 --------+ / /
     .4 * 2 = 0.8   keep the 0 ---------+ /
     .8 * 2 = 1.6   keep the 1 ----------+

Notice that the second four lines repeat the first four, and the next 
four would repeat them, and so on. Thus, our answer is:

     0.6 (base 10) = 0.10011001... (base 2)

For some more general information on converting fractions to other 
bases, check out:

   Fraction/Decimal Conversion to Other Bases
   http://mathforum.org/dr.math/problems/trinidad8.12.98.html   

For information on how computers represent fractions in binary, see:

   Floating-Point Binary Fractions
   http://mathforum.org/dr.math/problems/mairaj.7.19.99.html   

both from our Ask Dr. Math archives.
 
I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/