Floating-Point Binary Fractions

Date: 07/19/99 at 15:05:21
From: Amad Mairaj
Subject: Computers and Math

I have a question. Since computers think in terms of "1" and "0" how 
do they calculate and represent fractions such as 12.93?

How can you represent this number in binary? For example how would 
123.99 + 12.938 be added in binary? I am confused and would appreciate 
clarification.

Regards,
Amad

P.S. Love your website, best site in the WORLD! 

Date: 07/19/99 at 17:12:03
From: Doctor Peterson
Subject: Re: Computers and Math

Hi, Amad.

You can find some information about this sort of thing in our FAQ on 
bases; look for discussions of fractions:

   http://mathforum.org/dr.math/faq/faq.bases.html   

There are two ways to represent non-integer numbers in binary. One is 
called "fixed-point"; you just pretend there's a "binary point" at a 
certain place in the number. Then, for example, "10101" would 
represent

     101.01 = 5.25 (1*4 + 0*2 + 1*1 + 0*1/2 + 1*1/4).

More commonly today, numbers are stored in "floating point," where 
some of the bits are used to indicate where the binary point is. This 
is similar to scientific notation; 5.25 might be represented as 
something like "101010e0010", which would mean 1.0101 * 2^2. This is 
somewhat simplified, but should give you a general idea of how it 
works. If you'd like more details, write back!

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

Date: 07/21/99 at 16:48:28
From: (anonymous)
Subject: Re: Computers and Math

FPU or Floating Point Unit), hmmm... Can you clarify what 
'floating-point' means, in easy-to-understand terms? And how does it 
work with respect to fractions? Computers without FPUs, such as early 
286s, were also able to do fractions. How did they differ from CPUs 
with FPUs?

Thanks,
Amad   

Date: 07/21/99 at 23:25:23
From: Doctor Peterson
Subject: Re: Computers and Math

Hi, Amad.

Yes, the FPU is a part, in or alongside the CPU, that does 
floating-point computations. In early hardware, the CPU could only 
work with integers, so these calculations had to be done explicitly by 
software, which would use ordinary bit-manipulation instructions to 
multiply, convert to integers, and so on. Later, separate chips were 
made that would communicate with the CPU, accepting requests to do 
these calculations on their own, and eventually they were integrated 
into the CPU chip itself. Each step made floating-point calculations 
faster.

Floating point is essentially the binary equivalent of scientific 
notation. It simply means that we store the digits of the number and 
the location of the decimal (or binary) point separately, so that the 
point "floats," allowing us to represent numbers of different sizes 
without having to put the same number of digits after the point in 
every number.

In scientific notation in base ten, we first scale a number so that it 
is between 1 and 10, and then multiply that by the appropriate power 
of ten:

     123.45 = 1.2345 * 10^2 = 1.2345e2

In binary, we can write a number as the product of a number between 1 
and 2, and a power of two:

     5.25 base 10 = 101.01 = 1.0101 * 2^2 = 1.0101e10

(The exponent 2 is 10 base 2; each power of two shifts the binary 
point right one place.)

A computer works with such numbers the same way we work with 
scientific notation. To multiply, you multiply the fraction parts and 
add the exponent parts. To add, you have to convert them to use the 
same exponent, then add the fraction parts.

As I indicated in general terms, a floating-point number is stored as 
a string of bits, say 32 of them, with each set of bits assigned a 
different purpose. The first bit is typically a sign bit, which is set 
(1) to indicate a negative number. Then several bits (8 in a 
single-precision number) store the exponent, with 127 added to it for 
reasons I won't go into.

Finally, the rest of the bits (23 in single-precision) are used to 
store the fractional part; since this is always between 1 and 2, the 
ones bit is always 1 and doesn't have to be stored. So our number 5.25 
looks like this:

     S|E E E E E E E E|F F F F F F F F F F F F F F F F F F F F F F F
     0|1 0 0 0 0 0 0 1|0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     | \_____________/ \___________________________________________/
     |        |                             |
  positive   exp 2+127 = 129              frac 1.0101 -> 0101

If you're interested in details, here are some of many sites that tell 
about the standard floating point format:

IEEE floating point representations of real numbers - John David Stone
http://www.math.grinnell.edu/~stone/courses/fundamentals/IEEE-reals.html   

IEEE Standard 754 Floating Point Numbers - Steve Hollasch
http://research.microsoft.com/~hollasch/cgindex/coding/ieeefloat.html   

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