Definition of Floating Point Data

Date: 07/02/2001 at 19:53:01
From: Judy Strasbaugh
Subject: Definition of 'floating point' data?

What are 'floating point data'? How do they differ from an integer? 
What are some examples?

I looked up both 'floating point' and 'fixed point' in Webster's 
Dictionary, but the definitions do not provide examples that are 
meaningful to me.

Date: 07/02/2001 at 23:11:31
From: Doctor Peterson
Subject: Re: Definition of 'floating point' data?

Hi, Judy.

The Dr. Math archives include these answers:

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

   Fixed Point and Floating Point Numbers
   http://mathforum.org/dr.math/problems/huband.5.19.00.html   

I'll try to give a more basic explanation that is as clear as 
possible.

What you will see in a computer program (as opposed to what the 
computer itself sees) will look like this (an example in the C 
language):

    int a;
    float b;
    a = 12;
    b = 12.0;

I've told the computer that "a" is an integer and "b" is a floating 
point number. (Some languages would call the latter "real," which is a 
bit of a misnomer.) This tells the computer (actually, the compiler 
that turns my program into something the computer can read) to find a 
place to store the variable "a" as an integer, and a place to store 
"b" using floating point format. Then I told it to store the integer 
12 in a, and the decimal number 12.0 in b.

As a user, all you have to know is that an integer can hold a 
(positive or negative) whole number; a floating point variable can 
store a decimal value.

Internally, the computer will store the following binary number in a:

    00000000 00000000 00000000 00001100

This is 12 in base 2; it means 1*2^3 + 1*2^4. (I'm using "^" to 
indicate an exponent, and "*" for multiplication.)

In b, it will store

    01000001 01000000 00000000 00000000

This encodes the value

    +1.5 * 2^3 = 1.5 * 8 = 12

What it's done is to put 12 in the binary equivalent of scientific 
notation, by dividing 12 by 2 until the quotient is between 1 and 2, 
and then writing it as the product of this quotient, 1.5 (called the 
mantissa or significand), and a power of 2, namely 2 cubed, or 8. 
It's hard to see either 1.5 or 3 in there, but the first answer I 
pointed out in our archives will explain how it's encoded if you want 
to know. The important thing is just that it is encoded in a way that 
the computer can work with; otherwise, programs would be restricted to 
integers.

I used the same number in this example; if I had used 12.3, I would 
have been able to store that (correctly) only in floating point form. 
In an integer variable, only "12" would have been stored. Note, 
however, that 12 is valid floating point data; the difference is in 
how it is stored.

When data are stored in integer form, they use space as efficiently as 
possible; the range of values that can be stored in 32 bits (the usual 
size currently) is from -2,147,483,648 to +2,147,483,647. The value is 
stored exactly, allowing up to 10 digits of precision (decimal). The 
problem, of course, is that only whole numbers can be used.

When data are stored in floating point format, there is somewhat less 
precision, because some bits are used to store the exponent, leaving 
fewer to store the actual value of the mantissa. However, the range of 
numbers that can be stored is much greater: from -3.4028E+38 to 
3.4028E+38, either positive or negative, and numbers as small as 
1.1754E-38 can be stored with just as much precision. It is not exact, 
however, but holds only about 7 significant digits (decimal).

There are other integer and floating point formats in use, such as 64 
bits; the same principles still apply, but with different numbers.

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

Date: 07/03/2001 at 12:37:37
From: Strasbaugh, Judith
Subject: Re: Definition of 'floating point' data?

Thanks so much.  You've written the explanation so well, I actually
understand it!

Judy Strasbaugh