How precise is a computer?

kgun · #1 (**permalink**) 03-08-2008, 08:26 AM

The question in the heading can be divided in two.
1. How precise is a n-bit's computer?
2. How precise can you make software (simulations) to run on a computer?

I thought of the first question when I formulated this thread, but you are free to discuss the second. The heading of the thread could also be: "Pure math and computers, be careful."

An example:
The area of a circle is:

2 x Pi x radius x radius.

So the SQRT of the area is:

radius x SQRT(2 x Pi).

That is analytically (in pure math) there is no problem "squaring the circle."

Take a square with side lenght:

radius x SQRT(2 x Pi).

The square has area: 2 x Pi x radius x radius = The area of the circle Q.E.D.

But you are not able to square the circle on a computer, since you are not able to compute Pi, since Pi is a trancendental number.

Note:
Most mathematical functions implemented in comuter software, are not pure mathematical functions. Most often functions like the logarithmic function are represented with the first few Taylor series terms of the function, and that is as imprecise as saying that:

Pi = 3.14. It is not!

Related link:

http://www.sciencedaily.com/releases/2007/06/070614104042.htm

Conclusion: Pure math and computers, be careful.

cw1865 · #2 (**permalink**) 03-08-2008, 11:49 AM

Check out the normal distribution. There is no integral for the function! This one I remember quite clearly. For purposes of approximating the area under the curve by and large you rely on prepared tables. But they show you how they got the tables to begin with, essentially taking small rectangles and computing the area (which of course has an inherent error associated with it!) - nevertheless the professor demonstrated that at some point the computer, in an attempt for GREATER accuracy, lost the ability to calculate it (because the rectangle became SOOOO SMALL that the computer was returning an area of 0 for that particular rectangle); hence the computer would add 0 to the cumulative area and of course 0 is a terrible approximation of the area under the curve.....

Cool function, I am sure that you would like it....

Yes, PI, that crazy non-decimal number! The only way is to have a numerical system itself based on PI (which humans wouldn't like of course)

activeco · #3 (**permalink**) 03-08-2008, 02:40 PM

Quote:

Originally Posted by kgun

How precise can you make software (simulations) to run on a computer?

In the first place it depends on programming language and it's computing precision, which is often an arbitrary function.
In theory (and practice) you can work with numbers as big as your computer's memory plus hard disk size combined.

Arbitrary-precision arithmetic provides much slower computing than regular default precision, but if one really needs an extreme precision, it is possible to achieve it.
Whenever I need more precision for my small projects and ideas, I usually use Rexx (regina) language where I can define the precision with "NUMERIC DIGITS=" built-in function.

Jean-Luc · #4 (**permalink**) 03-08-2008, 03:39 PM

Quote:

Originally Posted by kgun

The area of a circle is:

2 x Pi x radius x radius.

Other mathematicians claim that the area of the circle is Pi x radius x radius....

Jean-Luc

kgun · #5 (**permalink**) 03-08-2008, 05:25 PM

Quote:

Originally Posted by Jean-Luc

Other mathematicians claim that the area of the circle is Pi x radius x radius....

Jean-Luc

One rep point since you discovered that. I was away from the computer visiting my daughter and reckognised the error some hours ago.

And, I am not a mathematician.

The precision of an n bit computer is (1/2)^n.

That means that the precision of a 32 bit computer is (1/2)^32

kgun · #6 (**permalink**) 03-08-2008, 06:08 PM

Quote:

Originally Posted by cw1865

Check out the normal distribution.

The Normal distribution is well behaved since the first and second order moments exist, that is the mean and the variance is finite.

But the Cauchy distribution that can seemingly fit the Normal distribution perfectly has an infinite variance. The tails of the Cauchy distribution is too heavy. The kurtosis is too high.