```On Thu, 28 Oct 2004 05:57:53 +0900, Jamis Buck <jgb3 / email.byu.edu> wrote:
> trans. (T. Onoma) wrote:
> > On Wednesday 27 October 2004 04:36 pm, Hal Fulton wrote:
> > | Jason DiCioccio wrote:
> > | > I'm confused now..  100.0 * 9.95 is clearly 995.  So what exactly is the
> > | > issue with floating point numbers is making this come out to
> > | > 994.999999999983 (or wahtever ;))?  If every language is plagued by this
> > | > problem, then I'd be curious to know the history behind it..
> > | >
> > | :) It's not history, it's math.
> > |
> > | Here's the short explanation.
> > |
> > | Remember repeating decimals, which you learned about in elementary school?
> > | For example, 1/3 = 0.33333... can't be expressed in a finite number of
> > | digits.
> >
> > Not exactly, '1/3' is finite.
>
> ... That's not what I learned in school. How many decimal digits does it
> take to express 1/3, then?

Reconfigure your misconceptions: decimal numeric notation has no basis
in reality. It is simply a way for humans to write down numbers.

Decimal, like binary, is a notation, a way to represent finite
numbers. It can not represent all finite numbers satisfactorily,
though; the notation has it's failings. One of the failings is shown
when representing finite rational numbers where the denominator is not
solely a multiple of powers of 2 and 5 (the factors of ten).

1/5 => 0.2
1/2 => 0.5
1/4 => 0.25
1/7 => 0.142857142857143.... (whoops)

In binary notation, you can only have finite representations of
rational numbers whose denominators are powers of two (the factors
of... two). The examples above translated into simple binary math:

1/101 => 0.0011001100110011.... (repeats)
1/10 => 0.1
1/100 => 0.01
1/111 => 0.0010010010010010.... (repeating)

To further confuse the subject. What you are representing when you
write the decimal number 42.23 is the rational number, 4223/100; aka
42 23/100, forty-two and twenty-three hundredths. That is a fraction;
a rational number. Decimal notation is simply a more convenient way of
writing it down, for making calculations or keeping records or
whatever.

In closing: Decimal numbers are made up! They aren't real! They are