A couple articles, and an argument based on *Binary*

Articles on the subject:

"Numbers such as 0.5 are sometimes regarded as repeating decimals since .5 = .50000... = .4999..."
( http://mathworld.wolfram.com/RepeatingDecimal.html )

.9999... is a repeating decimal, it is rational, which means it is expressible in a ratio. The ratio for .999... happens to be 1 / 1, or 1.


"... any nonzero number with a finite decimal expression (a decimal fraction) can be written in a second way as a recurring decimal. For example 3/4 = 0.75 = 0.750000000... = 0.74999999..."
( http://en.wikipedia.org/wiki/Repeating_decimal#The_case_of_0.99999... )



Also, let's not forget that when terminating decimals are converted between bases they can become infinite. As a simple example, which programmers should be somewhat familiar with, take the decimal value .1. In binary, .1 is expressed as 0.000110011001100... That's why you lose precision in the simple QB program:

FOR i = 1 TO 20
x = x + .1
PRINT x,
NEXT

The output from this program is:

  .1          .2          .3          .4          .5
  .6          .7          .8000001    .9000001   1
 1.1         1.2         1.3         1.4         1.5
 1.6         1.7         1.8         1.9         2

although, without considering base conversion, one would expect:

  .1      .2      .3      .4      .5
  .6      .7      .8      .9     1
 1.1     1.2     1.3     1.4     1.5
 1.6     1.7     1.8     1.9     2

The reason for the small foul-ups in the .8 and .9 values is because QBASIC needs to think in binary, and when it converts the .1 to binary it ends up with an unterminated decimal that it must terminate somewhere in order to conserve memory. So what it ends up doing is adding .100000000001 each time, rather than .1. I'm sure everyones run into this strange problem at some point in time.

Anyway, saying that 1 - .999... doesn't = 0 is like saying in binary 1/1010 - .000110011001100... doesn't = 0, when in fact, 1/1010 and .000110011001100... are really just different ways of expressing the same thing, which we can see when we convert them to decimal as 1/10 and .1.

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