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I would like to apologize to all professional mathematicians reading this for my clumsy approach to explaining the subject. I really am doing my best.
It is often said that one cannot represent numbers like by binary floating point numbers. That is, of course, true (assuming positional notation is implied, otherwise it is as simple as
). Unfortunately, in most cases, the advice ends here. Sometimes, there’s an example included, first to show that the process is correct:
It is then followed by the aforementioned impossible example, or a variation thereof:
This explanation is rather flimsy: sure, we can see that this method failed to achieve the wanted result, but does it prove anything? It would be much better to understand the rules governing it.
For this reason, one should remember how the positional notation works. Let’s suppose there is a number base
, where
,
,
,
and
are digits (in base
, each digit has an integer value between
and
, inclusive). The value of a number encoded this way is a sum of its digits, each multiplied by power of
respective to it’s position:
In other words:
From here we can easily posit that . It is important, because not only are all combinations of five digit numbers represented by this, but also it is clear that there is no possible five-or-less digit numerator that would not be represented by one of combinations.
In more general terms, any finite integer has finite positional base
representation
for any integer
.
With this fact in mind, it is easy to check any fraction for a finite representation in base : after reducing it to the lowest terms, one should check the prime factors of the denominator. If they are a subset of set of
‘s prime factors then it is possible to transform the fraction to the form of
and therefore this fraction has a finite base
representation.
For example, in base (prime factors:
(twice) and
) both
and
can be represented without any loss of precision, the values being
and
respectively. The representation of
is infinite, on the other hand:
.
This is because one can trivially transform them to the aformentioned form. In case of
:
As for it is not possible to transform it to the yearned form, because
is not a prime factor of
for any positive integer
.
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