On Wednesday, July 31, 2013 3:17:52 PM UTC-7, jonas.t...@gmail.com wrote> > > One question can fractions be expressed using other bases, and must they necessarly have ending nonerepeating pattern in base 10.
Yes. Here follow this please.
Let r be a Positive Real Rational Number. We know r = p/q, for some natural numbers p and q. No mention of base, as real ( and rational ) numbers are not defined in bases.
Now, suppose we write r in base 10 decimal expansion. We call this r_10. We can write r = r_10, as they represent the same Real Rational Number.
Now, since r is Rational, r_10 has two possibilities:
1. It terminates.
2. It repeats.
Next, change base to base m, and write r as r_m. In base m, r is still rational, as r = p/q. Of course, r is written as r_m and p and q as p_m and q_m, but they are still a Rational number and Natural Numbers. Hence, r_m has two options: 1. It terminates. 2. It repeats.
These are just results from the nature of the Real Numbers.