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Topic: Can a fraction have none noneending and nonerepeating decimal representation?
Replies: 108   Last Post: Aug 16, 2013 5:22 PM

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 Tucsondrew@me.com Posts: 1,075 Registered: 5/24/13
Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted: Jul 31, 2013 8:50 PM

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.

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.

ZG

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