MHB Repeating decimals (sic) in bases other than 10

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The discussion focuses on proving that the repeating decimal $.0222\ldots$ in base 3 is equivalent to $.1$ in base 3 and equals $\frac{1}{3}$ in base 10. The initial proof demonstrates this by expressing the repeating decimal as an infinite series and simplifying it to show it equals $\frac{1}{3}$. Another participant introduces a method using the property of repeating decimals, comparing it to the base 10 example of $0.\bar{9}=1$. A critique is raised regarding the validity of applying finite series rules to infinite series without proper justification. The conversation emphasizes the mathematical relationships between different bases and the nature of repeating decimals.
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Prove that $.0222\ldots$ (base 3) $= .1$ (base 3) $= \frac{1}{3}$ (base 10).First, we will show $.0222\ldots$ (base 3) $= \frac{1}{3}$ (base 10).
\begin{alignat*}{3}
2\left(\frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} + \cdots\right) & = & 2\sum_{n = 2}^{\infty}\left(\frac{1}{3}\right)^n\\
& = & \frac{2}{9}\sum_{n = 0}^{\infty}\left(\frac{1}{3}\right)^n\\
& = & \frac{2}{9}\frac{1}{1 - \frac{1}{3}}\\
& = & \frac{1}{3}
\end{alignat*}

I am having trouble with the second part.
 
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Re: base 3

This may not be what you have in mind, but this puts me in mind of a simple proof in base 10 that:

$\displaystyle 0.\bar{9}=1$

So, we have in base 3 (trinary):

$\displaystyle x=0.0\bar{2}$

Multiply by 3 in trinary:

$\displaystyle 10x=0.\bar{2}=0.2+x$

$\displaystyle 2x=0.2$

$\displaystyle x=0.1$
 
Re: base 3

MarkFL said:
This may not be what you have in mind, but this puts me in mind of a simple proof in base 10 that:

$\displaystyle 0.\bar{9}=1$

So, we have in base 3 (trinary):

$\displaystyle x=0.0\bar{2}$

Multiply by 3 in trinary:

$\displaystyle 10x=0.\bar{2}=0.2+x$

$\displaystyle 2x=0.2$

$\displaystyle x=0.1$

Your purported proof is invalid in any base in that it applies rules valid for finite series to infinite series without justifying their validity.

CB
 
How I solved it was with the fact that 3 in base 3 is 10.
$$
\frac{1}{10} = .1
$$
 
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