I am a mathematical putz – please be kind.
From what I know, a rational number is a non-imaginary number that can be written as $\frac{p}{q}$. A repeating number can also be a rational number (i.e. $\frac{1}{3} = 0.3333…$). Ok, so far, so good.
Ok, so now the question is to write $0.329999…$ (the 9 repeats) as a fraction and prove that it is rational, so here goes:
\begin{align}
1000x &= 329.9999…\\
100x &= 32.9999…\\
\end{align}
subtracting…
\begin{align}
900x &= 297 \\
x &= \frac{297}{300} \\
&= \frac{33}{100} \\
&= 0.33
\end{align}
Herein my dilemma: I don't think that $0.32999…$ is the same as $0.33$ (except in the limiting case), but the math tells me it is. Based on the definition of rationality does that mean that $0.32999…$ is actually irrational, since it cannot be presented as $\frac{p}{q}$? Does this therefore imply, that some irrational numbers can be written down with perfect clarity, like $0.32999…$?
Best Answer
That’s the problem when dealing with representation as decimal numbers. The point is that such representations are not unique but fractions are (up to equivalence). Although there is an error in your calculation, as decimal numbers, $0.33=0.32999\dots$.