Is $14.9999999$ ad infinitum $\lt 15$, $= 15$, or in between?
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2Can you find a real number between $14.\overline{9}$ and $15$? Would you expect to be able to if they were different? Also, see http://math.stackexchange.com/questions/11/does-999999999-1 – Eric Towers Jul 16 '14 at 04:28
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It is exactly $15$, in the same way that $0.999 \ldots = 1$. For my second affirmation, notice that $1/3 = 0.333\ldots$, then $1 = 3/3 = 3 \cdot 0.333\ldots = 0.999\ldots$.
Ivo Terek
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Definitely equal. It's a sum of a geometric series:
$$14.9999\cdots =14+\frac9{10}\left(1 + \frac1{10}+\frac1{10^2}+\cdots\right) $$
$$=14+\frac9{10}\left(\frac1{1-\frac1{10}}\right) $$
$$=14+\frac9{10}\left(\frac{10}9\right) $$
$$=14+1$$
$$=15$$
as desired.
MPW
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