0.999...=1/Proof 4
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Theorem
- $0.999 \ldots = 1$
Proof
We begin with the knowledge that:
\(\ds \frac 9 9\) | \(=\) | \(\ds \frac 1 1 = 1\) |
Now we divide $9$ by $9$ using the standard process of long division, only instead of stating that $90$ divided by $9$ is $10$, we say that it is "$9$ remainder $9$," yielding the following result:
- $ \require{enclose} \phantom{0..}0.9999\ldots\\ 9\enclose{longdiv}{9.0000\ldots}\\ \phantom{0}\underline{\phantom{0}8.1}\\ \phantom{000.}90\\ \phantom{0.}\underline{\phantom{00}81}\\ \phantom{0000.}90\\ \phantom{0.}\underline{\phantom{000}81}\\ \phantom{00000.}9\ldots\\ $
Thus, we are compelled to believe that:
- $0.999\ldots = \dfrac 9 9 = 1$
$\blacksquare$