Express $0.99999 \ldots$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates,discuss why the answer makes sense.
(A) Let $x = 0.9999 \ldots$ $(1)$
Multiply both sides by $10$,we have:
$10x = 9.9999 \ldots$ $(2)$
Subtracting equation $(1)$ from equation $(2)$,we get:
$10x - x = (9.9999 \ldots) - (0.9999 \ldots)$
$9x = 9$
$x = \frac{9}{9} = 1$
Thus,$0.9999 \ldots = 1$.
As $0.9999 \ldots$ continues infinitely,there is no gap between $1$ and $0.9999 \ldots$. Hence,both are equal.
Multiply both sides by $10$,we have:
$10x = 9.9999 \ldots$ $(2)$
Subtracting equation $(1)$ from equation $(2)$,we get:
$10x - x = (9.9999 \ldots) - (0.9999 \ldots)$
$9x = 9$
$x = \frac{9}{9} = 1$
Thus,$0.9999 \ldots = 1$.
As $0.9999 \ldots$ continues infinitely,there is no gap between $1$ and $0.9999 \ldots$. Hence,both are equal.
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