0.999.../Decimal multiplication by 10

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Multiplying an infinite decimal by 10 is just as simple as multiplying an finite decimal by 10: every digit shifts one space to the left.




If A = 0.a1a2a3 then 10 × A = a1.a2a3a4


We apply the definition of an infinite decimal as a series:

A = \sum_{n=0}^\infty \frac{a_n}{10^n}.

Next we apply the fact that a scalar multiple of a series can be computed term-by-term:

10A = \sum_{n=0}^\infty \frac{10a_n}{10^n} = \sum_{n=0}^\infty \frac{a_n}{10^{n-1}}.

Next we Shifting a series|shift the series:

10A = a_0 + \sum_{n=0}^\infty \frac{a_{n+1}}{10^{(n+1)-1}}.

But a0 = 0 by assumption, so we can simplify:

10A = \sum_{n=0}^\infty \frac{a_{n+1}}{10^n},

which is the desired result.