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.

Assumptions[edit]

Theorem[edit]

Statement

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

Proof

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

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

Next we Shifting a series|shift the series:

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

which is the desired result.