0.999.../Decimal multiplication by 10

< 0.999...

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:


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.