# 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.

## Assumptions edit

## Theorem edit

- Statement

If *A* = 0.*a*_{1}*a*_{2}*a*_{3}… then 10 × *A* = *a*_{1}.*a*_{2}*a*_{3}*a*_{4}…

- 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 shift the series:

But *a*_{0} = 0 by assumption, so we can simplify:

which is the desired result.