# 0.999.../Decimal multiplication by 10

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.

## Theorem

Statement

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

Proof

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:

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

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.