Last modified on 28 November 2011, at 08:48

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.

AssumptionsEdit

TheoremEdit

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}}.

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.