Last modified on 28 November 2011, at 07:51

0.999.../Decimal multiplication by a small number

Multiplication of infinite decimals is usually challenging because it involves a great deal of carrying. Fortunately, as in the cases of addition and subtraction, we are interested in identities that involve no carrying at all.

AssumptionsEdit

TheoremEdit

Statement

If there are two decimals A = a0.a1a2a3 and B = b0.b1b2b3 and an integer m such that for every index n, m × an = bn, then m × A = B.

Proof

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


B = \sum_{n=0}^\infty \frac{b_n}{10^n} = \sum_{n=0}^\infty m \frac{a_n}{10^n}.

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


B = m \sum_{n=0}^\infty \frac{a_n}{10^n} = mA.