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.

Assumptions

• Definition from series
• Term-by-term operations on series

Theorem

Statement

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

$B=m\sum _{n=0}^{\infty }{\frac {a_{n}}{10^{n}}}=mA.$