0.999.../Decimal multiplication by a small number

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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[edit]

Theorem[edit]

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.