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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.




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.


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


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