0.999.../Decimal addition and subtraction

Addition and subtraction of infinite decimals includes some easy problems and some hard problems. Even for finite decimals, identities without carrying are easy to verify (Template:Math), whereas calculations with long runs of carrying are relatively hard to perform (Template:Math). A similar phenomenon occurs for infinite decimals.

Fortunately, we will not be encountering any addition problems with carrying, so we can concentrate on a few simple proofs of identities without carrying.

Assumptions

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

Theorems

Addition by digits is correct

Statement

If there are three decimals Template:Math, Template:Math, and Template:Math such that for every index Template:Math, Template:Math, then Template:Math.

Proof

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

$C=\sum _{n=0}^{\mathrm{\infty }}\frac{c_n}{10^n}=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n+b_n}{10^n}.$

Next we apply the fact that sums of series can be computed term-by-term:

$C=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n}{10^n}+\sum _{n=0}^{\mathrm{\infty }}\frac{b_n}{10^n}=A+B.$

Subtraction by digits is correct

Statement

If there are three decimals Template:Math, Template:Math, and Template:Math such that for every index Template:Math, Template:Math, then Template:Math.

Proof

The proof is almost identical to the previous proof:

$C=\sum _{n=0}^{\mathrm{\infty }}\frac{c_n}{10^n}=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n-b_n}{10^n}=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n}{10^n}-\sum _{n=0}^{\mathrm{\infty }}\frac{b_n}{10^n}=A-B.$

The road not taken

If Template:Math and Template:Math are arbitrary infinite decimals, then it can be tricky to compute the decimal expansion of Template:Math. The problem is caused by the phenomenon of carrying from one digit to the next. To compute any given digit of Template:Math, one might need to inspect many more digits of Template:Math and Template:Math to make sure that their sum doesn't carry into the target digit.

This book does not explore the addition of arbitrary decimals, mostly because it is difficult and unnecessary.

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