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 ($123456 + 654321 = 777777$), whereas calculations with long runs of carrying are relatively hard to perform ($3456 + 6549 = ???$). 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

Addition by digits is correct
If there are three decimals $A = a_{0}.a_{1}a_{2}a_{3}…$, $B = b_{0}.b_{1}b_{2}b_{3}…$, and $C = c_{0}.c_{1}c_{2}c_{3}…$ such that for every index $n$, $a_{n} + b_{n} = c_{n}$, then $A + B = C$.
 * Statement

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



C = \sum_{n=0}^\infty \frac{c_n}{10^n} = \sum_{n=0}^\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}^\infty \frac{a_n}{10^n} + \sum_{n=0}^\infty \frac{b_n}{10^n} = A + B. $$

Subtraction by digits is correct
If there are three decimals $A = a_{0}.a_{1}a_{2}a_{3}…$, $B = b_{0}.b_{1}b_{2}b_{3}…$, and $C = c_{0}.c_{1}c_{2}c_{3}…$ such that for every index $n$, $a_{n} &minus; b_{n} = c_{n}$, then $A &minus; B = C$.
 * Statement

The proof is almost identical to the previous proof:
 * Proof



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

The road not taken
If $A$ and $B$ are arbitrary infinite decimals, then it can be tricky to compute the decimal expansion of $A + B = C$. The problem is caused by the phenomenon of carrying from one digit to the next. To compute any given digit of $C$, one might need to inspect many more digits of $A$ and $B$ 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.