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

Theorems

Addition by digits is correct

Statement

If there are three decimals $A = a 0 . a 1 a 2 a 3 \dots$ , $B = b 0 . b 1 b 2 b 3 \dots$ , and $C = c 0 . c 1 c 2 c 3 \dots$ such that for every index $n$ , $a n + b n = c n$ , then $A + B = C$ .

Proof

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

C=\sum _{n=0}^{\infty }{\frac {c_{n}}{10^{n}}}=\sum _{n=0}^{\infty }{\frac {a_{n}+b_{n}}{10^{n}}}.

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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

Statement

If there are three decimals $A = a 0 . a 1 a 2 a 3 \dots$ , $B = b 0 . b 1 b 2 b 3 \dots$ , and $C = c 0 . c 1 c 2 c 3 \dots$ such that for every index $n$ , $a n - b n = c n$ , then $A - B = C$ .

Proof

The proof is almost identical to the previous 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.

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