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.

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

Statement

If there are three decimalsA =a₀.a₁a₂a₃…,B =b₀.b₁b₂b₃…, andC =c₀.c₁c₂c₃… such that for every indexn,a_n +b_n =c_n,thenA +B =C.

Proof

We apply thedefinition of an infinite decimal as a series:

${\displaystyle C=\sum _{n=0}^{\mathrm{\infty }}\frac{c_n}{{10}^n}=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n+b_n}{{10}^n}.}$^{[ఉ 1]}

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

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

Statement

If there are three decimalsA =a₀.a₁a₂a₃…,B =b₀.b₁b₂b₃…, andC =c₀.c₁c₂c₃… such that for every indexn,a_n −b_n =c_n, thenA −B =C.

Proof

The proof is almost identical to the previous proof:

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

IfA andB are arbitrary infinite decimals, then it can be tricky to compute the decimal expansion ofA +B =C. The problem is caused by the phenomenon of carrying from one digit to the next. To compute any given digit ofC, one might need to inspect many more digits ofA andB 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.

• Book:0.999...

• Pages with reference errors
• No references for citations