TITLE: Proof of the standard algorithm for addition?
QUESTION [9 upvotes]: Can anyone present or point me toward a formal proof of the validity of the standard algorithm we all use for addition (line up your numbers one over the other, add the ones-place, carrying 'excess' digits to the next place-value, etc)?
This is what comes of reviewing set theory, elementary arithmetic, and proof theory at the same time :)

REPLY [5 votes]: If you know about polynomial arithmetic then integer addition is nothing but a special case of polynomial addition in $\rm\:\Bbb Z[x]/(x\!-\!b) = $ integer coefficient polynomials modulo $\rm\:x\!-\!b,\:$ where $\rm\:b\:$ is the radix. Indeed, mod $\rm\:x\!-\!b\:$ we can choose as a complete system of reps the polynomials whose coefficients lie in the interval $\rm[0,b\!-\!1]$. Then addition is simply polynomial addition, followed by the usual carry propagation to reduce the sum to normal form. This is an explicit constructive form of the isomorphism $\rm\:\Bbb Z \,\cong\, \Bbb Z[x]/(x\!-\!b)\:$ arising from the evaluation homomorphism $\rm\:f(x)\to f(b).$
For example, the computation $\ 298 + 97\, =\, 395\ $ in radix $\,10\:$ is as follows
$\qquad\begin{eqnarray}\rm mod\ x\!-\!10\!:\,\ \,2x^2\!+\!9x\!+\!8\,+\, 9x\!+\!7\!\!\! &\ \equiv\ &\rm 2x^2+18x+15\\
&\equiv\, &\rm 2x^2+18x+(x\!+\!5)\\
&\equiv\, &\rm 2x^2+19x+5\\
&\equiv\, &\rm 2x^2+(x\!+\!9)x+5\\
&\equiv\,&\rm 3x^2+9x+5\\
\end{eqnarray}$
Above, carry propagation amounts to interpreting $\rm\:10\!+\!n\,\equiv\, x\!+\!10\:$ as a rewrite rule, in order to rewrite the polynomial to one in normal form, i.e. with all coefficients in the interval $\,[0,9].$
Remark $\ $ For a more interesting example, see the following generalization for real numbers, where properly handling carry propagation is much trickier: $$ F. Faltin, N. Metropolis, B. Ross, G.-C. Rota, The real numbers as a wreath product. Advances in Math. 16 (1975), 278-304.