\begin{definition}[Definition:Dismal Addition]
'''Dismal addition''' is an operation defined on the natural numbers as follows.
Let $a, b \in \N$ be expressed in decimal notation:
:$\ds a = \sum_{i \mathop = 0}^n a_i 10^i = \sqbrk {a_n a_{n - 1} \ldots a_2 a_1 a_0}_{10}$
:$\ds b = \sum_{j \mathop = 0}^m b_j 10^j = \sqbrk {b_m b_{m - 1} \ldots b_2 b_1 b_0}_{10}$
The '''dismal sum''' of $a$ and $b$ is defined as:
:$\ds a + b = \sum_{k \mathop = 0}^{\max \set {m, n} } \max \set {a_k, b_k} 10^k$
That is, for each pair of corresponding digits, the maximum is taken.
\end{definition}