Definition:Dismal Addition

Definition
Dismal addition is an operation defined on the natural numbers as follows.

Let $a, b \in \N$ be expressed in decimal notation:


 * $\displaystyle a = \sum_{i \mathop = 0}^n a_i 10^i = \left[{a_n a_{n-1} \ldots a_2 a_1 a_0}\right]_{10}$
 * $\displaystyle b = \sum_{j \mathop = 0}^m b_j 10^j = \left[{b_m b_{m-1} \ldots b_2 b_1 b_0}\right]_{10}$

The dismal sum of $a$ and $b$ is defined as:


 * $\displaystyle a + b = \sum_{k \mathop = 0}^{\max \left\{{m, n}\right\}} \max \left\{{a_k, b_k}\right\} 10^k$

That is, for each pair of corresponding digits, the maximum is taken.