Is there a way to add positives and negatives with the same algorithm?

Question

It is taught to us in grade school that you can add (positive) numbers like this:

    5 2 3
    4 5 6
   -------
    5 7 9

But if we change it to 523 + (-456), we cannot use the same algorithm

    5  2  3
   -4 -5 -6  
   ---------
    1 -3 -3 

I know that I can evaluate that to 100 + -(30) + (-3) which gets 67, but in my situation, 17 is the maximum integer, and -17 is the minimum integer (excluding the actual numbers being added). Is there a way to do this where the 523 + 456 also works?

Answer 1

The reason why long long data type in c++ represents numbers range from $\ -2^{63}\ $ to $\ 2^{63}-1\ $ is because the representation used is $64$-bit twos complement. An analogous representations in decimal would be $n$-digit "tens complement", for some $\ n\ $, which could represent all the numbers in the range $\ -\frac{10^n}{2}\ $ to $\ \frac{10^n}{2}-1\ $, so it's not at all clear (not, at lest, to me) where your "maximum integer" of $17$ comes from, or what it is meant to be the maximum of.

Nevertheless, if you represent decimal numbers using sequences of positive and negative digits from $-9$ to $9$, as you have done, you can convert any representation to any equivalent one without needing to use any numbers outside the range $-9$ to $9$.

In your example, $\ 1\ -3\ -3\ $, for instance, first replace the rightmost digit by $7$ and decrease the next digit to the left by $1$ to get the equivalent representation $\ 1\ -4\ \ 7\ $. You don't even need to get the $7$ by subtracting $3$ from $10$, since you can get it by subtracting one less than $3$, namely $2$, from $9$.

Next, replace the new middle digit, $-4$, with $6$, and decrease the leftmost digit by $1$ to get the equivalent representation $\ 0\ 6\ 7\ $, from which the new leftmost digit $0$ can be deleted to give you $67$.