A strange operator on my new calculator

Question

Yesterday, I bought a new calculator in an antique shop. I wanted to try some basic operations to see if it was working well. Everything felt right, except for the + sign: it seemed like it was replaced with an obscure new operator!

At first, I tried these additions that seemed totally fine:

5 + 5 = 10
6 + 5 = 11
39 + 8 = 47
6 + 3 + 7 + 10 = 26

But then, things started to go slightly off with some additional tries:

6 + 4 = 12
7 + 3 + 10 = 21
8 + 4 + 9 = 22

So I computed more operations, and nothing made sense anymore!

2 + 2 = 10
1 + 2 + 3 + 4 = 1
5 + 5 + 5 + 5 + 5 = 130
3000 + 333 + 30 + 3 = 1020

My last hope to try to understand this operator was to compute a random big number to understand what's going on, but all I got was:

17603415 + 33 = ERROR

So, can you tell me what is this operator that replaced the + sign on my calculator?

Answer 1

I think the operation replacing + is

"represented in base"

That is, "a + b" is "a" (in base 10) represented in base "b".

Examples

6 + 5 = (1 x 5) + 1 = 11
6 + 3 + 7 + 10 = ((6 + 3) + 7) + 10 = (20 + 7) + 10 = 26 + 10 = 26
8 + 4 + 9 = (8 + 4) + 9 = 20 + 9 = 22
1 + 2 + 3 + 4 = 1 + 3 + 4 = 1 + 4 = 1
5 + 5 + 5 + 5 + 5 = 10 + 5 + 5 + 5 = 20 + 5 + 5 = 40 + 5 = 130
3000 + 333 + 30 + 3 = 93 + 30 + 3 = 33 + 3 = 1020

Finally the last result produces an error because

We would need to use additional characters beyond the regular digits to represent it.

In fact

If we use letters of the alphabet to represent additional digits, a=10, b=11,... then the result would be "error"