Defining an operation such that ##1+2+3=123##

[Saracen Rue]

Firstly, I'm aware that title doesn't really make sense but stick with me on this. I'm trying to find a way to define an operation which will "join" two numbers instead of adding them. So for example, $12+34=1234$. Ideally, it would be great if it also had something similar to sigma notation, like so:
$$\sum_{k=1}^{n} k= 12345...n$$

I'm sure this is actually something trivial that has been defined before, but I was finding it really difficult to search it up on Google (apparently "addition but instead of adding numbers just clomp together" doesn't yield very good results). So yeah, if anyone can tell me a better way to name this sort of maths it'd be greatly appreciated.

 

[PeroK]

This is called concatenation. You often have to program this kind of operation for strings or text.

 

[FactChecker]

In computer languages, this is often done but there is no standardized symbol that would be universally recognized in other fields. Since C++ is such a well-established language, it might be the best thing to mimic. It uses the plus '+' symbol. The trick would be to distinguish it from numerical addition. Consider using quotes to make it clear that you are treating the number as a text string: "12"+"34"="1234".
If this is to appear in a document, you should clearly define your notation and symbology in the document.

 

[pasmith]

I suppose you have $$\oplus : \mathbb{N}^2 \to \mathbb{N} : (a,b) \mapsto \begin{cases} 10^{1 + \lfloor\log_{10}(b)\rfloor}a + b, & b \neq 0, \\ 10a, & b = 0\end{cases}$$ which (if it does what I think it does) is associative but not commutative, and you need to decide whether $$\bigoplus_{n=0}^N a_n$$ means $a_0 \oplus a_1 \oplus \cdots \oplus a_N$ or $a_N \oplus a_{N-1} \oplus \cdots \oplus a_0$.

 

[jedishrfu]

In some computer languages, the addition operator took on different meaning dependent on the datatypes of items being added. It acted as normal addition for numeric types and as a concatenation operator for string types.

There were some purists who complained that since concatentation is not commutative that the addition operator shouldn't serve that function. However, using the addition operator for concatenation had become a defacto standard and was here to stay.

In Python, one can concat in several different ways:

- string1 + string2
- " ".join(string1, string2)
- ...

https://www.geeksforgeeks.org/python-string-concatenation/

and here's a larger set of multi language string concatentation examples:

https://en.wikipedia.org/wiki/Comparison_of_programming_languages_(string_functions)#Concatenation