5+3 = 28
9+1 = 810
8+6 = 214
5+4 = 19
What is the next number If we add 7+3?
They it is 710. How can this be?
Look for a pattern. In normal arithmetic, 5+3=8, which is the last digit of the number given by this "new addition". 9+1=10, which are the last two digits of the "new addition". 8+6=14, which again are the last two digits of the number they give. Finally, 5+4=9. So, each number they yield ends with the normal arithmetic. So, now you just need a pattern to find the first digit. 5-3 = 2, 9-1 = 8, 8-6 = 2, and 5-4 = 1. Hence, 7+3 could give 4 as the first digit (as 7-3 = 4 in normal addition). Then, it can be followed by 10, as 7+3=10. I would think 7+3 = 410. But, there are an infinite number of patterns. This is just one such pattern. I am sure there is a valid reason for 7+3 = 710.
It took me hours to play with this question.
I needed to combine A - B with A+B to get the final answer...so...
5 + 3 = (5-3) = 2, (5+3) = 8....28
9 + 1 = (9-1) = 8, (9+1) = 10...810
8 + 6 = (8 - 6) = 2, (8 + 6) = 14...214
etc....so...
7 + 3 = (7 - 3) = 4, (7 + 3) = 10...410
The answer is 410.
Note that this binary operator is not associative or commutative. (2+1)+1 = 13+1 = 1214 while 2+(1+1) = 2+2 = 4 (hence not associative) and 5+3=28 while 3+5=-28 (hence not commutative). However, it should always be the case that a + b = +/- (b + a) with a + b = b + a if and only if $\displaystyle a=b$ (I did not verify this, it just seems likely the case).
Usually, binary operators that are not associative or commutative are of relatively little practical use.