There is also the mod arithmetic. for example in mod 12, you would have 11 + 3 = 2. Now Z_2 is mod 2 arithmetic. In mod 2, you have only 1 or zero. And so 1+1= 0 (mod 2)
And there is something called a field with finite characteristic p. In the case where the field has characteristic 2, you will have 1+1 = 0 (the same as in Z_2.
And of course there are other non mathematical situations where 1+1 may not equal 2. Take for example a pile of sand. If I then scoop up one other pile of sand and place it on top of the given one pile of sand, you will still be left with one pile of sand.
And of course there is the case in special relativity physics where if you have two particles A and B both travelling at a certain high velocity speed the combined addition of the two speeds will not add up in the usual manner.
Suppose an object A is moving with a velocity v relative to an object B, and B is moving with a velocity u (in the same direction) relative to an object C. What is the velocity w of A relative to C?
Code:
v
B -------> A
u
C -------> B
w
C ----------------->A
Let 1 refer to the one unit = 0.9c, where c is the speed of light.
So we will have 1 unit plus 1 unit is 0.9c + 0,9c. But according to the addition laws of special relativity, 0.9c + 0,9c = 0.994c or 1.105 of our unit 0.9c.
In other words, 1 unit plus 1 unit is 1.105 units.
So 1 unit plus 1 unit does not equal 2 units in special relativity.