Its impossible for 2 + 2 to equal 5 if there is no modulus. This is because 2 = 1 + 1. Therefore, 2 + 2 = (1 + 1) + (1 + 1). 5 on the other hand is equal to 4 + 1. And since, 2 + 2 equals 4, this becomes (1 + 1) + (1 + 1) + 1. There is an additional 1 on the left which means that the two are not equal, thus, violating the principle of identity, it is similar to the laws of conservation in physics. To refute the proof provided by "Adam Smith" above me, one can see that in step eight where (x - 4) was divided from both sides. Since 2 + 2 = 4, step eight is really a division by zero which is impermissible in mathematics. Often dividing by zero or not providing a plus and minus sign after introducing a square root into an equation will allow one to derive statements such as 2 + 2 = 5 or 1 = 0. Because division by zero is undefined and square roots can be either the positive or negative case, assuming one or the other is fallacious reasoning.