There are no variations of 1 in mathematics. 1 is 1. 1.0999 is 1.0999. 1 does not equal 1.099, et al. 1 is defined as a natural number, 1.0999 is not. And while there are no "simple proofs" that 1+1=2, it can be proved and has been. I leave that as an exercise for you.
You are correct, inaccurate example, and sloppy thought. 1+1=2 can be proved absolute by the method of mathematics and such proofs define and perfect mathematical method. I should simply have said mathematics and its method is defined by the thought of man and cannot be proved absolute.