TITLE: Can equality be defined within a tolerance?
QUESTION [5 upvotes]: Can equality of two real number be defined within a tolerance? Say, 0 = 0.000009 given the tolerance to be 0.00001. I also assume that equality should be transitive, e.g. a = b, b = c,  then a = c. But, seems it is not always the truth. 
Sometimes, I want two real numbers to be considered as equal as long as they are close enough,  and also want this kind of equality to be transitive. Can it be possible?
(Not sure what tags should I put)

REPLY [2 votes]: Let $x, y$ be toleranced quantities.  That is $x = x_0 \pm \sigma_x$, and similarly for $y$.  Then $x + y = x_0 + y_0 \pm (\sigma_x + \sigma_y)$.  So as you can see the tolerance gets propogated to the result of calculation and increased.  This is refered to as propogation of uncertainty and has been studied under many operations including the basic arithmetic and general functions.