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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)

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up vote 7 down vote accepted

If it were... we could not have transitivity, for then everything would be equal.

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Equality within a range is a useful concept, but it gives up transitivity. The Archimediean property of the reals (or the rationals) guarantees that. If $z$ is your tolerance and you define $x \sim y$ by $|x-y| \lt z$ then for any two reals $a,b$ you can make a chain $a \sim a' \sim a'' \dots \sim b$ It just takes $\lceil \frac {b-a}z \rceil$ steps to get there.

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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.

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