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Often in physics you have to do maths with a finite amount of digits, e.g. $\pi = 3.14$, but this is not exact and without knowing the next digit this is only correct in the interval $[3.135,3.145]$. But does this mean that you can treat $\pi$ as an interval instead of a number, and say $\pi = [3.135,3.145]$? Could this be generalised for a new type of "numbers" as intervals?

Something like: $$ \vdots\\[0.5em] -2 =\,[-3,-2]\\ -1 =\,[-2,-1]\\ -0 =\,[-1,-0]\\ 0 =\,[0,1]\\ 1 =\,[1,2]\\[0.5em] \vdots $$ where $a+b = [a_1,a_2]+[b_1,b_2] = [a_1+b_1,a_2+b_2]$

I'm sure must be some name for this concept, and I'd like to read more about it.

Frank Vel
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  • What do you mean by "this is only correct in the interval $[3.135, 3.145]$? –  Feb 19 '15 at 12:53
  • @Ahmed I think it's a clunky way of saying that, if all you know is that $3.14$ is a correct roundoff of $\pi$ to two decimal places, all you really know is that $\pi \in [3.135,3.145]$. – Ian Feb 19 '15 at 12:56
  • By the way, a more physical approach to this will propagate uncertainty based on some probabilistic model of the quantity of interest. The standard way of doing this, which is commonly taught in physics classes without proof, can be derived by assuming that separate measurements are independent identically distributed Gaussian variables. – Ian Feb 19 '15 at 13:01

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There is a branch of numerics that uses intervals instead of numbers, see: here

Interesting is the connection to automated proofs.

If the software would be error-free the computed intervals could be interpreted as proven bounds. I think they market this as reliable computing now.

mvw
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