Suppose we for take example $\pi$ then one nice experiment to find an approximate value for it is the Buffon's Needle experiment. Suppose we are given any arbitary irrational number $\chi$, is it possible to find a physical experiment such that we can find a value of that number?
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Depending on what it means to be "given" an irrational number, https://mathoverflow.net/questions/223810/is-being-rational-decidable might be useful. – Connor Behan Oct 14 '22 at 19:00
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Limit of a seq in R which is not rational Ig @ConnorBehan – tryst with freedom Oct 14 '22 at 19:03
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Any analog computer? – Rol Oct 14 '22 at 23:47
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1Note that any number is an approximation in some sense to any other number. The question is rather vague in that sense. – StephenG - Help Ukraine Oct 14 '22 at 23:48
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1Indeed, $\pi$ is approximately 4. – Jon Custer Oct 14 '22 at 23:55
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You use your seq. to plot the number make a rectangle the other side 1 and the needle Experiment. All sqrt can be constructed physically, with rectangular triangles. there are mechanical integrators, so you can approximate ln(x) integrating 1/x. But also a computer is a physical instrument!
trula
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Use an experiment that measures $1\mathrm{\ s}$. Convert that to units of “ticks” where $1\mathrm{\ s}=\chi\mathrm{\ ticks}$. Then you have a measurement of $\chi$.
Similar experiments can be done with any dimensionful measurement.
Dale
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