Another amazing work by Hans Boehm back in 1999: a constructive real number calculator, https://www.hboehm.info/crcalc/ . In a sense, this is "as far as a computer can go" in the hierarchy of accuracy that includes floating-point, exact rational numbers, and then constructive reals.
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All algebraic numbers works fine. But also: https://en.wikipedia.org/wiki/Ring_of_periods … "The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable" But it's not been proved yet :( (Also, if the algorithm exists, it might be megasuperexponential...)
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Wow! That is a very broad class of numbers. More generally, I guess equality is decidable for any numbers for which a proof of equality or inequality exists, by proof search (since these computable reals can store their formulaic construction history).
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https://hackage.haskell.org/package/cyclotomic … is an option with all roots of unity.
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This seems perfect for computation geometry, because it can represent exact results of CSG intersection calculations on polyhedral rotated by rational angles.
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You can take arbitrarily many rationally independent reals and consider the field extension of the rationals by them.
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The way you compute in a quadratic field extenion is you take the quotient Q[x]/(x^2 + bx + c), which basically amounts to computing with a symbol x with the reduction relation x^2 = -bx - c any time you get an x^n term where n >= 2. This works for any algebraic number.
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For complex numbers: 'elementary' complex numbers have decidable equality if the Schanuel conjecture is true. https://www.sciencedirect.com/science/article/pii/S0747717197901577 … https://www.sciencedirect.com/science/article/pii/S0022404900001560?np=y … https://www.cs.nyu.edu/yap/zero/zero_files/sch.ps … https://cs.nyu.edu/yap/zero/zero_files/slides_richardson.pdf … http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=BDB41DFED3A91EB6B5C60E9BE43C81C2?doi=10.1.1.505.8882&rep=rep1&type=pdf … https://cs.nyu.edu/yap/zero/zero_files/slides_yap.pdf …
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