Timeline for How can a computer deal with real numbers
Current License: CC BY-SA 4.0
12 events
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| Feb 1, 2021 at 18:59 | vote | accept | Robert | ||
| Jan 29, 2021 at 21:50 | comment | added | Yuval Filmus | @JoelFan This is already mentioned briefly, under arbitrary accuracy. | |
| Jan 29, 2021 at 21:48 | comment | added | JoelFan | Besides floating point, fixed-point and symbolic representation, it's worth also noting en.wikipedia.org/wiki/Arbitrary-precision_arithmetic aka "bignum" representation. It still suffers from being an approximation but at much greater precision than either floating or fixed point. It can exceed margin of error in many practical applications, rendering it indistinguishable from the actual value. | |
| Jan 29, 2021 at 14:57 | comment | added | Yakk | @DanDoel +1 for "constructivist". If you want a "fun" read on it, springer.com/gp/book/9783642649059 where Bishop and Bridges rebuild much of Real Analysis in constructive logic. I find the intuitions gleaned a really helpful when working with doing computation on numbers (be they "real" real numbers or just IEEE floats) | |
| Jan 28, 2021 at 20:16 | comment | added | Dan Doel | Cauchy sequences (e.g.) are based on functions, and computers can deal with computable functions. The 'advantage' is that this is an actual definition of "the real numbers" (at least, one that would be recognized by a constructivist), and these other structures are not actually the real numbers, but presentations of more tractable subsets of the real numbers. The disadvantage is that many common operations on the real numbers are undecidable in general. I suppose the lesson is that directly using the actual real numbers is not really that important for most people. | |
| Jan 28, 2021 at 19:52 | comment | added | gidds | @DanDoel Aren't Cauchy sequences (and Dedekind cuts) based around infinite structures (sequences or sets)? So they couldn't be represented directly in a finite computer. (And if you're not storing them directly, but using some sort of summary or symbolic representation, then what benefit would that have over the ones discussed above?) | |
| Jan 28, 2021 at 13:53 | comment | added | GuiRitter |
Maxima (specially wxMaxima) can do it too: (%i1) sqrt(3)*((4/sqrt(3))-sqrt(3)) (%o1) 1
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| Jan 28, 2021 at 9:00 | comment | added | Federico Poloni | @gnasher729 This is known as "ball arithmetic". | |
| Jan 27, 2021 at 23:57 | comment | added | gnasher729 | What I rarely see discussed is representing a number as a pair (floating point number, best estimate of rounding error). In some cases this will give significantly better results than interval arithmetic. | |
| Jan 27, 2021 at 22:01 | history | edited | Yuval Filmus | CC BY-SA 4.0 |
added 496 characters in body
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| Jan 27, 2021 at 21:54 | comment | added | Dan Doel | There are ways of dealing with real numbers on computers, e.g. with Cauchy sequences or Dedekind cuts. But of course, this is not even what people are doing in these examples. They are simplifying symbolic expressions according to rules, just like Sage is. | |
| Jan 27, 2021 at 20:28 | history | answered | Yuval Filmus | CC BY-SA 4.0 |