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| What calculator do you use ? |
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| IanB:
--- Quote from: ejeffrey on April 23, 2012, 01:13:55 pm ---The problem is simply that (binary) floating point arithmetic of any precision is unsuitable for financial transactions because certain values are not representable. You can't represent $0.10 exactly in as floating point number. When written in base 2, the number 1/10 is an infinitely repeating fraction, so it can't be captured exactly with any finite width number. --- End quote --- This argument is entirely bogus. How do you represent $10/3 exactly in BCD? It's just as impossible as representing $0.10 in binary. It doesn't matter, you argue? Well what happens with interest calculations? Suppose you need to add 5.7% per annum to a balance on a monthly basis. Are you going to do that precisely in BCD? Not a chance! Ultimately the only way to get decently accurate answers is to use enough extra digits of precision (and careful arrangement of calculation order) so that rounding errors are insignificant. It ultimately makes no difference whether you do that in BCD or in binary. The results will be identical. For instance, this is what my (binary) calculator says about 200000 * (1 + 0.06/360)^(360*30): 1209748.03723 Apparently binary arithmetic works. |
| slateraptor:
--- Quote from: IanB on April 24, 2012, 01:41:20 am ---This argument is entirely bogus. How do you represent $10/3 exactly in BCD? It's just as impossible as representing $0.10 in binary. --- End quote --- Undoubtedly, eliminating error altogether is unrealistic, but that doesn't mean that the effort is futile. If it can be eliminated for select common cases, wouldn't it make sense, budget permitting, to design a system that has the potential to minimize compounded sources of error, viz. which has higher propensity for error: 16-decimal BCD or 16-bit binary? Seems to be the philosophy back in the day; in the case of HP, investing in a custom ASIC for their lineup was justified. Not so much the case today, where relatively formidable computational power is ubiquitous and quad precision can be had with little effort. |
| HLA-27b:
On a side note here is something that I wasn't aware of until recently. A NOVA film from 1985 about some of the more intricate and controversial aspects of mathematics. --- Quote ---For over a decade, Bertrand Russell tried to find a certainty through mathematics by reducing it to logic. In his massive work, Principia Mathematica, it took him 362 pages to prove that 1 + 1 = 2 Twenty years later, another mathematician, Kurt Gödel, proved that mathematics would never be completely certain. --- End quote --- What does the last sentence even mean? Now if a video from 1985 starts with rolling on the screen Fermat's Last Theorem The Goldbach's Conjecture The Riemann Hypothesis Classification Problem for 4-D Manifolds P?NP Problem Invariant Subspace Problem for Hilbert Spaces then you know it is about some serious stuff. But do these even matter? To me they do matter, in a certain way. They form a benchmark. If they don't matter for what I'm doing then I'm not doing something terribly fundamental or important, gives a sense of scale so to speak. Some of the things presented here you may have heard before, but I for example was not aware of the Bertrand Russel's paradox of catalogues and metacatalogues. It is also very interesting how mathematics can find out its own mistakes and limitations without external reference. also check out here for a summary and some more clues http://faculty.etsu.edu/gardnerr/Math-Mystery-Tour/mathematical-Mystery-Tour.htm |
| IanB:
--- Quote from: HAL-42b on April 24, 2012, 03:50:42 am --- --- Quote ---Twenty years later, another mathematician, Kurt Gödel, proved that mathematics would never be completely certain. --- End quote --- What does the last sentence even mean? --- End quote --- Maybe you already know this, but it means there are some true statements in any system of logic that cannot be proven. It comes from the idea that mathematicians had, that mathematics, being a pure system of abstract thought separated from reality, was one place where you could achieve a notion of certainty, where you could say "this statement is true and I can prove it so" for all true statements. Gödel showed that is was impossible, that you could never achieve such a thing. There will always be statements that are true, but which nevertheless cannot be proven to be true. Mathematics will always be incomplete. |
| HLA-27b:
--- Quote from: IanB on April 24, 2012, 04:06:24 am ---Maybe you already know this, but it means there are some true statements in any system of logic that cannot be proven. It comes from the idea that mathematicians had, that mathematics, being a pure system of abstract thought separated from reality, was one place where you could achieve a notion of certainty, where you could say "this statement is true and I can prove it so" for all true statements. Gödel showed that is was impossible, that you could never achieve such a thing. There will always be statements that are true, but which nevertheless cannot be proven to be true. Mathematics will always be incomplete. --- End quote --- It is funny how casually we talk about it. The guy went on and made certain that certainty is unmakable and we talk about it as if it was just a matter of putting in the man-hours. |
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