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How can any irrational number x be close to a rational number y? as long as x is irrational it seems to me it would still be infinitely far away from y. What no
by jfaucett 9y ago
How can any irrational number x be close to a rational number y? as long as x is irrational it seems to me it would still be infinitely far away from y. What notion of proximity are they talking about?
- jtolmar 9y agoPi is within 1% of 22/7. Edit: Another way of putting it, pi is not infinite. It's a number between 3.14 and 3.15.
- tgb 9y agoThat's not entirely what they mean either, because then every irrational number is "close" to a rational (just take its decimal expansion out to 200 digits - that's rational and very close to the irrational number). You also have to consider the size of the denominator of the rational you are comparing to: i.e. you want to say that irrationals are close being ration if they're close to be a "nice" rational. Where 1/2 is clearly nicer than 22/7 which is clearly nicer than 257/139.
- jfaucett 9y agothank you! That makes sense.
- asddkk 9y agoI think there's some information-theoretic ideas about compressibility or codelength that are relevant. The appeal, I think, is in being able to succinctly represent a random or irrational mathematical object with some other object that's not exactly the same, but is simpler to describe and equivalent to some high degree of similarity. Normally these ideas are applied to things that are thought of as random in a physically stochastic sense, but you could apply them to things that are random in an information-theoretic irrationality sense also. I'd say the polyhedra they discuss are kind of examples of this, maybe in reverse or something: they are simplified constructs that work as representations to some close extent. There's some interesting ties here to pseudorandom numbers, in that usually we think of them as approximating true randomness, even though they're entirely reproducible and predictable. This seems similar to me at some level.
- TeMPOraL 9y agoabs(x-y) being very low?
- jfaucett 9y agoYes but there must be a actual rigorous notion of this. Like epsilon delta limits in calculus. Close doesn't cut it mathematically, for example what would "very low" mean in your example? 1e-20 is very low, but not very low compared to 1e-30000.
- smichel17 9y agoThe article addresses this: > There’s no precise definition of a near miss. There can’t be. A hard and fast rule doesn’t make sense in the wobbly real world. For now, Kaplan relies on a rule of thumb when looking for new near-miss Johnson solids: “the real, mathematical error inherent in the solid is comparable to the practical error that comes from working with real-world materials and your imperfect hands.” In other words, if you succeed in building an impossible polyhedron—if it’s so close to being possible that you can fudge it—then that polyhedron is a near miss. In other parts of mathematics, a near miss is something that is close enough to surprise or fool you, a mathematical joke or prank.
- DonbunEf7 9y agoSince nobody's spelled it out directly: Let an approximant be a rational number p/q which is near the desired irrational x. Then, let p/q be a best approximant iff no other rational with denominator smaller than q is closer to x. It's a delightfully beautiful result that the best approximants for x are given by x's continued fraction: https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...
- bmm6o 9y agoThere is a rigorous notion about how well an irrational can be approximated by a rational, related to its continued fraction representation ( https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction ). I'm having trouble finding a better description than the note in that WP page, but the idea is if the cf representation has a "surprisingly large" value, then the rational number you get by truncating the sequence there will be a very good approximation, like 355/133 is for pi (see the comment in https://math.stackexchange.com/questions/435668/finding-the-value-of-a-continued-fraction https://math.stackexchange.com/questions/435668/finding-the-...).