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22/7 has you memorize four digits (including the position of the /) to get pi to four digits (including the position of the .), 355/113 has you memorize seven d
by handsclean 4y ago
22/7 has you memorize four digits (including the position of the /) to get pi to four digits (including the position of the .), 355/113 has you memorize seven digits to get pi to eight digits, 355/113-1/3748629 has you memorize 17 digits to get pi to 16 digits. I don’t get it. How is it easier than just memorizing pi? How is it surprising to get out about as many bits as you put in?
- a_t48 4y agoFor 355/113 you get error correction - if the answer you get isn't around 3.14 you misremebered.
- handsclean 4y agoOk, now you’re memorizing 11 digits, including 3.14, to get 8 error-corrected digits. Edit: changed 7 to 8, my mistake
- a_t48 4y agoMost people have the 3.14 down.
- handsclean 4y agoOk, then you’re memorizing seven additional digits to get four additional digits and error correction…
- sobriquet9 4y ago113\355 is easier to remember.
- tshaddox 4y agoI don’t think anyone is impressed by how easy it is to store the symbolic representation of the fraction versus the symbolic representation of the first n digits of pi, or suggesting that it’s a useful way for humans to memorize or calculate pi. What’s notable is simply how close the fraction approximates the irrational number pi given how low the fraction’s denominator is.
- Nomentatus 4y agoBut he just gave you another example of just that. So that feature simply isn't unique.
- tshaddox 4y agoWhat other example has a similarly low denominator but gives as good or better accuracy than 355/133? I’m under the impression that there aren’t any such examples.
- Nomentatus 4y agoHe gave "355/113-1/3748629 has you memorize 17 digits to get pi to 16 digits." This is a compression problem, a little more breakage here or there is to be expected. If e and every other irrational number showed similar breakage, then I might get interested.
- alanbernstein 4y ago22/7 and 355/113 are both special in that the approximation error, relative to the denominator side, is unusually low. Pi is special in that it has even one such small-denominator approximation.
- eru 4y ago> How is it easier than just memorizing pi? You can't 'just' memorize pi. You have to pick a number system first. And yes, if you work a lot with decimal numbers, then memorising an approximation for pi in that system will come comparatively easy.
- peeters 4y agoI agree from the perspective of how most humans could memorize pi. But that's exploiting the fact that humans treat decimal representations as having implicit denominators--3.14 is actually 314/100. So say you had a computer that didn't treat base 10 as special. Then an apples-to-apples comparison would be that 355/113 (9/7 = 16 binary digits) has the same precision as 3141592/1000000 (22/20 = 40 binary digits). 16 is a lot fewer than 40.
- romwell 4y ago> But that's exploiting the fact that humans treat decimal representations as having implicit denominators That's to say, we know which base we are using... which is a requirement for writing down numbers in the first place. >So say you had a computer that didn't treat base 10 as special Your computer treats base 2 as special. The apples-to-apples comparison would be comparing 355/113 to 11.0010010000111₂ (or a 16-bit float representation of Pi). 355/113 is a better approximation, but not as drastically as you wrote. Writing in binary, 355/113 = 101100011₂/1110001₂, and the equivalent one is 11.0010010000111111011₂ - hardly that big of a difference.
- peeters 4y agoI was using base 2 just for conveying the numbers of digits, i.e. showing that the digits in representation is related logarithmically to the magnitude of the base-agnostic integers in the fraction. I could have just as easily used natural log or log10 for evaluating the size of those numbers, because the important part is that it's not exploiting the implicit base in your denominator.
- bakul 4y ago1/pi == 113/355 - this is why I have remembered it since age 10 or so!
- zakki 4y agoBecause as you said, when you memorize 8 digits pi, your memory stop there. But with fraction it is possible to get next number as long as you want.
- Plutoberth 4y agoBut it'll be wrong. 355/113 is only useful up to n digits, not infinitely, and GP is saying that n is as big as the number of digits in "355/113".
- e12e 4y agoIf you're calculating by hand (ie: figuring out material needed for a winding staircase or something) - working with two integers / a ratio can mean you can easily simplify the math, and only deal with narly decimal precision after simplifying. Eg area of a half circle is: A = pi * r^2/2 Or, for example, radius 4 units (eg feet): 22 * 4^2 --------- 7 * 2 Simplifies to: 88/7. Easy to approximate, easy to work out to needed precision.
- e12e 4y agoErr, 22 * 8 isn't 88, but hopefully the point is obvious.
- alanbernstein 4y agoYou’re right, it’s odd that the article mentions memorization. That’s not what makes this interesting. Obviously bits is the relevant concept here from a modern information theory perspective. But “digits in fraction” is more interesting. 22/7 vs 314/100. 355/113 vs 3141592/1000000. Of course you don’t need to “remember” the denominator when it is a power of 10. But the error in the approximation is unusually low, compared to the size of the denominator. This is true when comparing to decimal approximations, or other “nearby” rational approximations. In that sense, these approximations should be “surprising” because they work much better than the decimal ones of similar size (or, much smaller than the decimal approximations of similar accuracy). But they also work better for pi than for many other common irrational numbers. This is also historically meaningful, as some calculation was done in fractions, e.g. in ancient Egypt. It is an interesting mathematical fact that 22/7 is an approximation to pi that is small enough to be easily discoverable, easy to compute with, and still practically useful.