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Very neat: naive attempts to memorize Pi with rational-number shortcuts (i.e., fractions with integers in the nominator and denominator, such as 22/7 and 355/11
by cs702 14y ago
Very neat: naive attempts to memorize Pi with rational-number shortcuts (i.e., fractions with integers in the nominator and denominator, such as 22/7 and 355/113) seem pointless, because getting more decimal digits of Pi right requires that one memorize a correspondingly larger number of digits in the numerator and/or denominator, defeating the purpose of these native attempts.
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PS. The author is offering a prize to the reader who finds the rational number which gets the most decimals of Pi right for every digit of such rational number that has to be memorized. (Note that only rational numbers are allowed -- that is, fractions with integers in the numerator and denominator. Using formulas or numbers that are not rational is not allowed in the competition.)
- kalid 14y agoI might be different, but 355/113 is very easy to remember and is not 7 unique segments of information. I just think "double the odds" 113355 We know we want a fraction, not a single number, so split down the middle: 113/355 And we know pi won't be less than 1, so flip it: 355/113. Knowing that the digits sequences are doubled lets you do some cheap, mental run length encoding. In a case where you need a hand calculation, spending 3 seconds to re-derive the sequence seems tolerable.
- 3rd3 14y agoUnwillingly, I just memorized a Pi approximation.
- boyter 14y agoAs did I. I came back here 6 hours after reading it to confirm that it is still stuck in my brain, I am guessing permanently. Although I already knew Pi to 6 decimal places so its not like I am gaining a lot of accuracy out of this.
- Evbn 14y agoRequired information for your approach: Repeat the digits Use odd digits Use increasing sequence Use three digits Split the result in half and divide Pi is greater than 1 But since humans have associative memory, these are easier to learn than a seemingly arbitrary digits sequence.
- jfoutz 14y agoRational approximations are good for finding reasonable answers without a calculator. My contrived example is the area of a circle, r=7. i can do 22 * 7 in my head and get 154, which is pretty darn close to 153.86. 7 is awkward (for me anyway) but it does offer opportunities to cancel.
- Jach 14y agoHow fast can you do 22 * 7 in your head? I get a lot of mileage out of simply approximating pi=3, and optionally adding in a factor of 10% later. For your example, I know 7^2=49 immediately, which I round to 50, 50 * 3=150 which is fairly close to 153.938. (Use more digits of pi!) I can quickly improve my estimate by adding 4.9. (Edit: It's also fairly trivial to get even better estimates quickly from here, but at this point it's probably faster to grab a cell phone / [favorite language] repl. From 154.9, subtract the additional 3 gained from using 50 instead of 49, now 151.9, add 4.9/2~=2.4, now 154.3, subtract 4.9 * 10% ~= .5, 153.8.)
- jfoutz 14y ago22 times a single digit is pretty easy, double the digit, shift the decimal, add the doubled number - but it's not trivial. As i said, a contrived example. The point is, you yourself stick to integer arithmetic, then try to fix it up with a 10% modifier at the end. People have been using pi for a long time. Easy access to calculators is, what, about 50 years now? I'll happily agree that rational representation is a historical artifact. But i still believe the vast majority of people doing arithmetic pre 1960 with floating point numbers did it like you and i do. They would put off the decimal representation as long as possible.
- recursive 14y ago> optionally adding in a factor of 10% later You're better off not adding anything. If you could add 5% instead, then you'd be much better still.
- Jach 14y agoIdeally you add 14%. Why do you say 5% (or nothing) is better than 10%?