http://davidbau.com/archives/2010/03/14/the_mystery_of_355113.html
March 14, 2010
Today's date is a good excuse to memorize a few more digits of 3.1415926535897932384626433832795....
And yet decimal approximations to pi are an artifact of our ten-fingered anatomy. Fractional approximations to pi are more satisfying, and they promise to teach us something more universal about pi.
We all know that 22/7 is a very good approximation to pi. But this well-known fraction is is actually 1/791 larger than a slightly less-well-known but much more mysterious rational approximation for pi: pi ~ 22/7 - 1/791 = 355/113.
What do you use when doing approximate calculations? Do you rely on a calculator or do you still feel comfortable doing mental arithmetic?
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(Score: 5, Funny) by Anonymous Coward on Sunday October 23 2022, @07:20PM (8 children)
A slide rule.
(Score: 4, Insightful) by Immerman on Sunday October 23 2022, @08:43PM (3 children)
Single-digit scientific notation. Maybe two digits if I'm feeling ornery. Any more precision than that will be completely wasted on an approximate calculation. In my head for relatively simple calculations, or on paper or probably calculator for more for more complicated calculations that invite careless mistakes.
As for pi - I've never seen any point in memorizing approximations. 3.14159 fits neatly into short term memory, and is much more precise than needed for pretty much any calculation I might possibly want to do - at least without a basic scientific calculator that includes trig functions, and pi to many more digits than I have any use for.
If you assume anyone who might want to use pi will recall at least 3.14 with ease, then memorizing additional digits is a much better mental investment than memorizing fractional approximations:
- 22/7 is three more digits to remember, while adding less than one digit of precision. But *maybe* it might occasionally be easier to do the math in your head.
- 355/113 is six more digits to remember, while adding only four digits of precision, and is almost guaranteed to make the math more complicated, so why would you bother?
(Score: 2) by krishnoid on Monday October 24 2022, @01:39AM (2 children)
It's a little easier when you think of it as 113355 -> 113/355 -> 355/113 . Doesn't make the math less complicated, though.
(Score: 2) by Immerman on Monday October 24 2022, @02:50PM
That's exactly what I did, for the sake of argument I didn't even attribute any mental load to keeping track of where the / should be.
113355 is six additional digits to remember, and it improves on 3.14 by
3.14159292...
as compared to the actual value
3.14159265...
four and a smidge more digits. If you're going to memorize another six arbitrary digits associated with pi, you're pushing 100x greater precision with ...159265
(Score: 2) by Immerman on Monday October 24 2022, @03:04PM
Oh, wait, I see what you did there. Somehow missed the order flipping - I blame the fact that I'm still waking up.
I suspect though that remembering the more complicated rules takes at least as mental investment as is saved by the easier digit pattern.
(Score: 4, Interesting) by crm114 on Sunday October 23 2022, @09:23PM
Well thanks.
For $20, one can get a vintage Pickett slide rule from Ebay. My dad taught me how to use it to check my math work in grade school (4-6th grade for non-Americans); later he let me program his TI-SR40.
Fun memories for a weekend evening.
And me? I just fire up "bc" in my terminal.
(Score: 3, Insightful) by legont on Sunday October 23 2022, @11:33PM (1 child)
E6-B
Have a simplified version of it on my watch dial.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by crm114 on Monday October 24 2022, @12:41AM
Actually have used one. Touche.
(Score: 2) by driverless on Monday October 24 2022, @04:27AM
A better writeup on this would have been coverage of Liu Hui's algorithm and Zu Chongzhi's discovery of this particular approximation over 1,500 years ago. Entering this on a phone so you'll have to look up the references yourself.
(Score: 4, Insightful) by Opportunist on Sunday October 23 2022, @09:00PM (8 children)
I know how to do it, I understand how things come to be, that doesn't mean I use it in everyday programming. I don't solve programming problems in assembler, I use python or C, depending on the goal. Likewise, I know where that elusive 3.1415926... comes from, but in everyday calculations, I press the Pi-labeled button on the calculator and be done with it.
(Score: 0, Troll) by Anonymous Coward on Sunday October 23 2022, @09:27PM (5 children)
FTFY. No wonder you need to use a calculator. :P
(Score: 2) by martyb on Sunday October 23 2022, @10:08PM (2 children)
The first 200 digits of pi (π) are:
3.
1415926535897932384626433832795028841971
6939937510582097494459230781640628620899
8628034825342117067982148086513282306647
0938446095505822317253594081284811174502
8410270193852110555964462294895493038196
Wit is intellect, dancing.
(Score: 2) by Rosco P. Coltrane on Sunday October 23 2022, @10:16PM (1 child)
Did you just prove that you knew 200 decimals of pi or that you can copy/paste something?
Also, you posted the first 201 digits of pi.
(Score: 4, Funny) by martyb on Monday October 24 2022, @05:26PM
Both.
I had memorized Pi to 40 places in High School. Early in my freshman year of college, I discovered that the guy across the hall (hi Rob!) had Pi memorized to 50 places. The battle was on! Using the college library, I found a book that had Pi to 1,000,000 places. I photocopied the first page and gave a copy to my dorm-mate. Wrote the first 200 places on my clipboard which I referred to then walking between classes and memorized more and more.
In typical knurd(*) fashion, the battle was on! We called a truce when we had each memorized Pi to 200 places. Sort of. When we'd go out to a party and had had several beers, one would approach the other an call out "Sobriety check!" and would together recite Pi to 200 places. Ascertaining that we had accomplishes that successfully, we decided that we were not yet sufficiently drunk and called out "DRINK!"
That was 40 years ago. I admit that I have forgotten several decimal places from disuse, but I can still recite Pi to 120 decimal place. (The preceding was, as you suspected, basically cut-and-paste from the internet.)
(*) "KNURD" is "DRUNK"... spelled backwards ;)
Wit is intellect, dancing.
(Score: 2) by Opportunist on Monday October 24 2022, @06:54AM
I cut it off there, I didn't round. I think the ... made that obvious.
(Score: 2) by FatPhil on Monday October 24 2022, @07:58AM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by Rosco P. Coltrane on Sunday October 23 2022, @10:01PM
From math.floor(math.pi*1e7)/1e7?
(Score: 3, Interesting) by looorg on Monday October 24 2022, @12:13PM
With modern architecture and assembly language division is just another instruction as far as I know, that said I don't do assembly on modern architecture so I don't really know for sure. But from what little I have noted and been told there are instructions for it now so it's not really harder to write anymore compared to other languages. That said if you go back in time or use some old stuff then yes doing division (and multiplication) in assembly is a lot more instructions, usually involving a lot of subtractions, additions and storing the carry until something equals zero or goes into the negative. In some regard I think it's a good way to solve problems as it's a lot of breaking it down to the smallest possible instruction compared to just writing a few lines.
Or possibly you had some more general comment in mind about the arcane nature of assembly languages. Which is true. But still also great and interesting if you know what you are doing. Still while it might be fast in the execution, certainly on limited and old platforms, it usually is not fast in the writing/coding.
(Score: 2) by looorg on Sunday October 23 2022, @09:20PM (9 children)
Why would I use an approximation? Giving an answer in fractions or with known symbols is fine. An answer in the form of a fraction is just as valid as another answer, and technically better then a decimal answer as it would be more accurate since you are not rounding it off eventually. If I have to give an answer with PI, or some other known symbol or quantity, just write that then. It's a valid symbol in an equation and can be one in an answer to. Using symbols is fine as long as everyone involved knows what it is, or you have clearly stated and defined whatever symbol you feel like using represents. In the case of PI it's the ratio of the circumference to the diameter so if you have your circle figure then just put those in then if you don't like writing PI in your equation. Then you don't have to approximate. But I, nor you, don't have to compute it out to digit or decimal form, that is for people that stopped taking math after elementary school and just doesn't know any better.
The only approximations I do in my head are the once I do in the store to figure out a approximately how much I am spending on groceries this week or how much this or that product actually costs if I get a some kind of discount but they for some reason refuse to put down the new price (you know the once -- take 3 pay for 2, 24% discount etc).
Also why are we discussing a 12 year old blog post? Why did it become relevant again?
(Score: 1) by Runaway1956 on Sunday October 23 2022, @10:14PM (2 children)
Hated fractions as a kid, won't use them as an adult. Back in elementary school, I discovered that I could convert fractions to decimals in my head, do the problem, and get the right answer. Dumb teacher insisted I show my work, so we never got along. I'd be happy if the whole world just forgot fractions, and did everything with decimals. Imagine my surprise and delight when I discovered the engineer's scale. No more fractions in building!! But why stop there? Let's just go metric, and be done with it!
Abortion is the number one killed of children in the United States.
(Score: 2) by looorg on Monday October 24 2022, @12:52PM
I didn't like it much or understand the point of it either back in elementary school. I blame the teachers and the curriculum. It's not really something they go into great detail about anymore as far as I understand it. But elementary school math is more about memorizing things, crunching the numbers and taking some information and plonking in the correct number in a known formula and then computing it. Valuable skills by themselves but it's a very small part of mathematics. Fractions became very useful at University when you started to do math. After all then it's no longer about just giving the answer to some equation. Also you notice, if it somehow got past you previously, that the real world doesn't give problems and answers in natural numbers and integers. Instead then it's more about working it and writing it and then fractions become so much more useful then decimal form.
That said either way you really should join the rest of us in the metric world. Now it seems you are just holding on to be obstinate.
(Score: 2) by kazzie on Monday October 24 2022, @05:06PM
There are a few areas I find fractions to be more convenient to work with: probability and multiple parallel resistances are two that come to mind.
An ex-teacher of mine used to do an annual high-flyers maths course, and worked through their topic (the name of which escapes my memory) using fractions to calculate results. Even though the audience were the more gifted students from local schools, he found they were uncomfortable using fractions, and later switched to adding decimals rather than adding fractions. I played classroom assistant some time after that switch, and found that the elegance of combining fractions made the conclusion of the work a little disjointed, and several students still got lost when asked to convert their values over to decimal.
These days I'm often stood at the front of a classroom doing some electroincs-related maths on a whiteboard. I don't stick slavishly to pre-calculated examples and notes, so I get a fair practice in mental (or pen-and-paper) maths as a result. I'm always very up-front with asking my students to double-check my work (not every mistake I make the board is intentional!), and will occasionally ask them to calculate a value for me when I've confused myself, or don't fancy the number-crunching myself. I usually manage to work out an approximate value to check against their figures though, often before they've managed to finish pressing buttons.
Now and again I'm asked how on earth I managed to work that out in my head. I'm always happy to take a brief detour to revise math skills that they may not have quite mastered at school...
(Score: 4, Touché) by Rosco P. Coltrane on Sunday October 23 2022, @10:20PM (3 children)
I think you'd immediately get a Fields medal if you can post the exact value of Pi as a fraction.
(Score: 2) by acid andy on Monday October 24 2022, @12:59AM
π / 1
Master of the science of the art of the science of art.
(Score: 2) by looorg on Monday October 24 2022, @12:30PM (1 child)
Firstly I'm to old for a Fields medal. Secondly I am not a big fan of pure mathematics and as I recall it those are usually the once that get a Fields medal. Thirdly I'm not that great a mathematician to begin with, not a child prodigy in that regard.
That said I don't think that would be of much interest to them, after all we already have a good enough definition of PI describing it as the ratio between a circles circumference and diameter. So what more is there? Sort of like I don't think they would be very interested if you somehow managed to calculate PI for some specific case in decimal form to it's last digit, or I guess that could be kind of interesting but I don't see it happening anytime soon or that it would somehow have any or many practical applications.
(Score: 2) by Rosco P. Coltrane on Monday October 24 2022, @01:26PM
Dude... I guarantee you, if you find the exact fraction for an irrational number, they'll be all over you you syphilis on a medieval monk! You'll be a superstar!
(Score: 0) by Anonymous Coward on Monday October 24 2022, @01:20AM
You really need to work on your voice to text program. It gives out some really stupid stuff.
Other than that, your posts are somewhat interesting.
(Score: 2) by aafcac on Monday October 24 2022, @03:57PM
There's plenty of times where 22/7 is close enough. I fail to see a situation where the longer 355/113 doesn't really. There's a bunch of times where you might want to estimate the size of a circle and 22/7 is probably going to be acceptable and likely something you can do without a calculator.
But, really, even cheap calculators these days will have a Pi button on them, so there's really no point in any approximation that isn't at least 7 digits of precision.
(Score: 3, Insightful) by Rosco P. Coltrane on Sunday October 23 2022, @09:46PM (1 child)
Fractional approximations of pi are cute, but they are of no practical value if you learn to recite a decent number of digits of pi and remember them for the rest of your life.
It's not that hard: after all, everybody learns multiplication tables that are much larger. Then, all you gotta do is punch that number you know by heart at full tilt on your calculator and you're done. No division, no remembering 6 digits to give you less precision than remembering 6 pi decimals directly.
(Score: 2) by legont on Sunday October 23 2022, @11:42PM
Кто и шутя и скоро пожелаетъ Пи узнать число ужъ знаетъ
That gives 11 digits, but requires Russian before Lenin
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 5, Funny) by theluggage on Sunday October 23 2022, @09:52PM
Intel Pentium…
(Score: 3, Touché) by Rosco P. Coltrane on Sunday October 23 2022, @10:10PM (2 children)
The Indiana Pi [wikipedia.org]. Much easier to remember.
(Score: 0) by Anonymous Coward on Monday October 24 2022, @01:29PM
Indiana Pi Bill [wikipedia.org]
corrected link (original missing an "L")
(Score: 2) by aafcac on Monday October 24 2022, @04:01PM
And here I thought they were finally making a movie about Dr. Jones younger and somewhat nerdier brother's adventures in forensic accounting.
(Score: 2) by oumuamua on Monday October 24 2022, @12:21AM (2 children)
You know, Pi day, or next best Tau day:
https://www.internationaldays.co/item-detail/tau-day-2022/r/recsYn2CcFP28xIIi [internationaldays.co]
Find all the stuff going on for any particular day:
https://www.internationaldays.co/today [internationaldays.co]
BTW be sure to eat a bologna sandwich tomorrow:
https://www.internationaldays.co/item-detail/bologna-day-2022/r/recmDfnvgVgf25fZt [internationaldays.co]
(Score: 4, Touché) by janrinok on Monday October 24 2022, @06:32AM (1 child)
Subs get submitted when they get submitted....
We were not going to sit on it until next year. And as Pi day only works with the US system of writing dates for most of us it is irrelevant when we publish it. :)
(Score: 3, Funny) by kazzie on Monday October 24 2022, @05:08PM
Sometimes irrelevant, always irrational.
(Score: 3, Interesting) by krishnoid on Monday October 24 2022, @01:51AM
Per the article, 355/113 - 1/3748629 (handy mnemonic: no repeated digits in the second fraction) is much better, and you can memorize e as 2.7 1828 1828 45 90 45. So both of those are better than 1 in 1E12. Wish I'd known that before I memorized so many more digits of pi.
(Score: 2, Interesting) by pTamok on Monday October 24 2022, @01:05PM
I know my 7 times table better than my 113 times table. If I'm doing mental calculations to check if a result from elsewhere looks about right, then the simpler version is adequate.
Having a feel for numbers, and checking results from calculators is a good habit to have. I can sum a column of numbers faster manually/mentally than using a calculator, and by writing down intermediate results, I can have more confidence in the result, and recheck quickly. If I ask someone else to check a total for me, they often pull up a calculator on a mobile phone, do a lot of rapid tapping, and come up with a wrong answer, either because they have missed an entry or mistyped/tapped; and to recheck, they have to do it all over again.
(Score: 2) by Freeman on Monday October 24 2022, @03:39PM
In the event that whatever built-in calculator in the OS I am using doesn't work for me. Python will do the job.
Joshua 1:9 "Be strong and of a good courage; be not afraid, neither be thou dismayed: for the Lord thy God is with thee"
(Score: 1, Interesting) by Anonymous Coward on Monday October 24 2022, @11:09PM
If anyone is curious why certain irrational numbers have good rational approximations... voila! [youtube.com]