Finding Pi by Archimedes' Method
ฝัง
- เผยแพร่เมื่อ 8 เม.ย. 2012
- Archimedes approximated the value of Pi by starting with the fact that a regular hexagon inscribed in a unit circle has a perimeter of 6. He then found a method for finding the perimeter of a polygon with twice as many sides. Applying his method repeatedly, he found the perimeter of a 12, 24, 48, and 96 sided polygon. Using the perimeter as an approximation for the circumference of a circle he was able to derive an approximation for Pi equivalent to 3.14. This video uses a somewhat simpler method of doing the same thing and carries it out to polygons with millions of sides. All that is needed to understand the calculation is knowledge of the Pythagorean Theorem.
For a followup (showing both the inscribed and circumscribed polygons) see • Finding Pi by Archimed...
For a good explanation of Archimedes actual steps used to carry out his calculation see
itech.fgcu.edu/faculty/clinds...
See my website: mathwithoutborders.com
Whenever I'm feeling down on myself I just come here and read all the rave reviews!!! I'm just doing what I do. Thank you all for your appreciative comments.
Thank you for taking the time to make the video. I really enjoyed it.
You taught me something new! Thank you very much!
The idea of pi being infinite drove me nuts until I found this video and discovered that it is infinite In the sense of the ability to decrease the digitized granularity represented to a higher resolution infinitely. I can rest at peace now.
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@@jeremybarbarek9665
When I was a schoolboy a wise person said to me, "Whatever it is that you end up doing in life, do it well."
Remembering those words I do what comes easily and naturally to me. I have gained a lot from you doing what you do. Thank you!
it took me 40yrs to understand how PI came to life from this video. my teacher said just take his word on the value of PI. thanks you for this video and thanks for your hard work
Well honestly, your professor kind of did that to help you. People often question "why is pi that value" or "why is e that value" which is not bad, but there is also no explanation to it other than "that is just what it is, the nature of the thing produced that value". Usually, the existence of these numbers are already explained through their definition, but it is what it is, their definition. This video just shows what other scenarios that has the nature of pi, not explaining where pi comes from.
*which is not a bad question to ask
@@dan-gy4vu ehh i heartily disagree with this answer. it kinda discourages other students from asking those "why" questions, because those "why" questions are what drives scientific innovation and discovery in the first place. it's in our human nature and it would be destructive to suppress it. i started suppressing my "why" questions my freshman year of high school and a couple years later i deeply regretted my decision, because i later went back and found out all the beautiful math i missed. i think this is partly the reason why so many math students hate math in the first place: their teachers don't take the time to help them understand the beauty behind all the formulas and the numbers that seem so random and difficult to understand. it truly is amazing, but students see it as boring and useless because they're almost never taught the fundamentals. keep asking why, no matter what anyone tells you, and never, ever just "accept" things for what they are. we wouldn't be as far as we are in science if that was the collective mindset. hope i helped :)
@@goofygoober6211 I understand and I 1000% agree with what you have said. I don't want encourage to stop asking why; but sometimes we have to accept that math is messy and sometimes is random. there is no answer why Pi is that number, it is what it is. A student can definitely ask "why pi is that?" and the only answer you can give is " because that so happens to be the ratio of the circumference of a circle to its diameter". I guess I should change my wording into "always ask why, but sometimes you just have to accept there is no why and it is what it is because the universe doesn't care about clean numbers and if humans can comprehend what's happening".
@@dan-gy4vu Little more close to the truth would be to say that because we have (most of us) ten fingers.
This is very clear. A great overview of Archimedes' method combined with effective use of GeoGebra and a spreadsheet. I like, in particular, how the software used does not obscure in any way the mathematics being presented.
Thank You, 4390 views and no comments saying thanks to you, is sad.
Your work is awesome, thanks for combining mathematics with computer resources, specially with open source resources.
Yes, Archimedes was a genius far, far ahead of his time. Just imagine doing that longhand using Roman numerals!!!
@François Miville Any source?
@François Miville any sources on your suggestion that the romans killed him for divulging secret knowledge to the small city? Or is that just something your assuming. And how do you know he did nothing new?
Yeah..and he has many discoveries too..and invention like the simple tools..so his not fucos on that pi thing.
A bit of necroposting, anyway Archimedes was Greek and used the Greek numerals, not Romans. The difficulty anyways was similar to using Romans numeral. A link to the Greek system: www.foundalis.com/lan/grknum.htm#ancient
He used Greek representations of numbers, which was based on place value like ours. However another big hurdle for him was they didn't have the equivalent of decimal fractions. He had to cope with the calculations using common fractions. Try that in your spare time!
The thing that I should have known at the age of 15, I came to know about it at the age of 60 !!!!! 45 years behind. , but better late than never! !! Thanks for your hard work and for today's technology.
What would Archimedes say or think if he saw how fast you found the value of pi using a spreadsheet.
Those things wouldn't exist if he hadn't calculated pi back then the way he did. You will find that geometry is the foundation of most (if not all) great scientific innovations and inventions.
EDUARDO12348 Cheater!
you missed the point
We have spreadsheets and CAS everywhere but no one is using it to make students/or others work faster.
EDUARDO12348 n(
around 14 minutes, he means the number of digits -- which leads us to a concept called the "machine epsilon" -- the smallest value of a number that can be shown.
THANK YOU, THANK YOU VERY MUCH! I searched all over the net for something to help me understand Archimedes method and yours is the only one that was VERY WELL TAUGHT.
Best video on Pi on the net.
3/14/15 is just over. (It's just past midnight). Since it was "pi day" and there was so much attention on the topic, I tried (and saw videos of) different methods. 1. Adding fractions: 4/1 - 4/3 + 4/5 ... . First in a spreadsheet; then with an HP calculator program. 2. Monte Carlo Simulation. (Another video). 3. Archimedes. In just 23 rows... such precision!!! So many decimals, that are so costly (data-wise) to get through the other methods. Amazing! Thanks for such an excellent presentation of the topic. Bests!
>how to find pi by archimedes method
>(pulls up an excel spreadsheet)
He showed you Archimedes' mathematical method with the aid of a worksheet. It would be like criticizing him for using a pencil.
@@RR-mp7hw pretty sure he's joking...
Li Hua Where's the humor or cleverness in the "joke," please?
@@jasoncall3731 are you saying you have neither?
Li Hua Dear Yes. Please explain to me why, in 2019, I should be amused at someone using a computer to avoid longhand calculation. I want to be clever and funny like you.
You showed such an elegant and coherent method to described the finding of pi. Thank you so much.
355/113
accurate to 6th digit after comma.
And circuit circumference is C=2r(pi)
Pi has been a mystery to me. At school, we were taught that Pi equals 3.1416. However, we were never told how this value was derived. From this video, I have learnt how to find the value of Pi using the Archimedes method. I followed the instruction and put the values and functions onto my Excel spreadsheet. In an instant, the computer worked out the perimeter of a polygon with over 25 million sides and the value of Pi accurate to 15 digits. It was like magic! Wow, it was mind-blowing! Thank you so much.
There is no more to say except: thank you for taking the time and the patience to upload this video.Merci encore mille fois.
Very nicely done. Really appreciate you taking the time to explain everything completely, leaving nothing out. To think that Archimedes worked out the method, and did the maths, all that time ago, is quite humbling.
Beautifully done - and explained with crystal clarity. Kudos!!
Best video on Archimedes' approach 👍 You explain things well.
Very well done. I like that it doesn't use trigonometry like some solutions do and that it leverages the spreadsheet to do calculation.
Archimedes lived so long ago, he had to rely on Lotus 1-2-3!
Trigonometry (at least in radians) is dependent on the value of pi so that wouldn't even make sense.
lol especially since trig is derived from pi itself. that, my friend, is what would be called a circular argument
They he didn’t show you the hard work of actually calculating all of those square roots.
OUTSTAAANDING Presentation! I am an Excel user, and this is by far is one of the best demonstrations of the calculating the value of Pi. Thank you.
Thank you for pulling everything together. I appreciate your critical thinking around this.
Excellent video explaining something I never took the time to learn and just took for granted!
I really really loved this video!!! It is so well scripted and structured, showing us a real & practical!!! method on how we can actually calculate pi!
Thank you so much!!! :D
Oh this is truly, truly, beautiful. I admire your Excel approach. Thank you for this video!!
I just got this in my recommendations and I'm not disappointed.
This was so satisfying!
I was looking for this video for a couple of weeks now. I did my ninth grade 7 years ago but wanted to go through it again just to grasp the concepts. Except this time around, I want the concepts to be more clear and make more sense so I am using all the resources I have to learn whatever I can. Thanks again for this explanation.
Great to see this Archimedes method fot Pi. I am pleased because I arrived at this method myself some years ago.
Man...you are blessing to humanity by explaining this on youtube...thanks :D
This was so great - enjoyed hearing the finer details!
I just discovered your wonderful video.
It was a fantastic review of Archimedes 200 BC
Method of calculating pi
Using simple geometry and a modern spreadsheet.
Actually, I used Microsoft Excel 2000.
I was a math major in 1970
And my daughter is a math teacher.
I'll have to show this to her.
Thanks for posting this!!
I have added a link in the notes above to an article explaining the actual (historical) steps Archimedes used in his calculation, using a theorem about angle bisectors. He used two separate procedures for inscribed and circumscribed polygons. Keep in mind he usd common fractions because he did not have access to decimal numbers for fractions. My method starts from the same point, follows the same general pattern, and achieves the same result, but I think would be easier for students to understand and carry out with a calculator and/or spreadsheet.
For circular area, pi is okay. But for circumference the constant is more than pi, because in tiny angles the length of tan base line and sin base line are practically identical, WHILE there is still space between both tan and sin base lines. So the arc becomes longer even than tan base line.
Says the link is unavailable!
@@jaafars.mahdawi6911 I know. I have looked for that article elsewhere with no luck so far. let me know if you find it or the equivalent.
Nice demonstration of Archimedes' method! If only they had Excel in ancient Greece, Archimedes could have computed what took him a lifetime in just 5 minutes!
Great presentation on Arch.'s method followed by an entertaining and instructive spreadsheet romp. Thank you! I shall dream tonight of 25-million-sided polygons.
Great job - I was surprised how hard it was to find an explanation of how the Archimedes method worked without trig (sin/cos/tan), finally stumbled across this video. I realized the method in the first minute (starting with a hexagon, d'oh), but let the video run while writing this comment so you'd get an extra "view"!
Hey man, thank you very much for this video! I've always wanted to know how Archimedes did this calculation and I didn't find anywhere else. Thank you for your dedication. Regards from Brazil!😄
the best demostration of pi, thank you
Great video. Used PI throughout college and throughout Calculus courses, but took PI for granted. Nice to learn. I was able to do it on MS Word.
Exceptionally clear explanation! Cheers 😄.
thanks for this video i write a compartment work about pi and archimedes and i didnt understood how he is getting pi, but i saw your video and now i know how he do it. thanks a lot :)
Very good tutorial, no other youtube tutorial was this good!
Well done. I'm adding this video as a favorite under my Math videos. Thank you.
Loved your method. Thanks for sharing
The best explanation ever provided
Wonderfully clear explanation of the method Archimedes used, using the hexagon to start with and polygons of increasing number of sides. What I don’t understand is how did he get the accurate square-root of all the numbers he would have had to compute as he went along? I tried this by hand a few months ago and that is the problem that stumped me.
Man, you're so awesome, you made it so simple
thank u! the best explanation videos out there
Thanks for this amazing and comprehensive video! :D
Beautiful explanation friend. Very helpful.
thank you so much, i have an assignment on this and this video clears everything up. Again thank you.
simple and perfect explaination. thanks
Beautiful explanation!
thank you for this great explanation
very helpful and easy to understand, Thanks a lot.
Nice work very well explained
Very neat!
It should be remarked that, while Archimedes *did* work this inscribed-polygon method, to find a lower bound for π, he also used a circumscribed-polygon method to arrive at an upper bound.
And not having access to modern computation methods, or even decimal notation, he used those results to enclose π between a pair of fractions:
223/71 < π < 22/7
In decimal form, that looks like:
3.14084507... < π < 3.14285714...
Interestingly, the simple average of those bounds is a pretty good approximation: 3.14185...
Fred
ffggddss do you know how Archimedes calculated the roots?
@@thomaspaine5601 That's a really good question; no, I don't. (Some other commenters might have that knowledge - anybody?)
He didn't, for instance, have what we now call Arabic numerals, or algebra, all of which would only be invented centuries later.
But there's a problem he famously posed, called the "cattle problem," that involves very large numbers, and winds up, in the way it must be solved, requiring a solution for Pell's Equation,
b² = na² + 1
in positive integers, a and b, where n is a positive, non-square integer. In Archimedes' cattle problem, n turns out to be a many-digit number, putting the answer thoroughly beyond any possibility of solution without automated calculating machinery. And in the way he poses the problem, anyone with any sense of that difficulty, can clearly see Archimedes' tongue planted in his cheek, trying not to break out in uproarious laughter.
But the point is, he probably had ways to solve Pell's Equation for at least some reasonably-sized n's.
And such a solution provides a very good way to approximate √n with a rational number:
√n ≈ b/a
[In fact, this fraction will always be a slight overestimate, b/a > √n. Sometimes, n has a solution for the "negative Pell's Equation,"
b² = na² - 1
which will always give a slight underestimate, b/a < √n.]
So it's my suspicion that this was involved somewhere along the line. Essentially, he would have to take the radical expressions he got, and find upper and lower bounds in rational numbers for them, using Pell's Equation solutions.
NB: The name, Pell's Equation, came from a misattribution by Leonhard Euler, of an analysis of the problem by John Pell, who was merely recording the work of others, notably Brouncker. The problem itself is much older, including its treatment in India centuries before.
Fred
@@ffggddss The thing is he cannot keep taking rounded numbers forward for continued iterations of his method because it will drift further away from the actual limit. He would also need to know the level of precision he is expecting by the time he gets to 96 sides and be working in excess of that accuracy. How could he possibly know? So one is left wondering if his method is sound in principle, indeed a manifestation of genius, but that he did not, in fact, have the 'tools' to actually carrying it out.
@@thomaspaine5601 He was, I'm sure, very aware of accumulation of errors, and I imagine what he did was to keep strict track of upper and lower bounds throughout any given lengthy calculation. This was well within his intellectual abilities.
I'm not an Archimedes scholar, but any blunder of this sort would most certainly have been pointed out by others in the intervening centuries, especially considering that this is perhaps the most famous calculation in the history of mathematics.
Fred
@@thomaspaine5601 It's not clear in the literature exactly how he did it, but I know how he could in principle have done it. Try applying the divide and average method with fractions! It works out nicely.
Illustrating the numberness in potential possibilities, of e-Pi-i interference positioning, resonance of temporal Eternity-now location, the Interference/probability vectors interval, and the Superposition-point Singularity Hologram.., of pure relative motion (implied by superimposed/axial-tangential relative number values/resonance), QM-TIMESPACE., Time Duration Timing Calculus.
Everything happens all at once in a complicated mess of superimposed AM-FM continuous creation connection Principle from which the cause-effect of Time functionalism has to be extracted/abstracted. Physics Mathematics by discovery/measurements and reiteration methodology, ..and reverse process, Pure Dynamic Mathematical (disproof) Abstractions methodology, as demonstrated.
Very useful demonstration, thanks.
That was a great explanation! Thanks!!
Thank you sir.
Your video was very helpful!
very fine video, but i miss a proof for the lenght of the hexagon-side… consider angles and the euclidean thm. about the sum of the angles in a square (and add a proof for that too)….
that problem is a true masterpiece!
Finally someone who describes how archimedes found the circumference of a circle without knowing pi!
Great explanation! Archimedes is one smart dude!
Very well done, you explained it perfectly understandable even for a math-dummy like me.
Thx
This is amazing, thank you for making this!
Good job. Very fast way of calculating pi!
Thank you very much. Very enlightening.
Well done man, you nailed it !!!
Simply elegant.
Thank you for this calculation to determine pi
Thanks you so much! I know this is more than three years old but you helped me a lot. Thank you :D
Thanks for your help. I think nothing beats the old chalk board, ruler, and the math teachers finger, or wooden pointer to focus attention on the exactitudes of spacial problems. Laser pointers, and computer cursors, just don't get the job done. Not only that, when you have a mathematics dwarf like me to try and teach, "good luck" you can always beat them with the pointer sick. Try that with a laser pointer.
Thanks! Great explanation!!
Very well explained.
thanks for the concise info
Very helpful. Thank you very much.
I have to agree with a previous post, this is the best pi video on TH-cam. I've seen some horrible ones.
Love u so much, it was easy and nice explaining. You saved me from homework
very clear, very helpful , thanks
Great video. Thank you
Very nice!
ty, wonderfully explained
In Excel 2007, you will eventually get to a situation where the pi approximation jumps around. I'm nearly sure it is because the numbers involved are becoming so large that Excel is rounding in a detrimental way (stack overflow sorta deal). Also, you can do all this in one equation:
Y=sqrt(1/2 - sqrt(1-x^2)/2)
and just iterate from x to y.
Computer solutions always eventually run into truncation error.
Thank you, David! In the past, when I've tried to figure this out, I always tried to find the areas. That was too hard for me (without more motivation). If I'd pointed myself this direction, I bet I could have figured it out. (Darn it!) I just put it on excel. How satisfying to see pi come up. I might do it in python so I can get more digits.
I did this method originally in class with a hand-held calculator and a table of values on the chalk board (yes...chalk!). I only later added the spreadsheet. The idea is to use the tech to eliminate the tedium that obscures the math. The method here is not the spreadsheet, it is the algorithm.
Great video. Thanks, MWB.
Thank you for making this Video - Came back here to save the Video and learn more about Spreadsheets
Right. Spreadsheets are not just for financial calculations. They constitute a "visible" programming language.
@@davidschandler48 ..Thank you again David and for your web site.
Mathematics can be so visual ( and fun ) like with the fibonacci sequence, & the golden ratio .. they have even found 'shapes' and forms in the 3X+1 puzzle .. I must investigate more on the mandelbrot set.
Fair Play to our old friend Archimedes - doing the calculation by hand !! Did they even had log tables back in 200 BC ?
@@peterpauldonoghue7024 The hardest part for Archimedes was finding the square roots, given that he didn't have decimal numbers to work with. He was a bit obscure in how he did it, but I figured out a way, using ordinary ratio-type fractions. Take that as a challenge!
Love it! Keep up good work,
Excellent! Thank you.
nice. Good information. Kudos
Imagine how amazing the education system would be if we taught stuff like this at a younger age. Anyone can understand this topic, even toddlers. Leading them to an actual understanding of mathematics in the future rather than memorizing that pi is simply a number
It hurts my soul and existence that we cant find exact value of anything in this universe. That our brain is struggling with term of "infinite".Our whole life and existence is approximation of reality. I was always pushed with my curiosity and while I learned more I proportionally felt more sad and miserable about life and universe.
Thank you for this excellent video.
Perhaps the investigation of transcendental numbers can give hope instead of sadness. The irrationality of pi is not a defect in the universe or existence. It does remind one to stay humble when making universal claims. It is beautiful that the numbers that organize the physicalworld, pi, phi, e, etc. are all knowable and real but exceed the limitations of our decimal system. They point to the reality which is only accessible by reason and inexhaustible in its nature.
Great video. The trouble with modern maths education is not doing lots of examples like this before being introduced to trig.
Very well done sir
RIPi Archimedes
very helpful. so much thank u. really very helpful
I got a excel tutorial in this as well lol
Archimedes only calculate a range of pi. I’m going to copy/paste a summary of how he did it. Then you’ll have ‘the rest of the story’.
A similar approach was used by Zu Chongzhi (429-501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method-but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for π, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.
Absolutely lovely, simple and clear.
Except for that moment at 3:22 where you put S1 next to a line segment (but it refers to the whole line) and S1/2 next to another segment (but it refers to only the segment).
This notation is very very misleading and ambiguous.
tq so much for tis video,it help me a lot
Here are SIX ways how to find the area of a circle of radius equal to one(1) where the area would be equal to pi.
1. summing up the triangles to obtain the area.
2. starting with a square to add the main trainables. to find the total area
3. Starting with a hexagon to add the main triangles to find the total area
4. Using polygons with different numbers of sides and finding the area.
5 PRINT "SUMMING OF SMALL TRIANGLES "
N=2:A=1:B=1 : AREA=0
FOR X=1 TO 20
AREA =AREA + (2*N^X)*A*B/2
A=0.5*SQR(A*A + B*B)
B= 1-SQR(1-A*A)
PRINT AREA
NEXT X
PRINT :PRINT PI
STOP
10 PRINT "SQUARE START"
A=1 B=1
FOR X= 2 TO 14
A=0.5*SQR(A*A + B*B)
B= 1-SQR(1-A*A)
AREA=(2*2^X)*(1/2)*A*SQR(1-A*A)
PRINT, 2^X;" "; AREA
NEXT X
STOP
PRINT " "PI
20 PRINT " HEXAGONAL START"
A=1 B=0
FOR X=1 TO 14
A=0.5*SQR(A*A + B*B)
B= 1-SQR(1-A*A)
AREA=(2*3*2^X)*(1/2)*A*(1-B)
PRINT, 2^X;" "; AREA
NEXT X
PRINT " "PI
STOP
60 PRINT "OTHER HEXAGONS"
FOR N=200 TO 220
P= 2*PI/(2*N^2)
A= SIN(P)
H= COS(P)
AREA = (2*N^2)*A*H/2
PRINT AREA
NEXT N
PRINT PI
STOP
80 FOR X= 40000 TO 40100
P= X*SIN(PI/(X))
PRINT P
NEXT X
PRINT PI
STOP
100 P=.0000001
FOR X= 0 TO 1 STEP P
AREA= AREA + (SQR( 1-X^2))*P
NEXT X
PRINT 4*AREA
PRINT PI
thank you for this clear explaination !!!!! you are helping mr with my homework rip
Great!
I did a bit same
For any angle A
Arc (A) = N* Arc(A/N)
Arc(A) ~= 4* Sin(A/4).......Using trigonometry
a + b = R.....R is the radius
a = R * Cos(A/2)
b = R*( 1 - Cos(A/2))...............1
Si = R* Sin(A/2)
Si = LK*Cos(Angle KLJ)
LK*Cos(Angle KLJ) = R* Sin(A/2).............2
b = LK* Sin(Angle KLJ) = LK* sqrt(1- Cos(Angle KLJ) ^2)
b^2 = LK^2 *(1- Cos(Angle KLJ) ^2)
2*LK ~ = ARC (A)
For A = 180 Deg
ARC (A) = Sin(180/4) = 4*Sin(45°) = 4*(1/sqrt(2)) =2 sqrt(2) =2.8184
= 8*Sin(45/2) =8* 0.382683432 = 3.061467459
=16*Sin(45/4) = 3.121445152
Sin(A/2) = sqrt((1-Cos(A))/2 always
Trig depends on pi, so this argument is circular.
Awesome video!!!!!!