Continued Fractions - Professor John Barrow

[@roberthillier80]

That was an excellent lecture by Professor Barrow. Thank you for making it available.

[@ffggddss]

I really found this exposition highly informative and thorough. But at around 23 to 24 min, I have a slight quibble with his phrasing of the fact that large numbers in the cfe indicate a very good rational approximation (RA): Yes, they do; but more precisely, what they indicate, is that truncating the cfe immediately *before* a large number, gives an unusually *efficient* RA. You can reasonably define such representational efficiency (RE) as: • inverse absolute fractional error (the bigger that inverse, the better) • divided by the product of numerator and denominator (the bigger that product, the worse, because you're using more "numerical real-estate" in your representation). This (or perhaps, its logarithm) is a good way to quantify just how good a given RA is, and a table of such values would bolster his points. This also makes for a quantifiable comparison with decimal representations, where you can divide inverse fractional error by the (truncated) decimal representation with the decimal point removed; e.g., RE(π, 3.14) = (π / |π - 3.14 |) / 314 = π / (314·|π - 3.14 |) = 1 / (314·|1 - 3.14/π |) ≈ 6.3 RE(π, 3.1416) = 1 / (31416·|1 - 3.1416/π |) ≈ 13.6 If you do this for 3/1, 22/7, 333/106, and 355/113, as RA's for π, e.g., you'll see that 3/1 is good, 22/7 is better, 333/106 is pretty crummy, and 355/113 is exceptionally good. And you'll also see that the RE is close to the first omitted number in the cfe: RE(π, 3/1) = 1 / (3·1·|1 - 3/π |) ≈ 7.4 RE(π, 22/7) = 1 / (22·7·|1 - 22/7π |) ≈ 16.1 RE(π, 333/106) = 1 / (333·106·|1 - 333/106π |) ≈ 1.07 RE(π, 355/113) = 1 / (355·113·|1 - 355/113π |) ≈ 293 Now there is a credible sense in which truncated cfe's always yield the "best" RA's of a given real number, x - namely, they are closest to x of any fraction with equal or smaller denominator. But truncating just before a "1" will produce a "best" RA that can often be substantially improved by including just one more term. I have a remedy for this circumstance which I will post in a follow-up comment.

[@ffggddss]

Following up my earlier comment (please read that one first): What is presented in this lecture is the 'traditional' way of doing cfe's - record the integer part, invert the fractionsl part (which is just the current number minus its integer part); repeat. This is historically accurate ("It's how we've always done it!"); and it gives terms which are all positive integers, and which, when truncated and converted to "simple" fractions, give "best" rational approximations (RA) to a given real number, x, in the sense that there is no fraction with equal or smaller denominator, that is closer to x. But "best" isn't always so good; and is sometimes "crummy," as I've pointed out in my previous post. What I propose is a modification of that process, in which you take, not the integer part, but the *nearest* integer, to the quantity at hand, and then subtract that to obtain your new "current" value. This widens the pool of cfe terms to include negative numbers; but it also removes 1 and -1 from that pool. And by eliminating the 1's, it also avoids the "crummies." Under this new rule, the cfe sequence for π goes from: [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, ...] to: [3; 7, 16, -294, 3, -4, 5, -15, ...] Note that the simple fractions generated by the portions of both sequence I've shown, are identical! 80,143,857/25,510,582 And the new rule produced that fraction considerably quicker. This will also change the holder of the "most irrational number" title - φ = [2; -3, 3, -3, 3, ...] with (-3, 3) as the repeating part; √2 = [1, 2, 2, 2, 2, ...] with (2) as the repeating part. That is the smallest sequence of terms possible under the new rule (in which any ±2 must be followed by a term of like sign). So now √2 takes the "most irrational" title. And this is backed up by the representational efficiency (RE) calculation I defined in my earlier post. Also of interest: the new form gives e = [3; -4, 2, 5, -2, -7, 2, 9, -2, -11, ...] - - - also a rather nice, predictable pattern, although it starts out a bit chaotically. If you break the rules a bit at the start, you can get the more pleasing form: e = [1; 1, -2, -3, 2, 5, -2, -7, 2, 9, -2, -11, ...] A caveat: If you've fiddled with this new rule, you will have noticed that, even sticking to positive x values, any given partial fraction (PF) can be either +/+ or - / -. So it's to be understood that as a final step in finding the PF, you take absolute values.

[@rskissack]

Thoroughly enjoyable!! Thank you very much for posting this!

[@DaveSalwinski]

What he said about the series 55 min in is incorrect. It is an open problem whether or not that series converges. Its convergence would imply the irrationality measure of pi is

[@DaveSalwinski]

Its called the Flint Hills series by the way

[@briandenzer]

Thoroughly enjoyable, and informative. I've been struggling with the logic of how to calculate gear teeth for, precisely the application Huygens was trying to resolve. Nowhere in weeks (months) of occasional research have I found a better presentation. Nicely done. Thank you. I am, perhaps with modest intellectual means over here in math-challenged United States, to balance the deficit to, perhaps 2.99 out of every 2 people having difficulty with fractions (or 9/6 people, ha ha ).

[@jimtayler555]

Great seminar.

[@BubbaYoga]

Fascinating. Thank you.

[@TheHumanParacite]

Excellent seminar :)

[@sradtke314159]

Most interesting presentation.

[@UnderAcheiver1]

So at 12:45 we see what you write it as when all the numerators are one. How do you write it if they aren't?

[@junkdubious]

Do continued fractions say anything specifically about transcendentals or just about irrationality in general?

[@CORD1BUS]

transcendental continued fraction integer sequences are sometimes periodical (e = [2; 1, 2, 1, 1, 4, 1, 1, 8, 1, ...... ,1, 2n, 1,....]) but sometimes seemingly completely random (pi = [3; 7, 15, 1, 292, 1, 1, 1, .....]

[@bonbonpony]

If this is a "second lecture", then where's the first one? Is there any playlist of these lectures in order?

[@GreshamCollege]

This lecture is in a series entitled 'Maths is Everywhere' from Professor Barrow's Geometry series 2010/11. The first lecture of the series can be found here: kzbin.info/www/bejne/e3jZaKSliMtoesk. A search of the lecture title on our website will direct you to a link to the series it belongs to. We are working to make our KZbin playlists more detailed in the future. I hope this answers your question!

[@bonbonpony]

GreshamCollege Yes indeed, thank you very much :) Very good lectures.

[@rob876]

why is 104384/33215 such a bad approximation for π? This gives 3.14267650158... According to the slide at 21:38, it should be a very good approximation. It should be 104348/33215 (= 3.14159265392...)

[@SKyrim190]

That is a typo from his part. The partial fraction in that step should be: 104 348/33 215, noticed that he swapped two numbers in the numerator. Also, this fraction gives pi with an error in the 10th decimal place. Quite good!

[@lunafoxfire]

About the closing statements: so maybe all those crazy New Age people going on about the Golden Ratio and the harmonics of the universe are actually unwittingly on to something? Pfff haha. In all seriousness though, this was a great lecture about a fascinating facet of mathematics.

[@markmoody8418]

and guess what pie is so amazing one thing it does is change itself from a none prime inteiger into a prime on its own.

[@markmoody8418]

the fractions are half the problems they are generalization if you have the correct pie out to the 11 digit the results are way more accurate than with the fractions and this bit of difference is everything

[@Etothe2iPi]

5:30 This obviously doesn't work for b=3/2.

[@DanDart]

I hope you have fixed your mic setup in the last 5 years

[@mshioty]

Round / Like a circle in a spiral / Like a wheel within a wheel / Never ending. ... Like a door that keeps revolving. In a half forgotten dream, ... Spiral out, keep going.

[@MrJonnal]

that's not the continued fraction for pi 15:46 should be [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14....]

[@MrPoutsesMple]

True: mathworld.wolfram.com/PiContinuedFraction.html

[@FlockOfHawks]

@@MrPoutsesMple Thank you friend , for sharing this site ! I never knew it existed until i read it here , and now i'm hooked !

[@rrr00bb1]

The aliens will use τ instead of π.

[@Naomi_Boyd]

Hot Damn! I wish I could give that two thumbs up.

[@layloshoyiqulova9231]

lptidjbvhh Iul I I djyg hm I ouzjhf I Nj X d Mon b Iao'clock

[@thomasolson7447]

You're wrong. The secret message is in the square root of three.

[@FlockOfHawks]

nah it's in a bit of unused dna

[@EleFlameMax]

Yawn.