Infinite Fractions - Numberphile

Matt Parker on Stern-Brocot numbers, fractions and rational numbers.
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NUMBERPHILE

Videos by Brady Haran

[monkey]

"Read them like a spiral"

Or regularly.  Left to right, top to bottom.

[NNOTM]

Reading it in a spiral seems like an unnecessarily unusual way of saying that you read it from top to bottom and left to right. Very cool, though.

[ksng]

"That's cool." - Brady 2014

[CodenameJD]

Man, this seemed so cool that I started plotting it out myself... and the number of patterns that can be found in it are so fantastic. The gaps between each appearance of a 1 give the pattern 1, 3, 7, 15, 31; increasing binary numbers-1, which could also be expressed that the gaps between the numbers in that pattern follow 2, 4, 8, 16. The gaps between each appearance of a 2 follow 2, 5, 11, 23, 47, and the gaps between the numbers of that pattern follow 3, 6, 12, 24.

At that point I predicted that the gaps between appearances of 3 would increase by 4, 8, 16, 32; but instead found they followed the pattern 1, 2, 3, 5, 7, 11, 15, 23 31 - gaps increasing 1, 1, 2, 2, 4, 4, 8, 8.


Haven't worked out the pattern with the gaps between 4s yet. That goes 5, 2, 11, 5, 23, 11. First number that gaps decrease. Would likely need to carry on the sequence further to find the pattern.


But with all of these patterns that show up... it's just awesome :D

[jesselong]

Having all 3 of them 'rules', simplified, never repeating, and all fractions... is amazing... I don't understand a lot of the things brought up on this channel, but this specific video gave me chills. 

[johnnicolo]

If I had known these guys earlier I would have been a math major. Love this stuff.

[ckmishn]

This method is so much better than the traditional diagonalization approach. Just think how much faster you can reach infinity this way....

[subscribefornoreason]

I am in love with the Stern-Brocot sequence.

[MrPsychicNoodles]

Lol your ways of writing 5 is really inefficient Matt (4:52)

[alexanderfarrugia]

In 3:11, there's a mistake. The phrase 'the (set of) rationals are the same size as the (set of) reals' should be 'the (set of) rationals are the same size as the (set of) natural numbers'. Cantor proved that the reals are not the 'same size' as the rationals, but are uncountably infinite, whereas the rationals are countably infinite (and the sequence of this video is a proof of this fact!).

[lukasdon]

Can you do an episode on the Continuum Hypothesis? I was very intrigued by the fact that it cannot be proven false nor correct. Is this true? And how can we make sense of a hypothesis that falls outside of our known truth values?

[elliottmcollins]

Just spent 10 minutes making graphs with the Stern-Brocot sequence and the associated sequence of fractions. Totally awesome.

[Kaepsele]

Yes, a spiral is much more obvious than just read them top to bottom, left to right :D

[jamma]

The title of this video made me think that he was going to talk about infinite continued fractions, and possibly even some Diophantine approximation.

It was still interesting, but you should do a video on that!

[Natalie-cxcp]

I just got the super pack ( I think that's what it's called) from mathsgear, for Christmas. It was the best present EVER especially as the book was signed! Merry Christmas everyone. And thank you Matt and the rest of the numberphile "gang" merry Christmas! 🎅🎁🎄

[tozmom]

Matt has an amazing cheeky grin that appears when he's enjoying the magic of maths. :-D

[toast_recon]

Matt says "The video showed that the rationals are the same size as the reals" but then later the paper says "Everyone knows that the rationals are countable". 

We know that the reals are uncountable, so I'm guessing Matt misspoke, meaning to say "the rationals are the same size as the naturals" (unless I've got something wrong here)

[BatteryAcid]

Matt, you'll be happy to hear that I found an actual use for the Fibonacci sequence. The largest numerator in a given row of the tree representation of the Stern-Brocot numbers is equal to F(n+1), where the top-most row is n=1.

[ToadJimmy]

Fantastic! Came across this on a wiki-walk a while ago. Still super awesome. Or more awesome. Started fangirling on the inside when I recognised the sequence. Now I understand it a little more, and that's a great feeling. Maybe my favorite sequence. Thanks Brady & Matt for all this goodness.

[stevieinselby]

I like the way that each line on the fraction tree has rotational symmetry.