5:03 as someone who somehow loves everything involving Phi, Fibonacci numbers, exponentials, Euler's constant and so on, I LOVE that continue fraction :D so simple and elegant, yet holding SO MUCH information and relations. WOW
One of the first to work out a thorough system for CF arithmetic was Bill Gosper (apparently in 1972, according to info I found on Wikipedia). If you search him up with 'continued fractions' as part of the search, you should find some good information/sources on his original CF arithmetic work. It's highly likely that (please correct me if I'm wrong) the method described here and in the other linked video are at least indirectly inspired by Gosper's work. However, personally, I found reading Gosper's description just a little too difficult for me to try it out myself. (Probably I was just too impatient with myself?) And I found your video very helpful for overcoming that barrier. Thank you! Saved your video to a private playlist so I don't lose track of it when (hopefully soon!) I revisit CF arithmetic and want to figure out how to implement it on my own. I often like to try these kinds of things by hand on my own to really ensure that I completely understand it. This video will allow me to do that for sure! Great work, and I'd love to see more videos on CFs (and other related topics) in the future, if/when you delve deeper into them yourself! One little preference/critique: Although another commenter said they appreciated that you didn't show 'overly' much 'rigour' (e.g. not showing more D-hat expansions), I actually prefer videos that err more on the side of showing 'more than strictly necessary', for the purposes of helping more people really 'get it' by being able to see a more complete/thorough example. For those who 'get it' with just minimal 'rigour', IMHO they can just speed up the video playback to skim past the 'extra' detail. But if that extra detail is not in the video at all (or at the very least linked to in some other video perhaps, or link to a more thorough write-up in the description), then slowing down the video playback doesn't magically make that detail appear, for those who might need it. This is similar to the problem (well, I feel it is sometimes a problem) when a professor or textbook decides to leave the 'details of the proof' as an 'exercise for the reader'. While, yes, it is important for readers/viewers to get into the math and work on it directly themselves, such 'exercises for the reader' assume a certain level of experience/confidence in the audience of all possible readers and -- especially for something like a RUclips video, I would say -- the actual interested audience often can have people with much less experience/confidence than is needed to be able to tackle such 'exercises for the reader'. In such cases, that portion of the potential audience just gets left behind. Whereas if the details are made explicit, then this provides an opportunity for the reader to 'catch up' to the topic by studying the details themselves, eventually understanding how the professor/author got from A to B. This is what happened to me with Gosper's work, for example. And I imagine that some of the elisions in this video may have made some people's eyes get lost in the sea of 'strange symbols'. Just my two cents! Thanks again for making this video on this topic! Cheers!
I developed my method before reading about Gosper's, but yes the linked video does discuss that one along with others. I also had some trouble following the sources. I appreciate the feedback. One of my main purposes making these videos is for entertainment, and I think at some point working out in detail loses value for the average viewer. lt would be nice to have extra detail available in notes or an extended cut for viewers like you who want that, but that would also take extra time to prepare which is beyond me at the moment.
I am quite familiar with continued fraction, from their definition, to properties and their advanced applications, and your initial claim about finding the continued fraction of a multiplication was very intriguing. However, I got to admit that at some point I just stopped following the computations, as they seemed as just collections of letters, subscripts and superscripts. I suggest that at the very least you should keep a "small" definition of the various notations that you use at the corner of the screen. Maybe something like a=continued fraction of a with coefficients a_i = lim n^a_j / d^a_j = lim A_j, the same for b, and then something for the N,D and their hat version. While you remember all of the notations and the definitions since you studied it and worked on this video for a long time, it is much harder for someone who just view this video, even if you know a lot about continued fractions.
Any continued fraction can be expressed as the product of matrices with column vectors [a_i,1] and [1,0] If you calculate the eigenvalues of the product then you will find the value of the continued fraction… iirc. I’d done some messing around with these years ago and I’d need to find my notes.
Really good video, well explained but it would be helpful if what was on screen was more clearly labelled. That way it is easier to follow and you do not have to rely wholly on what is said to figure out what is going on. Would lead to less pausing and rewinding, more understanding
The upload schedule is fast af! I still need to finish the last series, but this lovely brown background showed up in my notifications... it must be hard keeping such a calm voice after bloopers 😂
Thanks! No, this is just a standalone video. Although I am planning to study continued fractions more in the future and I'm reserving this brown color scheme for the topic
I need to force myself to work on that, it's likely that there are still stuff to do and it might be convenient in lots of situations. I'm wondering how you can translate it on a ring. Would you be able to find a sequence of embedded invertible values that converge to...idk, it has to be an integer. Would it show some periodicity? (likely/necessarily). If only I could think correctly more than 30s a day.
I like to switch color schemes for different topics, and I think it's fun to have my pfp match the most recent video. It doesn't having any other meaning
I can do this but for only periodic simple continued. I hope this video doesn't convert these into their roots and add them and finds a more clever work around.
This approach works for non periodic simple continued fractions which is excellent but it seem very inefficient and doesn't really prove that you have the correct answer. Solving this algebraically using recursion because they're simple periodic continued fractions seems to be the best approach.
Any continued fraction can be expressed as the product of matrices with column vectors [a_i,1] and [1,0] If you calculate the eigenvalues of the product then you will find the value of the continued fraction… iirc. I’d done some messing around with these years ago and I’d need to find my notes.
5:03 as someone who somehow loves everything involving Phi, Fibonacci numbers, exponentials, Euler's constant and so on, I LOVE that continue fraction :D so simple and elegant, yet holding SO MUCH information and relations. WOW
I appreciate the level of rigor in this video. Very thorough, but not overly so. (Going in to show the first D hat equivalence, but not the other two)
23:14 "Heghhhh fu-"
Truer words have never been spoken
Ooh, I never really thought about actually calculating using CF, although I did make a small webapp to help one convert between decimal and CF.
A webapp that converts between decimal and CF? Link please!
It creates funny results if you divide two irrationals to produce a rational
3:53 MR. 305 MR. WORLDWIDE WE DOIN CONTINUED FRACTIONS LEMME JUMP IN THIS TRACK AND SPIT SOME ARITHMETIC IN THIS HOUSE
One of the first to work out a thorough system for CF arithmetic was Bill Gosper (apparently in 1972, according to info I found on Wikipedia). If you search him up with 'continued fractions' as part of the search, you should find some good information/sources on his original CF arithmetic work.
It's highly likely that (please correct me if I'm wrong) the method described here and in the other linked video are at least indirectly inspired by Gosper's work.
However, personally, I found reading Gosper's description just a little too difficult for me to try it out myself. (Probably I was just too impatient with myself?) And I found your video very helpful for overcoming that barrier. Thank you!
Saved your video to a private playlist so I don't lose track of it when (hopefully soon!) I revisit CF arithmetic and want to figure out how to implement it on my own. I often like to try these kinds of things by hand on my own to really ensure that I completely understand it. This video will allow me to do that for sure!
Great work, and I'd love to see more videos on CFs (and other related topics) in the future, if/when you delve deeper into them yourself!
One little preference/critique:
Although another commenter said they appreciated that you didn't show 'overly' much 'rigour' (e.g. not showing more D-hat expansions), I actually prefer videos that err more on the side of showing 'more than strictly necessary', for the purposes of helping more people really 'get it' by being able to see a more complete/thorough example.
For those who 'get it' with just minimal 'rigour', IMHO they can just speed up the video playback to skim past the 'extra' detail. But if that extra detail is not in the video at all (or at the very least linked to in some other video perhaps, or link to a more thorough write-up in the description), then slowing down the video playback doesn't magically make that detail appear, for those who might need it.
This is similar to the problem (well, I feel it is sometimes a problem) when a professor or textbook decides to leave the 'details of the proof' as an 'exercise for the reader'. While, yes, it is important for readers/viewers to get into the math and work on it directly themselves, such 'exercises for the reader' assume a certain level of experience/confidence in the audience of all possible readers and -- especially for something like a RUclips video, I would say -- the actual interested audience often can have people with much less experience/confidence than is needed to be able to tackle such 'exercises for the reader'. In such cases, that portion of the potential audience just gets left behind. Whereas if the details are made explicit, then this provides an opportunity for the reader to 'catch up' to the topic by studying the details themselves, eventually understanding how the professor/author got from A to B.
This is what happened to me with Gosper's work, for example. And I imagine that some of the elisions in this video may have made some people's eyes get lost in the sea of 'strange symbols'.
Just my two cents!
Thanks again for making this video on this topic! Cheers!
I developed my method before reading about Gosper's, but yes the linked video does discuss that one along with others. I also had some trouble following the sources.
I appreciate the feedback. One of my main purposes making these videos is for entertainment, and I think at some point working out in detail loses value for the average viewer. lt would be nice to have extra detail available in notes or an extended cut for viewers like you who want that, but that would also take extra time to prepare which is beyond me at the moment.
I remember seeing matrix methods to do streaming computations with continued fractions in some old plaintext source... If only I could remember where!
I am quite familiar with continued fraction, from their definition, to properties and their advanced applications, and your initial claim about finding the continued fraction of a multiplication was very intriguing. However, I got to admit that at some point I just stopped following the computations, as they seemed as just collections of letters, subscripts and superscripts.
I suggest that at the very least you should keep a "small" definition of the various notations that you use at the corner of the screen. Maybe something like a=continued fraction of a with coefficients a_i = lim n^a_j / d^a_j = lim A_j, the same for b, and then something for the N,D and their hat version. While you remember all of the notations and the definitions since you studied it and worked on this video for a long time, it is much harder for someone who just view this video, even if you know a lot about continued fractions.
Thank you for the feedback. I like the idea of showing eeminders of those definitions
Any continued fraction can be expressed as the product of matrices with column vectors [a_i,1] and [1,0]
If you calculate the eigenvalues of the product then you will find the value of the continued fraction… iirc. I’d done some messing around with these years ago and I’d need to find my notes.
Really good video, well explained but it would be helpful if what was on screen was more clearly labelled. That way it is easier to follow and you do not have to rely wholly on what is said to figure out what is going on. Would lead to less pausing and rewinding, more understanding
i wonder if there is a version of this for generalized continued fractions
Is this going to be a series of videos?? I love this, thanks for all the creativity and effort you put into these videos.
The upload schedule is fast af! I still need to finish the last series, but this lovely brown background showed up in my notifications... it must be hard keeping such a calm voice after bloopers 😂
Thanks! No, this is just a standalone video. Although I am planning to study continued fractions more in the future and I'm reserving this brown color scheme for the topic
YESSSS CONTINUED FRACTIONS MY BELOVED!!!!
I need to force myself to work on that, it's likely that there are still stuff to do and it might be convenient in lots of situations.
I'm wondering how you can translate it on a ring. Would you be able to find a sequence of embedded invertible values that converge to...idk, it has to be an integer. Would it show some periodicity? (likely/necessarily). If only I could think correctly more than 30s a day.
Wait, no one's gonna mention thr change of pfp? It's the same palette of the video. Does it mean anything?
I like to switch color schemes for different topics, and I think it's fun to have my pfp match the most recent video. It doesn't having any other meaning
What's the computational complexity of this method?
computation complexity is *yes*
I had to look “negative continued fraction” up. The concept doesn't make much sense to me
10:01 that non binary friend of yours that's going to the gym and getting pretty swole
I can do this but for only periodic simple continued. I hope this video doesn't convert these into their roots and add them and finds a more clever work around.
This approach works for non periodic simple continued fractions which is excellent but it seem very inefficient and doesn't really prove that you have the correct answer. Solving this algebraically using recursion because they're simple periodic continued fractions seems to be the best approach.
But finding a method for non quadratics solutions of continued fractions without error seems to still be a mystery
I’d say the answer is (rad5)(2rad5)= 10.
{2n+2n ➖ }+{1n+1n ➖ }={4n^2+2n^2}=6n^4 +{4n+4n ➖ }+{1n+1n ➖}=6n^4+{8n^2+2n^2}={6n^4+10n^4}=16n^8+{4n+4n ➖ }+{1n+1n ➖ }=16n^8+{8n^2+2n^2}={16n^8+10n^4}=26n^12+ {4n+4n ➖ }+{1n+1n ➖ }=26n^12+{8n^2+2n^2}={26n^12+10n^4}=36n^16 +{4n+4n ➖}+{1n+1n ➖ }=36n^16+{8n^2+2n^2}={36n^16+10n^4}=46n^20+{4n+4n ➖}+{1n+1n ➖}=46n^20+{8n^2+2n^2}={46n^20+10n^4}=56n^24+{4n+4n ➖ }+{1n+1n ➖ }=56n^24+{8n^2+2n^2}={56n^24+10n^4}=66n^28+{4n+4n ➖}+{1n+1n ➖ }=66n^28+{8n^2+2n^2}={66n^28+10n^4}=76n^32 .{3n+3n ➖}+{1n+1n ➖ }={6n^2+2n^2}=8n^4+{6n+6n ➖ }+{1n+1n ➖ }=8n^4+{12n^2+2n^2}={8n^4+14n^4}=22n^8+{ 6n+6n ➖ }+{1n+1n ➖ }=22n^8+{12n^2+2n^2}={22n^8+14n^2}=36n^12+ {6n+6n ➖ }+{1n+1n ➖ }=36n^12+{12n^2+2n^2}={36n^12+14n^4}=50x^16+{6n+6n ➖}+{1n+1n ➖ }=50n^16+{12n^2+2n^2}={50n^16+14n^4}=50n^20 +{6n +6n ➖ }=50n^20+{12n^2+2n^2}={50n^20+14n^4}=64n^24+{6n+6n ➖}+{1n+1n ➖ }=64n^24+{12n^2+2n^2}{64n^24+14n^4}=78n^28+{6n+6n ➖}+{1n+1n ➖ }=78n^28+{12n^2+2n^2}={78n^28+14n^4}=94n^32 {76n^32•94n^32}=7144n^1024 100^700^144n^512^512 100^10^70^144n^128^128 100^10^70^12^12n^64^64 10^10^10^7^10^12^12n^4^16^4^16 10^10^10^7^10^12^12n^4^4^4^4^4^4 2^5^2^5^2^5^7^2^5^3^4^3^4n2^2^2^2^2^2^2^2^2^2^2^2 1^1^1^1^1^1^7^1^1^1^3^2^2^3^2^2n^1^1^1^1^1^1^1^1^1^1^1^1 1^1^1^1^1^3^1^2 3^2n (n ➖ 3n+2).that is longest one i have written yet i think
Any continued fraction can be expressed as the product of matrices with column vectors [a_i,1] and [1,0]
If you calculate the eigenvalues of the product then you will find the value of the continued fraction… iirc. I’d done some messing around with these years ago and I’d need to find my notes.