Infinite Continued Fractions, simple or not?

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What Are Continued Fractions?
Continued Fractions,
Write sqrt(2) as a continued fraction, infinite simple continued fraction
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[@danielpeter725]

Its just the opposite of normal mathematics, instead of making equation simpler we just make it more complicated.

[@blackpenredpen]

Daniel Peter pretty much! : )

[@alan2here]

Apparently some continued fractions have reasonably large solution sets? More than 2 answers. I'd also love to see some mathematical constants beyond phi and the often mentioned ones and get some more intuition for how the series of integers results in this impressive diversity of common constants.

[@goragabdula8203]

Absolutely! My friend once said the following: "The point of math is to transform huge equation to have 1,0, pi or x " So, it works vice versa too)

[@jagrutivispute7600]

Yes it's not simplification it's complication

[@MrConverse]

Perhaps, but it’s also much more fun this way!

[@gocrazy432]

The problem with the sqrt2 continued fraction is you don't have to go past the point you wrote it in the denominator any more than you don't have to complete ramanujans sum of infinite nested rooted perfect squares. We already knew sqrt2 doesn't terminate. It doesn't matter that that the fractions continue infinitely as even rational numbers have infinite decimal notations and see ramanujan finite AND infinite representation of an integer. That didn't really prove anything.

[@yoavcarmel1245]

because 1+1=2 :)

[@blackpenredpen]

Yoav Carmel yes!! That's very important.

[@U014B]

But did you know that 2 = 1 + 1?

[@alejrandom6592]

@@U014B :0

[@shivamchouhan5077]

@@U014B really, thank you so much bro I didn't know Can you give me a proof

[@dhruvtalwar7250]

@@blackpenredpen hey sir, if we equal the given fraction as x we can right 2+ 1/(2+1)....to infinity as x+1 . Then we can simply right the repeating unit as x+1 too . We get the equation x^2 equal to 2 there . Neglecting the negative the answer is root 2 . Is this mathematically correct

[@vipulshukla6345]

7:34 That correctional 2️⃣ on the left side! ❤️

[@JustATest01]

Nobody could have seen it

[@CornishMiner]

The golden ratio is very elegant as a continued fraction.

[@adude6568]

Indeed it is. I found it to be equal to 1 + 1 / (1 + 1 / (1 + 1 / (1 + ...)))

[@themathsgeek8528]

Yes!

[@Psi_Fan123]

Yes, phi=1+1/phi then you go "hey look, I can replace that with this!" And then you get phi=1+1/(1+1/phi) then after a while Phi=1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(.........)))))))) Also a neat thing about phi is that phi^2=phi+1, there is another number with this property and it exists because of the equation (x^2)-x-1=0, As when you go and do this you get (1±sqrt(5))/2 and then you get phi=(1+sqrt(5))/2 and the other one that is (1-sqrt(5))/2, because sqrt(5) is bigger than 1, you get that the other number that satisfies x^2=x+1 is actually a negative number

[@BigDBrian]

when you take 2 = 1 + 2/2 you can also substitute for the 2 in the numerator, creating a fraction tower diagonally upwards. No idea about the uses for this , just seems cool

[@PrincessEev]

Or even nuttier - substitute it in both the numerator and denominator. :p

[@drakkenfan2]

And then substitute the fraction for the 2s in the new fractions infinity times

[@alkankondo89]

Ah, yes - continued fractions: the subject of my master's thesis, which I recently completed! I learned so much about simple continued fractions and their properties, and this video is a great introduction to the subject!

[@blackpenredpen]

alkankondo89 yay!!!! I like CFs too. And congrats on your masters degree!!

[@alkankondo89]

Thank you very much! It took a lot of work to get that degree, but I'm glad to have gotten it. Now, I'm continuing to watch your videos to keep up my math skills!!

[@pauljackson3491]

Can you upload you thesis or it it illegal or you don't want?

[@alkankondo89]

I appreciate your interest! I'm not really comfortable posting my thesis online, but if you want, you can search "even-integer continued fractions and the Farey Tree" to get an idea of what my thesis covered. There is a paper by Ian Short and Mairi Walker that I cited heavily in my thesis; that paper should be easy to find with a Google search.

[@hossamabdulsalam1514]

Sir, Please solve this problem. 3^(cot(x))^2 = 9*sin(x)*cos(x) Find the values of x. Many time I wrote this problem 🙏🙏

[@AccuphaseMan]

if you can't solve a problem that easy, i feel sorry for you

[@devanshtripathi7879]

@@AccuphaseMan ohhh if thats the case write solution then flex off if u cant then shut up and move on

[@mrocto329]

@@AccuphaseMan haven't solved it yet? It's been 2 years kid, I'm starting to feel sorry for YOU

[@NeerajSharmah2c2o4]

It's there a continued fraction for pi?

[@stevethecatcouch6532]

Yes, but it's ugly. The continued fraction for phi is much prettier.

[@NeerajSharmah2c2o4]

@@stevethecatcouch6532 I know the one for phi.

[@ffggddss]

All true. There are many interesting side roads in this territory. For instance, what if the b's (numerators) are kept at 1, but the a's (denominators) are allowed to be integers of either sign? (Short ans: You get more efficient CF's! Ex: φ = [2; -3,3,-3,3,...] ) Some day, I'm gonna hafta get off my duff and make a video about these fascinating gems myself... Fred

[@icespirit]

the one for pi is: [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, ...] (ugly)

[@blue_tetris]

The continued fraction for pi also doesn't repeat or form patterns. One of the reasons is because it is transcendental (but the proof is fairly complex).

[@yuval2be]

I read more about continued fractions, especially infinite contined fractions. But I didn't understand how to find the value of a given infinite continued fraction with known sequence. For example: [1,2,3,4...] or [1,4,9,16,25,36...]. If there is a known sequence does it mean the value is algebric number? I understood that because "pi" is Transcendental number there is not a pattern for the continued fraction.

[@colejohnson66]

e is transcendental as well, but it does have a recognizable pattern: [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] After the first 2, it begins going [1,2n,1] (every second term in the group of three is two more than the last group’s second term)

[@adamhrankowski1298]

How about changing the numerator 2's as well?

[@ImKurono]

Before class started my Calc 3 teacher was talking about this on Wednesday

[@blackpenredpen]

Oh cool!!

[@chriswebster24]

In the past, apparently, this man proved that the square root of 2 is irrational. That's how I know he's very wise.

[@BeltranAmenabar]

Can you make a video about simplifying continued fractions to get the original number back?

[@blackpenredpen]

Beltrán Amenábar yup. That will be my future videos.

[@BeltranAmenabar]

blackpenredpen Yay!!

[@Goku17yen]

seems very similar to the thing you see when first introduced to series, where you add together the infinite sum of a repeated decimal or something of that sort :D

[@Eternap]

Goku??Even you can understand this?! :p

[@blackpenredpen]

Of course, he is a super saiyan!

[@Goku17yen]

Eternal Entity lmao

[@adamoksiuta4715]

Do you know what is result of infinite continued fraction when you have only ones (1)? :P

[@m4nde]

it’s the golden ratio 1.618033… (phi) 1/phi + 1 = phi x = 1+1/(1+1/(1+1/(1+…) 1/x + 1= x (:

[@adamoksiuta4715]

@@m4nde Correct :)

[@axbs4863]

alright heres my work to solve for that continued fraction: 1+(1/(2+1/(2+....) take out that initial (1+) term for now set 1/(2+1/(2+.... = x replace the repeating terms with x to get: x = 1/(2+x) multiply both sides by (2+x) to get x^2 + 2x = 1 subtract by 1 on both sides to get a quadratic; x^2 + 2x - 1 = 0 quadratic formula: x = (-2 +/- sqrt(4+4))/2 simplfiy to get -1 +/- sqrt(8)/2 add that (+1) term back in to cancel out that (-1) term x = sqrt(8)/2 = 2sqrt(2)/2 = *sqrt(2)*

[@blackholesun4942]

Thanks, this worked nicely and seems to use recursion

[@jumpman8282]

This was fun. Especially since I about a week ago saw Virgin Rock's reaction video to Tool's song "Lateralis", whose musical structure is based on the Fibonacci sequence. What I learned from that video is that the ratio between two consecutive Fibonacci numbers approximates the golden ratio and the further down the sequence we go the better the approximation will be. What I managed to figure out is that the "Fibonacci ratio" can be written as the continued fraction 1 + 1/(1 + 1/(1 + ... but I couldn't satisfactorily show that this continued fraction is equal to the golden ratio, i.e., (1 + √5)/2. But then this video popped up in my feed and it painfully dawned on me that all I needed to do was to write (1 + √5)/2 as a continued fraction.

[@Spectrojamz]

6:18. How many times do you want me to do this; seriously I don't know🤣😂😂

[@flamingpaper7751]

What about a continued fraction for functions, such as x/ln(x/ln(x/...))?

[@icespirit]

then x would have to be bigger than infinity (I think)

[@ThePianofreaky]

If you call the whole thing y, you can rewrite it as y = x/ln(y) . Now, solve for y. y * ln(y) = x ln(y^y) = x y^y = e^x y is the super root of e^x. That's how you can solve these problems in general.

[@icespirit]

sorry i was sleepy and i didnt want to think :p

[@brainlessbot3699]

@@ThePianofreaky does it converge for every value of x?

[@That_One_Guy...]

Actually continued fraction also appear at polynomial division (when the denominator is higher degree than the remainder)

[@VIJAYSWARUP]

Can you make an infinite continued fraction video for pi ?

[@sandhinfk]

If you just cut the video part and keep the audio part, you can make a 10 hour video on “I see this square root of 2, then i can put this into that, ......., now I see this square root of 2 again, then i can put this into that again............”

[@ישראלורטהימר]

I have a question you say that because 2=1+(2/1+(2/1+... have 2 on the up size it is not a irr so I can do the following 2= 1+1=1+2/2=1+1/2/2 2=1+1/2/2-------------------------------------------------------^ 2=1+1/((1+1/2/2)/2) and then you have the ones on the top. so what i got wrong

[@arnavtripathi273]

I did it another way by adding 1 on both sides of eqn then we get x+1 = 2 + 1/(2+1/2.......... If we write it like x+1 = 2+ 1/(x+1) ... solving for x we'll get x=±√2 and we can also conclude that the result will be positive since all stuff that's being added is positive.

[@EnimossRandy]

you are God

[@jonathanaguillon5701]

You know, that stuff (continued fraction) is the initial algorithm to help me find new set of gear, it is vital to detemine the number of teeth of drive and driven gear.

[@borisburd2951]

i dont know why but the continued fraction of sqrt of 2 made me laugh haha its so elegant

[@jacoboribilik3253]

Great math channel, im so glad to have come across it.

[@aymanabdellatief1572]

Do you have similar patterns for sqrt(3) , sqrt(5), and other irrational square roots? What about cube roots and higher order roots?

[@Patapom3]

Great! Would have loved to see Phi developed as a continued fraction as well...

[@ethanchandler3934]

Cant we say 2 is irrational since it can be written as an infinite fraction?😂

[@sam.asdfjkl]

They proposed a wrong theory if it's infinite continues its irrational? Lol

[@sreejithunni3267]

I love mathematics and you..😘😘😘😘

[@morgengabe1]

I feel like it's circular to define number with a continued fraction that contains it, no? Is that appropriate? this is my first real experience with them. Sorry for silly questions.

[@deusvult5738]

Well, it's more like a series. Has stuff to do with limits

[@dr.rahulgupta7573]

Very good explanation green pen blue pen red pen ( gp bprp) .Thanks . DrRahul Rohtak Haryana India

[@Qermaq]

Rather than "simplify" the objective is "obfuscate".

[@toufik9632]

A=2 A-2=0 (A-2)(A+3)=0 A²+A-6=0 A(A+1)=6 A=(6/A)-1 And repeat and repeat😅

[@jonahansen]

I totally did not understand the point of this. Seems like a solution in search of a problem.

[@q..to..bigbrother6768]

I don't get but am I the only one that got 3 all the time the he was writing the formula ... ( 6:33 )

[@MercenarySmash]

Dang looking good with that fresh haircut

[@Rekko82]

0 = 0 /1 = 0/(1+0/1) = 0/(1+0/(1+0/1)) or something like that.

[@yoavcarmel1245]

not simple tho :)

[@sanchitkumar6550]

What is the source from where this topic has been taken

[@pauselab5569]

proving that they must converge at a fixed point is insanely difficult. made me learn banach's fixed point theorem.

[@rajatmeena7341]

That's great

[@star_ms]

Can all algebraic irrationals be written as infinite CFs?

[@neformalik7414]

Now we know what he dresses when he goes on dates :)

[@donwald3436]

7:37 You're not supposed to use white-out on your monitor.

[@ffggddss]

Thanks! CFs are a pet topic of mine. This video should help raise interest in them; it is very basic, and right away leads any inquisitive mind to ask a myriad of other questions. _So like a siren, drawing in all unwary sailors to her lair..._ Fred

[@vector8310]

Great comment! I actually purchased Oskar Perron's German language Die Lehrer von den Kettenbruchen because I've caught the continued fraction bug.

[@Abigmass]

Why need to put 1 on the fraction

[@wkingston1248]

When your teacher says your answer doesn't need to be in its simplest form an you give him a pagn of fractions😂 As a side note what is the actual purpose of writing fractions like this.

[@Bidisha_Nath]

Really useful... Thanks... From. Assam, India

[@heribertobarahona7695]

Continued Fractions are so fun especially with irrational numbers!!! I just Log in to Brilliant, looks like an awesome page!!!

[@blackpenredpen]

I like CFs a lot too!! Just really hard to write it neatly tho. : )

[@heribertobarahona7695]

Yeah!! You're totally right in that aspect.

[@meghdipshah9615]

How can we prove root 3 is irrational by this method numerator becomes other than 1 and continue so by this is root 3 is rational

[@dammuraja8010]

Student from india. Lovely explaination sir, enjoyed.😄

[@fCauneau]

May look tricky for math lovers, but the paradox is that it makes some of the structure of numbers accessible for people who only know in arithmetics...

[@dmavbchy]

Given a continued fraction, how I deduce what number/function is it?

[@우울바이러스]

I learned English as math

[@rssl5500]

I think it’s sqrt 2 not sure

[@shubhampaswan3286]

Greenpenredpen :-) Yay ...

[@jimjam1948]

Hey bprp can you discuss the erdos-straus thereom next. The theory behind it is pretty simple and easy to understand even for young children. Pleaseeeeèe

[@blackpenredpen]

Ok, I will try! I have many things planned already (such as the Goldbach's Conjecture) but that is certainly an interesting one too! Thanks!

[@jimjam1948]

@@blackpenredpen yesss that's great

[@urvpatel829]

Hey I'm 😀from India but why I enjoy your lecture (I don't know)

[@samarthsai9530]

I don't know why but I can smell some metallic ratios. #festive season

[@snejpu2508]

YAY!!!

[@lazerdemowolf]

I have an alternative solution using a quadratic formula.

[@lazerdemowolf]

By letting the original function be f(x). I can say that f(x)= 2+(very Long fraction)-1. Because the fraction just repeating to infinity I can replace the long fraction with 1/(f(x)+1). Then I will solve algebraically and get (f(x))^2=2. Since f(x) contains no negative terms, then f(x) must be positive, hence it is equal to root 2.

[@danielbenyair300]

6:33 the silver ratio equal to 1+sqrt2 Note: the ... at the end is wrong!! There has to be sqrt2 at the end!!!

[@blackholesun4942]

My guess is root 2

[@SuperDeadparrot]

How would you take a derivative of a continued fraction?

[@SuperDeadparrot]

Is the representation unique?

[@nellvincervantes3223]

Physics? Mechanical vibration? Can you explain it pls?

[@jihyepark9139]

He talks like my Math Teacher in highschool.

[@renatoshigueto9291]

Thanks man, i was have a difficult problem for understand this and now.. i get it, thanks

[@jagrutivispute7600]

Thank you so much before I wasn't knowing how to turn a fraction into the continuous fraction form but after watching video now I come to know thank you again

[@niruvibes]

i hate hurlstone

[@alan2here]

Greatest compositing ever :)

[@breakzxz648]

since sqrt root of 2 is equal to (1 + sqrt of 2 -1) then u cannot multiply (sqrt of 2 - 1) only thats my opinion correct me if im wrong, its still cool though getting that infinite sequence 👍

[@antimatter2376]

If you divide it by the same thing, it's ok

[@11.anoodaygoswami74]

Srinivasa ramanujan ❤️

[@CharIie83]

wont the value change as you go down the ladder of fractions? I know the numbers stay the same, but the values are changing, or?

[@inakibolivar664]

No, they don’t change at any point

[@renren4236]

But, I am just thinking what can we do with such things

[@rongzhang9283]

your big help

[@sudhakarunakaran5033]

Waste

[@janv.8538]

8:55 cant you plug in this term on the top of the fraction, which would make it infinetely complicated?

[@mountainc1027]

Oh my god it looks like the one shown in the video but "reflected" in the x-direction

[@parthsushamachavan915]

10^10^10 wow

[@Jom1331]

2 = 1/1/2 And you can repear it forever

[@jcb3393]

It's neat and all, but is there any practical application for continued fractions, infinite or otherwise?

[@danmccarron0]

the answer is yes. the fact that it terminates after finite number of iterations for rational numbers is a perfect way to test for closure of a subset of the infinite group of elements obeying this property and that could correspond to something physically real, like, say, the number of accessible states of the system of discrete energy levels (a behavior of quantum mechanics) or something like that. or it could demonstrate how many times you have to shuffle a certain set of elements before you get back what you started with. Only elements whose values are integer or fractional numbers will return back to the starting point and that is determined specifically by the number of times you can iterate a continued fraction. I'm sure there are plenty of applications other than this but these come to mind.

[@vector8310]

Yours is a first-rate reply.

[@sawyerandrobbie]

Super interesting

[@kingbeauregard]

Continued fractions are cool, and are a good way to find rational approximations for numbers with lots of decimal places. Using continued fractions, even an oaf like me was able to independently derive that 355/113 is a freakishly good approximation for pi (3.1415929 instead of 3.1415927). 355/113 is easy to remember, by the way, if you think denominator and then numerator: one one three three five five.

[@danieltiema]

Very helpful, thankyou...

[@ranjeetarani2155]

Those are easy

[@takyc7883]

This was actually such a fun video lol

[@amilo5]

Bring those numbers down down !

[@SuperYoonHo]

awesome!

[@hamsterdam1942]

8:40 that's pretty much eat? What do you mean?

[@ShaniYadav-dk1ns]

Nice sir

[@bernardfinucane2061]

So an irrational number is just a rational number with an infinite denominator. That's why you can keep descending forever. xD

[@blackpenredpen]

Yes!!

[@blackpenredpen]

: )

[@drjagtarsingh3498]

Nice

[@YorangeJuice]

Cool vid

[@moshadj]

So for the last example of the continued fraction of 2 you can also use the relation for 2 on the numerator as well.

[@jcb3393]

But then that just reverts back to a series in the numerator and the same series in the denominator, which is x/x which is 1. So we are back to 1+1=2.

[@marco_mate5181]

Itsnanda wrong attività the begin

[@LUISFARIASPINO]

anybody has a good book about this???

[@maxfraguas]

Thank you very much!

[@Zealot0630]

can you prove that the series converges to square root of 2 please.

[@blackpenredpen]

Zang MingJie ok. Future video

[@canaDavid1]

Okay, Integer = continued fraction with 1 term Rational = continued fraction with finite terms Algebraic = infinite fraction, but repeating numbers Transcendental = infinite fraction, not repeating Is this correct?

[@mrgreenskypiano]

No, one of pi's continued fractions has repeating terms or a pattern of terms

[@canaDavid1]

@@mrgreenskypiano which one?

[@canaDavid1]

@@mrgreenskypiano also, i was referring to the continued fractions with 1 as all numerators

[@mrgreenskypiano]

@@canaDavid1 Ah, ok, it isn't a simple cf