Lucas Numbers - Numberphile

Part Three of Golden Ratio Trilogy
Why the Lucas Numbers are better than the Fibonacci Sequence.
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Proof: • Golden Proof - Numberp...
Lucas Numbers: • Lucas Numbers - Number...
Extra bit: • Golden Ratio and Fibon...
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Thumbnail caricature of Lucas by Knut Johannes Håland

[@martinshoosterman]

-Unnecessary precision. -Prints a mile of pi.

[@aldenwilner3300]

Prints a mile of pi. Thus illustrating necessary precision. :-)

[@Azrage]

martinshoosterman Don’t forget “pointless and over the top”

[@DeathBringer769]

The rounding thing comes from an exact relation -- look up Binet formulas.

[@Triantalex]

??

[@klobiforpresident2254]

"There's some amazing things that drop out of rounding" like .14159265 ...

[@martinshoosterman]

this comment is underrated.

[@tommyglodo4495]

Klobi for President I’m confused but u want to understand the joke

[@cularre9544]

"roundind", like a circle, and pi goes 3.14159265

[@mihailmilev9909]

Parker Maths

[@mihailmilev9909]

​@@cularre9544thank you...... now I might remember thoose last three numbers lol

[@thcoura]

Side comment: how funny to see that the visual mark of the channel is the type of paper used. Rather unique I have to say.

[@skyiloh7460]

Brown paper™

[@mihailmilev9909]

Right lol

[@lpsp442]

Look at how satisfied he is when he says "Golden Ratio". This guy loves maths.

[@martinshoosterman]

no, he said the fibonacci sequence was over rated Martin.

[@Ub3rSk1llz]

he says he's a massive sceptic of the golden ratio, watch it again.

[@joe-360]

@@martinshoosterman u

[@Triantalex]

??

[@Ethan-mm9yk]

So, to get a Lucas number, you have to take the Parker Square of φ. Interesting.

[@anawesomepet]

142857 is a parker 1/7th. 142857x7=999999

[@345derder]

Matt, you've done it again I've fallen in love with yet another series of numbers

[@secularmonk5176]

4:12 In defense of rounding: "People link mathematics with unnecessary precision ... with things that are pointless and over-the-top." What, like printing out a million places of pi? (ducks)

[@TheChemistryShack]

Actually 5 trillion digits

[@aboubacaramine8689]

Well you gotta have some fun from time to time.

[@dylanrambow2704]

There was a reason for finding so many places of pi--they wanted to know whether or not the decimal sequence ever repeated, whether pi was truly irrational. And now we know that it is indeed irrational. That wasn't done just to nerdily rattle off as many decimal places as possible.

[@zwz.zdenek]

Dylan Rambow We can devise algorithms for numbers that could never fit into this universe. To determine if pi is transcendental, one must make a smart proof. No amount of computing power will create a mathematical proof.

[@andie_pants]

I am a lowly stats 1 student, but given that we bounce from decimal place to decimal place depending on the problem, I like his essential notion that after a certain point, we just don't give a flying f---.

[@shield543]

"I'm going to say L-oo-cas instead of L-oo-ca because I'm lazy" - Matt logic

[@GrandMoffTarkinsTeaDispenser]

Nice to see you Gauss.

[@fatsquirrel75]

I'm going to say it in English, rather than read it in English, realise its a french word, translate it and say it in French. Speaking words in your native language is definitely lazier.

[@DeathBringer769]

I wonder how Matt chooses to say the loanword in English from French "foyer" as well, lol...

[@papergamesproductions]

Immediately calls him l-oo-ca after saying that.

[@thorodinson6649]

@@DeathBringer769 longer ago

[@alonamaloh]

Call P=(1+sqrt(5))/2 and Q=(1-sqrt(5))/2, the two solutions of x^2-x-1=0. The Lucas sequence is simply P^n+Q^n, which happens to be all integers. Since Q is between -1 and 1, Q^n converges to 0, and that's why rounding works (for n>=2, that is).

[@pianodries]

phi^1=phi phi^2=phi+1 phi^3=2phi+1 phi^4=3phi+2 phi^5=5phi+3 phi^6=8phi+5 ... I think the Fibonacci sequence definetly is very much linked to phi actually...

[@rastrisfrustreslosgomez544]

hahahahahahahahaha would you dare say so? XD

[@Guztav1337]

Wow, this is a great finding actually

[@twistedgwazi5727]

shrdlu But it was already discovered?

[@paulprescod6150]

He did a new video about this property after someone pointed it to him on Reddit.

[@TruthNerds]

Thanks, that is very interesting. BTW, the Lucas numbers also have a precise identity: L(n) = phi^n + (1-phi)^n and the Fibonacci numbers have a similar one: F(n) = (phi^n - (1-phi)^n) / sqrt(5) There is also a direct identity: L(n) = F(n+1) + F(n-1) Oh, and finally, from the closed form of F(n) above, follows: F(n) = round(phi^n / sqrt(5)) because the "error term" (1-phi)^n / sqrt 5 is always less than 1/2.

[@Robi2009]

0:18 - it just occured to me: Fibonacci was really named Bonacci, so his nickname is Fi-Bonaccci, you can spell it Phi-Bonacci or even Φ-Bonacci. He has Golden Ratio in his name!

[@richlawton1995]

i love matt, think he's by far the best numberphile 'guest'

[@noahlapsley5122]

Hey Brady. I really like what you did with these newest three videos, how you broke them up and put a fun thumbnail with each. I could totally see all three of these as one long video, but I think that separating them allows for you to think about each concept/topic and process what you just watched before going onto the next one. Maybe do this in future videos. Anyway, keep making great videos.

[@jabelltulsa]

"almosty" is pretty much my favorite word now. Thanks Brady.

[@UltraLuigi2401]

The amount that it misses by for the nth Lucas number (where 2 is the 0th one, of course), is exactly equal to the conjugate golden ratio ((1-sqrt(5))/2) to the nth power. So to exactly get the nth Lucas number, you add the nth powers of both ratios.

[@007bistromath]

I am curious: do different Fibonacci-type series approach phi at different rates? If so, what's the fastest known? Is it possible to construct any other kind of series that will do it faster?

[@BricksOfAwesome]

I would argue against lucas numbers being superior, and here's why. If you take ɸ^1 you get ɸ+0, if you take ɸ^2 you get ɸ+1, now if you take ɸ^3, if you do some algebra given that ɸ^2 = ɸ+1, you get 2ɸ+1. ɸ^4 is 3ɸ+2, ɸ^5 is 5ɸ+3, and so on. It is very clear that ɸ^n = ɸF(n) + F(n-1). Is this something he hasn't come across, because this is mathematically exact, no rounding involved, and you get the fibonacci sequence from powers of the golden ratio, when you simplify it down.

[@hitobite]

Getting the numbers from rounding seems more magical, than making a decomposition. There is actually an exact decomposition of the lucas numbers to show that why the rounding gives these numbers: L_n= ɸ^n + (- ɸ)^(-n). But you're right, that doesn't mean the lucas numbers are superior. Since they both have a similar property and Matt actually didn't mention L_0 is approximated by ɸ^1, while L_1 equals ɸ^0.

[@prototypesoup]

ok then tell me what the 100th number in the Fibonacci sequence is, using your formula. Then tell me the 100th Lucas number is using the method from the video.

[@hitobite]

Bricks Of Awesome To be fair, the Lucas number show up in the decomposition of (2ɸ+1)*ɸ^n, while rounding ɸ^n/sqrt(5) gives the Fibonacci numbers.

[@BricksOfAwesome]

hitobite Although that is interesting stuff, I still think of ɸ^n giving fibonacci numbers is superior just because it's derived more simply. Fibonacci numbers seem more like the base line you go from for these types of series. I prefer Fibonacci numbers, but I get someone would prefer Lucas numbers.

[@hitobite]

Ow, my last comment was meant for Brandon Kekahuna. Bricks Of Awesome I would have to agree with Matt that the Fibonacci numbers are a bit overrated. I wouldn't say that the Lucas numbers are better.

[@TheYoshi463]

in fact if you use the golden ratio again to write phi^n you can find Fibonacci again. Nothing spectacular about Lucas Numbers: phi^1= phi + 0 phi^2= phi + 1 phi^3= 2phi + 1 phi^4= 3phi + 2 phi^5= 5phi + 3 ...

[@maximsokol4146]

Flewn not bad bro

[@braindead-borderlinefemini2900]

Same goes for any fibonacci style sequence, write out p subscript n for terms a and b(where a is the first term, like 1 in the fibonacci sequence, and b is the second term) and you find p subscript n=x (subscript n) *a+x (subscript n+1)*b when the starting series is a,b,p subscript 1, p subscript 2,... and x subscript n refers to the nth number of the fibonacci sequence

[@mohanlalchoudhary872]

I too noticed the same thing ; was just about to comment bout it

[@TheDoubleTea]

Phi^6 = 8phi + 5

[@ALLNAMESTAKEN911oku]

I think it's also interesting to note that both the Fibonacci and Lucas numbers can be expressed exactly in terms of the Golden ratio without any need for rounding.

[@technowey]

I like that if Phi is the Golden ratio, then the sequence starting: 1, 1 + Phi, 2 + Phi, 3 + 2 Phi, ... Is just a sequence of the powers of Phi.

[@LeoStaley]

I love his completely off-the-cuff defense of imprecision through rounding.

[@RedInferno112]

This is an awesome video! Golden Ratio is one of my favorite areas of mathematics.

[@mightyNosewings]

Ah, but Matt, I can one-up you. Recall that φ possesses the curious property that φ^2 = φ + 1. With that in mind, let's look at some powers of φ. φ^1 = 1φ φ^2 = 1φ + 1 φ^3 = φ*φ^2 = φ(φ + 1) = φ^2 + φ = 2φ + 1 φ^4 = 3φ + 2 φ^5 = 5φ + 3 φ^6 = 8φ + 5 And so on. It is clear that φ^n = fib(n)φ + fib(n - 1). I think this is a much more intimate connection than the one Matt displayed with the Lucas numbers.

[@CookieDynamics]

It seems that Lucas numbers get closer and closer to Phi^n as they go.... 3 - phi^2 ~= 0.4 4 - phi^3 ~= 0.2 7 - phi^4 ~= 0.15 11 - phi^5 ~= 0.1 So I assume the "error" will get smaller and smaller.... Which means that after a few interactions, this sequence can estimate with incredibly good precision the "n" power of phi. It's amazing since the sequence itself is made by adding natural numbers. The sum of natural numbers is being used to calculate the n-power or an irrational number. ---------- Edit: On the other hand, that's probably just a coincidence. If what I said at the beginning of this comment is true, then phi^n must have a tendency to become a natural number as it goes to infinity. That doesn't seem true, but I haven't tried it out.

[@hitobite]

L_n = phi^n + (-phi)^(-n), the last term goes to zero. So yes, phi^n goes closer and closer to a natural number.

[@CookieDynamics]

Johnny Luken I guess lack of self-esteem took a hold of me on that comment. Since this wasn't mentioned in the video I ended up assuming I was speaking non-sense. Thankfully, hitobite showed up and proved that I was right, which is nice.

[@gerwindox7499]

hm hm. Let me clarify something. we have a function f(n) = f(n-1) + f(n-2). This function is defined by its first two values, any other value is dependent on those two alone. Mathematically spoken, this function has two degrees of freedom. Now, this function has special solutions in the form of a^n: the two roots of a^2 - a - 1 = 0 (as substitution in the definition tells us that a^(n+2) = a^(n+1) + a^n). But those two numbers are exactly phi and -1/phi (which equals 1-phi). This means that phi^n is a contestant for f, as is (1-phi)^n. More interestingly, any linear combination a*phi^n + b*(1-phi)^n will satisfy the conditions on f. And this combination has two degrees of freedom as well. Furthermore, you can always solve f(0) and f(1) to be your preferred starting numbers, like f(0) = 2308 and f(1) = 4261. This equation is always solvable, and the solution will always yield a function that satisfies both your initial condition as the function definition, so it is an explicit form of your function f. If we set f(0) = 2 and f(1) = 1, then a = 1 and b = -1, which proofs that: phi^n - (1-phi)^n are exactly the Lucas numbers. Now, as hitobite stated: 1. f(n) is an integer, by induction. 2. (1-phi)^n < 0.5 for n > 1 so phi^n rounded to the nearest integer indeed equals the n-th Lucas number. EDIT: this also proves that for any number q = a+sqrt(b), q^n 'has the tendency to become an integer when n goes to infinity', AND you can calculate these integers with recursion... if and only if abs(a - sqrt(b)) < 1 So you want to calculate q = 2+sqrt(7)? this number is a solution of x^2 - 4x - 3 = 0. Also, 2 - sqrt(7) > 2 - sqrt(9) = -1, so abs(q) < 1 Now we create the recursive function for which q^n satisfies the condition, which is: f(n) = 4f(n-1) + 3f(n-2) We can start with any two values we want, as the secondary term, (2-sqrt(7))^n, will go to 0. So we start with a decent approximation: f(0) = 1 f(1) = 4 so we get: 1, 4,19,88,409,1900,... And after a quick calculation, I estimate 2 + sqrt(7) to be 1900/409 = 4,645477 2 + sqrt(7) is actually equal to 4,645751, which means my answer is to 3 digits correct.

[@frogkabobs]

The reason is that the n=1th lucas number is given by φ^n + (-φ)^-n and as n approaches infinity (-φ)^-n approaches 0 so φ^n approaches a lucas number

[@0x8055]

You are correct. Let phi be the golden ratio. One can show that x(n)=A*phi^n+B*(-1/phi)^n solves x(n) = x(n-1)+x(n-2) For Lucas sequence we have the system of equations: x(0)=A+B=2 x(1)=A*phi+B*(-1/phi)=1..................... Solve that as exercice :) The solution is A=B=1, so x(n)=phi^n+(1/phi)^n. As phi>1, when n goes to infinite, (-1/phi)^n goes to zero and x(n) goes to phi^n as you said

[@chrisharrison763]

The more you feature Matt, the more I've gotten comfortable with his stye (which is cynical but often amusing). He's a good asset to the Numberphile team.

[@JordanMetroidManiac]

Phi, the golden ratio, has other interesting properties. It can be used to calculate the exact value of pi, and pi can be used to calculate the exact value of e. pi = 5 * arccos (phi / 2) e = (-1) ^ (1 / [pi * i]) To calculate e, you must use a calculator that can deal with imaginary numbers, because e is the "pi i" root of negative one, where 3.1415927... is the coefficient of i. Just search "(-1) ^ (1 / [pi * i])" on Google. You'll get e exactly. Google's search engine is a mighty fine calculator itself. You can prove that these equations are true by checking to see if you get zero after taking the calculated value of pi and subtracting the "real" pi from it. The same process can be done with e. The Google calculator is precise to the 308th digit, so it's pretty safe to say that these equations are 100% accurate.

[@coopergates9680]

+Jordan Fischer But Pi = 6 * arcsin(1/2) and arcsin and arccos are more easily calculated for inputs near zero. Tenth root of e = 1+1/10+1/(100*2!)+1/(1000*3!)+1/(10000*4!)+1/(100000*5!)+.....1/(10^n*n!) so n begins at 0. Converges faster than the usual factorial sum, then raise to the tenth power.

[@prosincr]

That last sentence though...

[@ganondorfchampin]

No calculator can calculate those numbers exactly because calculators generate rational numbers, while those numbers are all irrational. Your idea of a proof is also awful, because it doesn't matter how accurate the approximation is, it's still just an approximation. You need to use calculus which doesn't convert the numbers into rational approximations in order to actually prove the formulas.

[@coopergates9680]

You know what he meant, calculators can get within 0.000000001% of the exact answer. Given the infinite series I mentioned, though, I wonder why he chose to use complex numbers to derive e when he didn't need to.

[@ganondorfchampin]

Cooper Gates Because the formula using complex numbers is a simply expressed, finite formula. It's not an approximation, it's an identity. It's better to use complex numbers than to use an infinite series for those reasons.

[@xCorvus7x]

Consistently getting within rounding range of every power of the Golden Ratio is still pretty impressive.

[@kevinslater4126]

I'm going to invent my own sequence so I can be known for something. How about you start with 1. And then you repeat the number 1. So start with 1 and then the second number is 1 and the third number is another 1 and so on and you just keep repeating that for infinity. What do I win?

[@gersomfrendy1893]

No research has been done - not that it has to be done - to get that 'sequence', so no, Kevin. You are not getting anything.

[@kevinslater4126]

Nooooooooooooooo!

[@AaronHollander314]

7 minute abs

[@turtlellamacow]

someone beat you to it -- that's sequence A000012 in the OEIS :(

[@alienplatypus7712]

Also that sequence is ζ(0) so it's sum is technically 1/2. Yes, 1/2=1+1+1+1+1+1+1+1... Cheeky, though objectionable proof. (Better ones exist) c=1+1+1+1+1+1... 2c= 2 +2 +2... -c=1-1+1-1+1-1... -c= 1-1+1-1+1... -2c=1 c=-1/2

[@coolboyrocx]

So I have the "Math Metal" song about phi that Phil Moriarty did with Dave from Boyinaband on my iPod and I quite enjoy it. I never really pay much attention to Phi, but when Matt started writing the digits I instantly found that in my head I could list them along without trying. So there we go, music does make things easier to remember.

[@jmich7]

People link mathematics with unnecessary precision, says Matt : and that is exact!

[@Mattihew1]

I know this destroys the beautifulness of the rounding and the Lucas numbers, but here it is anyway. Using the quadratic equation used to define phi (which I'll write as p to be readable), you get that p² = p+1. Multiplying this by p^(n-2), a real number, you will get that for all n, p^n = p^(n-1)+p^(n-2). Therefore, starting a sequence with 1 and p, you will get the sequence 1, p, p², p³, ... The ratio of these numbers is always the golden ratio, and so you don't even need to round the powers of phi to get a sequence that satisfies the criterion that the ratio converges to infinity. In fact, it can be seen as the "best series" because it converges to the golden ratio instantly. I guess rounding makes the numbers more everyday and down-to-earth, but in my opinion, the unrounded sequence is still the better one...

[@kristhetrader5029]

Great series of videos, definitely worth doing a series of videos on rounding!

[@alastairbateman6365]

As much as it sticks in my craw to agree with old Matt that the Lucas numbers relate more to the Golden Ratio than the Fibonacci numbers do, surely an old pro like him can do better than rounding up/down to fit to an already known sequence. So lets help him out! Let A=GR and B=GR-1 or 1/GR ( they are both the same). Then A^1-B^1=1 : A^2+B^2=3 : A^3-B^3=4 : A^4+B^4=7 : A^5-B^5=11 : A^6+B^6=18 & so on & so forth. Voila the Lucas numbers which are the difference of the ODD powers and the SUM of the EVEN ones. What about a link between the triangular & the Fibonacci/Lucas numbers.

[@coopergates9680]

Is the last sentence a tease, since triangular numbers grow quadratically and the golden ratio sequences exponentially?

[@alastairbateman6365]

Hello once again. Since you've taken the trouble to ask I'm more than pleased to tell you. Take the two consecutive triangular numbers 21 and 28. Expand them as a Fibonacci series to give 49, 77, 126, 203, .... . Divide by 7 to give 3, 4, 7, 11, 18, 29, .... the Lucas Numders alias a Fibonacci derivative. Is it of any great significance? I doubt it but presumably the two curves are simultaneous at the points 21 & 28.

[@coopergates9680]

Ah, because 21 and 28 happen to be multiples of 7. Another consecutive pair would be 36 and 45, both divisible by 9, and dividing up gives the members 4 and 5 in 3, 1, 4, 5, 9, 14, 23, ... (less precise for phi approximation than Lucas or Fibonacci) Moving further into the triangular numbers yields lower ratios between consecutive terms unless some are skipped, same goes for square numbers. It just so happens that 5, 2, 7, 9, 16, 25, 41, 66, 107, ... contains three consecutive square numbers (hint: Use Pythagorean triples if you want more cases), and that is probably the such sequence with the smallest starting terms. It also gives a fun joke because 5, 2, 7, ... eventually reaches 999801, 1617712. Is phi roughly 1.617712? Not really, much closer to 1617712 / 999801. Shift the decimal point and divide.

[@alastairbateman6365]

+Cooper Gates. Actually old Matt is misguided. The golden ratio relates equally to both the Fibonacci & Lucas numbers, the powers of the GR being calculated in terms of quantities defined by two consecutive terms of the fibonnaci sequence.

[@coopergates9680]

Since phi = (1 + Sqrt(5)) / 2, it doesn't have to be derived from such a sequence, and to make phi's powers tend toward the Fibonaccis instead of Lucas numbers, multiply each by (5 + Sqrt(5)) / 10.

[@OMH1314]

Awesome series of vids! Clear, insightful and entertaining.

[@backwashjoe7864]

It’s fitting that Matt’s Golden Ratio Trilogy shown in the end screen has 4 videos. 😊

[@eriktempelman2097]

Thanks for sharing! I knew the Lucas numbers, but that they link to Phi powers was new for me. Amazing properties...

[@likjhnfkjsbn]

vids like these are why i love this channel!!!

[@AdamPFarnsworth]

oh man "Lucas" numbers? In the last video I thought he said "Ludacris" numbers! But I gotta, kn-kn-kn-know what-what's your fan-ta-ta-sy

[@ZanoniSnowflake963]

Love you answer at the end because really mother earth & the multiversal laws, seasons, cycles, planet orbit etc. etc. aren't exactly persice hence ovals over circles. At least thats our understanding. Man likes things to be to linear with a beginning and an end when we live in a cyclical world where the end is the beginning and the beginning is the end the its a spectrum and an almosty fuzzy point, kinda like rounding has to be chosen! THANK YOU for this video we have only recently discovered a Lucas Sequence and more accurately the Lucas Numbers and are wanting to learn more on the comparisons and or maybe better worded how these and the Fibonacci and be used together. Blessings

[@HunterJE]

I think one answer to Brady's "isn't rounding kind of imprecise" complaint is to think to the previous video and remember that the "nth term/(n-1)th term" definition of the golden ratio we started from is already the _limit as n approaches infinity_ and that if you divide any two finite adjacent terms you will at best get an approximation of φ (one that gets better the bigger those two terms are); similarly the powers of φ more and more closely approximate the lucas numbers as the powers get higher (and start pretty dang close)...

[@Nanaki404]

Although it could be considered "cheating" since it uses real numbers instead of integers, but the sequence starting with (1, phi) is so beautiful. 1 + phi = phi^2, so the next number is phi^2. Then, you have the same thing, just multiplied by phi, so you get phi^3 and so on... In the end, you have 1, phi, phi^2, phi^3, ... and the Nth number is phi^N (considering 1 is the "0th" number). It's like Lucas numbers, but without rounding !

[@ThePeaceableKingdom]

An Engineer knows that EVERY number has already been rounded...

[@Nebch12]

What about 0? What about pi? What about i?

[@ThePeaceableKingdom]

Nebch12 Not sure what you're asking... pi is easily rounded off - practically every real world engineering application of pi uses an approximation. Same with zero - if you've lost your job and the mortgage is due, you may well say "I don't have any money" when there's actually 73 cents in your checking account... i is a little different. i would be difficult to round off to, or approximate. In that respect, it's more a number property than an actual quantity, isn't it? Taken out of its context as a factor of a negative square, what distinguishes one of them as an imaginary number? A seven could be a diameter or a circumference or a hypotenuse, but without the context to define its purpose, one couldn't tell which one a given instance of seven was...

[@TheTck90]

ThePeaceableKingdom I'm not an engineer and yet I got the joke better than most guys in this comment section^^

[@ThePeaceableKingdom]

TheTck90 Thanks!

[@ThePeaceableKingdom]

mwbhome e Thanks! I've never heard that one!

[@cdplayer2397]

4:31 brings back memories of when I did maths in school and would add as many digits to endless decimal numbers as would fit on my calculator's screen, my flawless logic being "more numbers = more maths = superior maths; Q.E.D. I am the best at maths."

[@ieuan2010]

I thought the Lucas numbers were: 4, 5, 6, 1, 2, 3, 7 !?

[@Nicoder6884]

What?

[@ieuan2010]

+Nicolino Will .Star Wars!

[@__nog642]

4, 5, 6, 7

[@mercronniel3122]

What? Is that a factorial in the end?

[@__nog642]

Martin Josip Kocijan 8 and 9 haven't been made yet.

[@unvergebeneid]

There are currently two people who clicked their way from one video to a hidden video and all the way to this hidden video ... and then disliked it. What is wrong with you people?!

[@adrianbiber5340]

The rounding is what makes it applicable to Nature, and perhaps why Nature *decided* to use it.

[@harry8175ritchie]

rounding is just a new flavor of mathematics! it would be nice to see more videos on the 'laziness' of mathematics and the interesting things that pop out! (using the term 'laziness' loosely)!

[@mathmethman]

I'm not going to defend the Fibonacci numbers as 'better' than the Lucas numbers, but I was curious as to why they appear in nature, and whether you could derive them from the golden ratio. So I set up a spreadsheet to try and find the best rational approximations to the golden ratio. On each line I increased the denominator by one, then found the 'best' numerator (by rounding) and compared the resulting ratio with the golden ratio. Then I went down the list (as far as the denominator being 60) and picked out any ratio which was more accurate than the previous best. This is what I found: 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55

[@salmachi9836]

Love you all numberphilers

[@jackconway7465]

Matt didn’t address the fact that the powers of the golden ratio are the two previous powers added together. The Lucas numbers just disguise this far more interesting property

[@scottdiesen3731]

this is the most amazing thing i have ever seen

[@calereece2483]

The Golden Fibonacci Numbers: The exact sequence for φ^n would be starting with the numbers 1,φ: 1, φ, (1+φ), (1+2φ), (2+3φ), (3+5φ), (5+8φ), (8+13φ), ..., (ƒ + [ƒ+2]φ) And when one does this the coefficients (ƒ) are the Fibonacci sequence.

[@agmessier]

A more compelling argument for the Lucas numbers, rather than rounding (which seems hand-wavey), is to consider that as the exponent increases, the difference between the result and the corresponding Lucas number integer approaches zero. phi^13=521.0019, for instance, and each successive power is even closer to a Lucas number.

[@robertschlesinger1342]

The Lucas Numbers are quite interesting. My first book on the Fibonacci Numbers was actually a booklet on The Fibonacci and Lucas Numbers.

[@Ctrl-h]

pi in base 706245 is (using the same principal as hexadecimal) is 3."99999" . allow "99999" to be equivalent to a single digit that equals that number in that base system. this is a wonderfully accurate representation of pi with minimal memorization

[@cambrown5633]

By extension, if you start the sequence 1, Φ... you get the exact powers of Φ: 1, Φ, Φ^2, Φ^3, Φ^4 And this goes backward, if you start with any two consecutive powers of Φ in order (e.g. Φ^-10, Φ^-9) it will generate all the powers exactly.

[@dmitriysilov4837]

Love your channel! It seems to me however that these numbers are not "superior" because of the power property, as the power property of the Lucas numbers is a necessary consequence of the fact that the quotient between two subsequent numbers in any sequence in the "Fibonacci family" (don't what is the real name) converges to the golden ratio as n goes to infinity. You can see it already in the video, where the errors are decreasing as n increases. Therefore, the nth term of a Fibonacci sequence with any starting terms can eventually be written pretty accurately as c*phi^n (where phi is the Golden Ratio and provided n is big enough). In case of the classic Fibonacci sequence, this c is equal to about 0.7243606... (according to my excel sheet)! It's just convenient beauty that the Lucas numbers have a c equal to 1. That is to say - they are both equally beautiful! No discrimination in mathematics madafakkaaa. Unless you're solving quadratic equations of course.

[@jimzamerski]

The further in you go, the closer phi^n approaches the next term (less gets rounded off)

[@MrAlcod]

I have discovered a cool explanation of why the Lucas series begins 2,1 , it will sound slightly strange to begin but I'm sure it makes sense))). The 2 means that in this series each number is the sum of the 2 previous numbers, And the 1 is 1 less than 2 to the power of 1. I have discovered some other Ratios associated with series' where each number is the sum of its 3,4,5 or 6 predecessors, when we take the powers of these ratios which extend towards 2, we have an equivalent Lucas series for each ratio. So the powers of the third ratio (1.839286755) round towards the 3,1,3 series, the 3 means each number is the sum of its 3 predecessors and the 1 and 3 are 1 less than 2 to the 1 and 2 squared. The 4th Ratio rounds to the 4,1,3,7, series then 5,1,3,7,15 and 6,1,3,7,15,31.My spreadsheet wont allow me any further so my progress has slowed somewhat, I'm not sure if the tenth powers rounds to the series beginning 1 or 10. This is another reason why the Lucas series is King as it ties these other ratios together with phi.

[@willemDiscool]

The Fibonacci sequence actually has another link to the golden ratio: you can calculate the Fibonacci numbers using this formula: F(n)=(φ^n-(-φ)^(-n))/sqrt(5) where F(n) is the nth Fibonacci number and φ is the "major" golden ratio (1.6180...). This formula will work so that for every integer value of n, Fn will be an integer (even for negative numbers). When n increases, (-φ)^(-n) will converge to 0, which also leads to the fact that the ratio F(n)/F(n-1) converges to φ.

[@teyxen]

+gewoonWILLEM Every sequence of this form also has a similar formula. a(n+1) = ( ( c + b(φ)^1 )*φ^n + ( c - bφ )*(-φ)^n ) / sqrt(5) Where b and c are the starting values of your sequence. In fact, using this you find that the formula for the Lucas numbers is even simpler than that for the fibonacci: a(n+1) = φ^n + (-φ)^(-n).

[@sztefenpierdziszewski1996]

I noticed, that as we raise phi to higher powers, the outcome is getting closer and closer to whole numbers, while other numbers multiplied by themselves seem to give pretty random decimal expansions. Why's that? Is phi the only number behaving this way?

[@ByronIgoe]

Or, start the sequence with 1 and Φ. 1 + Φ = Φ^2. Φ + Φ^2 = Φ^3...

[@2ghBrahhs]

James Grimes should make a video on using the Lucas Sequence to find a prime

[@culwin]

Another interesting observation - the decimals for the powers are the same as the decimals for each multiple-1. ie. look at the decimals for 2phi and phi^3, 3phi and phi^4, and so on

[@Etothe2iPi]

The golden ratio, written as a continued fraction (see wiki) reads ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,…]. Its rational approximations are 1/1, 3/2, 5/3, 8/5, ... which consist of the Fibonacci numbers. So the two ARE closely related. That's why both appear in nature.

[@Centar1964]

The most important part of this video starts at 4:20....this is proof that nature "rounds"....I hadn't considered this before but it is an eye opener...

[@Summertail]

3:49 You might even call it a Parker Square. :)

[@willypataponk]

This is absolutely amazing !

[@Italiankid1029]

I will believe literally anything Matt tells me. It's not like he is lying or anything in this video, just saying. Matt's the man.

[@ugurcansayan]

This video made me think about that "Any numbers when summed give you the golden ratio" And I tried to find the "magic / logic" behind it. Here is the result, //SPOILERS// it isn't surprising //SPOILERS END// U(1) = a U(2) = b U(3) = a + b U(4) = a + 2b U(5) = 2a + 3b U(6) = 3a + 5b … U(11) = 34a + 55b U(12) = 55a + 89b So, when you do the algebra (or "four basic operations"), somehow, anynumbers will give you Fibaonacci sequence back, so you will have golden ratio again. (Yes, it works with also Lucas Numbers, too.)

[@ZipplyZane]

Yeah, that's the direction I went, too, but then I saw the Matt's actual proof, which doesn't involve any of that at all. It just uses X(n) = X(n-1) + X(n-2). Golden Proof - Numberphile

[@juanchiflado11]

This

[@ugurcansayan]

ZipplyZane I know, what they shown as X there is what I shown here as U. The other part is just an algebraic example of the explanation. :)

[@mevludindauti27]

lucas = zeros (100,1); lucas (1) = 2; lucas (2) = 1; for i=3:100 lucas (i) = lucas (i-1) + lucas (i-2); end phi = (1+sqrt(5))/2; t = zeros (100,1); for i=1:100 t(i)=round(phi^(i-1)); end v==t I executed this MatLab code and the result is that from a point the rounding terms are not same as the Lucas numbers( except for the first two terms),Is that from the MatLab error in rounding on decimals? Is there any proof that rounding powers of phi gets you Lucas Numbers?

[@matrixstuff3512]

If phi is the golden ratio and phiconj = (1-sqrt(5))/2 = (1-phi) =-1/phi, The sequence is given exactly with no rounding by: L_n = phi^(n-1) + phiconj^(n-1) In fact any sequence that is generated by U_{n+1}= U_{n} +U_{n-1} can be expressed as U_n = a*phi^n + b*phiconj^n; where a and be are constants determined by the first two terms in the sequence. In this case a=1/phi and b = phi.

[@matrixstuff3512]

If we start the series with 1, 3, then L_n = phi^n + phiconj^n which to me seems like a nicer result

[@lingomaniac88]

+Kyle W Or just zero-index the sequence. In other words, L_0 = 2, and then L_1 = 1 and L_2 = 3 will follow naturally. That closed-form formula also shows why the rounding thing works, as the phiconj^n part will vanish as n gets large (since |phiconj| < 1).

[@morgangraley1049]

As a US citizen, I love the fact I can multiply my distance measured in miles by phi(ish) to help my more rational (HA!) friends who utilize km for distance instead.

[@disgruntledtoons]

If you look at the successively higher powers of phi, you notice that the difference between the value and the nearest integer grows smaller, and in fact the difference is a power of the inverse of phi. Lucas numbers follow the formula phi^n + (-phi^(-n)).

[@Cooliotso]

Wow, finally something I understand! But this is really awesome!

[@carlnickerson8950]

A thought about the "rounding" issue. Couldn't you say that phi^n and the lucas sequence are each the 'limit' of the other? As phi^n continues, the rounding adjustment gets smaller and smaller approaching 0, The ratio of the lucas sequence, as it continues, aproaches phi with increasing accuracy. (adjustment approaches 0) I don't have time to calculate it right now, but it would be intersting to compare the degree of accuracy for each process at each step. A) is the adjustment the same at each step? B) if not, is there a formula that generalizes the difference? heading off to work... please have answer by 5:30pm EST ;-)

[@shadyparadox]

You can get similar rounding results if you generalize the rule to X_n = A * X_n-1 + X_n-2 for some integer A. Using the same technique as in the proof video, the generalized golden ratio values will be of the form (A + sqrt(A^2 + 4)) / 2. The familiar golden ratio comes from the special case A = 1. But if you try A = 2 for example, you get a ratio of 1 + sqrt(2). The powers of this value round to 2, 6, 14, 34, 82, 198, 478, ..., which are the values of a Lucas-like sequence in which double the previous term is added to the term before it. (The ratios of consecutive terms do converge to 1 + sqrt(2) as expected.) For A = 3, you get a ratio of (3 + sqrt(13)) / 2. Once again, the powers of this value round to 3, 11, 36, 119, 393, 1298, 4287, ... , another valid sequence (tripling this time before adding the term before it). In fact, the first two terms are always A, A^2 + 2. (Except the rounding doesn't work for the first term when A = 1. Hmm.) Anyway, I don't know why this happens, I'm just playing around in Excel.

[@VACATETHE48]

Speaking of series, how about a video explaining the property of the addition of consecutive odd integers equaling squares (2^2=4=1+3, 3^2=1+3+5=9, 4^2=1+3+5+7=16 etc.) Never understood why this behaves like this.

[@EmonEconomist]

Great series of videos, and I would love to see a proof of this. Do the powers of phi more closely approach the corresponding terms of the Lucas sequence as it gets higher, or do they diverge?

[@aleksandersabak]

The exact formula for Lucas numbers is phi^n + psi^n where psi = 1 - phi = -1/phi. The "psi^n" term will tend to 0 as n increases, therefore phi^n will be closer and closer to a whole Lucas number. To derive the formula you can start with the definition: f(n) = f(n-1) + f(n-2) and get it from there with methods I barely understand how to use, and definitely not enough to explain them.

[@FeralAxiom]

Interesting comment on exactness in mathematics. I certainly prefer nice, exact quantites (the square root of 2 is just that; the number that, when multiplied by itself, equals 2, not some fractional estimate), but I can see there are things to be gained from looking at things in a more estimate-y way...maybe.

[@Ctrl-h]

pi in base 706245 is (using the same principal as hexadecimal) is 3."99999" . allow "99999" to be equivalent to a single digit that equals that number in that base system

[@kg583]

Loved the video(s)! I was wondering if you guys could do a video on the Collatz Conjecture (3n+1 conjecture). I read about it online but it didn't give too much information in the article.

[@IceMetalPunk]

Perhaps you should mention that it only works if you start at the first power, since phi^0 = 1, not 2 :P

[@lexinaut]

Those Lucas Numbers are wild! What kind of coffee are they drinking? Fun video that stimulates my curiosity very very deeply!

[@trukkstop1]

Rounding powers of Phi [the golden ratio] to get the Lucas numbers, as shown in this video, must work for any base, like binary or tertiary representation of numbers, and not just for numbers represented as decimals (base 10). Perhaps a simpler statement of this result would be: The integer closest to (Phi)^N is the N'th Lucas number.

[@chrisg3030]

It's well known that you get from the Fibonacci to the Lucas sequence by swapping round 1 and 2 in the Fibonacci 1 2 3 5 8 13 . . . so you get the Lucas 2 1 3 4 7 11. . . if you keep to the same addition rule. What's not so obvious is that the 4 in this new, Lucas, sequence, doesn't just coincidentally equal 2(1x2). Try swapping round another consecutive Fibonacci pair, say 2 and 3 to get 3 2 5 7 12 19 . . . and note that 2(3x2) = 12. Note too that 12 is four along from 3 in the new sequence, and 3 is at index 4 in the original Fibonacci, (and the other way round for 2). Swap Fibonaccis 5 and 3 round and see you get 2(5x3) = 30 in the new sequence, and so on. Can anyone express this identity elegantly?

[@coopergates9680]

4 is the 4th Lucas number, but 12 is the 5th number in the next sequence you used. You also squared the 1st Lucas number but squared the second term in your second sequence (the 2, not the 3). What about 3, 1, 4, 5, 9, 14, 23, 37, 60, ...? It just so happens that 3*4*5 = 60. The next two terms are prime (97 and 157).

[@chrisg3030]

Shall we call this the Gates sequence? In this case it isn't a pair of immediately consecutive Fibonacci terms that are swapped round but a pair of terms, 3 and 1, separated by another, and which are then brought together as a consecutive pair to be added. This produces some interesting results too, especially if we bear in mind that the term to the left is -2. We then have two terms, -2 and 3, whose squares occur later on in the sequence, namely 4 and 9. Do this next-but-one-swap with other terms and we get the same result, for example the numbers 5 and 13, usually separated by 8 in the Fibonacci sequence. We get -8 13 5 18 23 41 64 105 169 379 . . . , in which -8 and 13 lead to their squares.

[@christianaustin782]

I know this is kinda old, but as you raise Phi to higher powers, it gets extremely close to whole numbers. like by the time you raise it to the 25th power or something it's about within 0.000003 of a whole number

[@culwin]

The nice thing is you don't actually need to round - if you don't round, it's still a sequence where each number equals the previous 2.

[@КонстантинЩепин-р8г]

It is a well known fact that Fibonacci numbers arise from continious fraction decomposition of golden ratio thus their ratios are the best possible approximations of golden ratio. (any other rational number with denominator less or equal to corresponding Fibonacci number is further from golden ration)

[@madhatternick]

He makes a really good point about rounding. It brings to mind the translation from pure mathematics to real world scenarios. For example, nature cannot produce a fraction of a protein, a fraction of a cell, a fraction of an organism, a fraction of a population, or a fraction of a biosphere. Rounding brings ideas to reality.

[@Randomz-jk4hk]

FUN FACT: If you take the roots of the quadratic equation x^2=x+1 and call them a and b. Then, *nth Lucas number = a^n + b^n* For example, 0th Lucas number = a^0+b^0 = 1+1=2 1st Lucas number = a^1+b^1 = 1 2nd Lucas number = a^2+b^2 = 3 and so on... Also, one of the roots of that equation is infact the golden ratio. EDIT: I just realised that I accidentally proved why each Lucas number rounds off to phi^n As one of the roots is phi, let a=phi. So, b= 1-phi which is approx -0.6 So, if we want to find each Lucas number, say 4th Lucas number = phi^4 + (-0.6)^4 So, phi^4 = 4th Lucas number - (-0.6)^4 But, as (-0.6)^4 is definitely less than 0.5 and as the 4th Lucas number is definitely a natural number, removing (-0.6)^4 from the 4th Lucas number is not going to decrease it below 0.5 less than the number itself. Meaning, that if we happen to round up this difference, we will end up with the 4th Lucas Number. This means that phi^4 rounded up gives the 4th Lucas Number. Infact, phi^n will round off to the nth Lucas Number on one condition: (-0.6)^n must be less than 0.5, which will be true for all n>1. So, phi^n will have to round off to the nth Lucas Number. Also, the proof of nth Lucas Number = a^n + b^n is pretty easy, using Induction. All you have to do is find a+b=1 and a^2+b^2=3, then, show that as x^2=x+1, x^(n+2) = x^(n+1) + x^n [Multiplying by n] As a and b satisfy this equation, we get: a^(n+2) + b^(n+2) = a^(n+1)+b^(n+1) + a^n+b^n This is the exact way how we generate Lucas numbers; by adding consecutive terms to get the next term. So, this sequence is clearly the Lucas numbers.

[@Ubeogesh]

So could rounding be a topic for a next video? Are there some cool tricks, problems or patterns with rounding?

[@inchinaxp8663]

"I'm gonna say Lukas. If it wasn't for Luka" LOL

[@ishwar8119]

L_n starts 1,3,4,7... Matt shifted the Lucas numbers by one place.

[@reverendjim7066]

my immediate thought is there any relationship between "left overs" after rounding? Some pattern to phi^n-round(phi^n) ? the "round" being a pseudo computer program command to round the number, not sure how that should be written in math speak :-)

[@ganondorfchampin]

The mathematical operations are called floor and ceiling, and are represented by || like absolute value, only with feet or hats respectively. Rounding can be derived from those operations.

[@nicholasosullivan831]

I write the unnecessary decimal places, because I enjoy having that amount of accuracy and I also like to see how big the impact of rounding is.

[@jeanterre1134]

Well, if you want Xn = φ^n, then choose X0 = 1, and X1 = φ. Since φ² = φ + 1, then φ^n = φ^(n-1) + φ^(n-2), and this way we indeed have Xn = X(n-1) + X(n-2), and Xn = φ^n.

[@SanderClaassen]

So, if the Lucas numbers are 1, 3, 4, 7, etc, when he says that φ^2 rounds up to 3, φ^3 rounds up to 4, φ^3 rounds up to 7, etc to conclude that the powers of φ are (rounded) the same as the Lucas numbers, he ignores the exception of φ itself. φ=φ^1=1.61... which would have to be rounded up to 2, which is not equal to 1, the first Lucas number. He should just have mentioned that this rule only doesn't apply for the first Lucas number. For the rest, I find this amazing, especially the fact that φ^n approaches the n-th Lucas number, more and more closely as n gets larger. When you think of this as follows: φ=1/2+1/2(sqrt(5)). Then φ^n, which has a lot of terms containing sqrt(5), becomes an integer, when n goes to infinity. Isn't that beautiful?

[@thechemuns74]

Anyway, we round to get phi in Fibonacci too. 1/1 is 1, 2/1 is 2, 3/2 is 1.5, 5/3 is 1.666... (and it starts to look closer to phi), and so on. Only with n that tends to infinity we can say that Fn/Fn-1 = phi...

[@rcksnxc361]

Honestly I find rounding very useful I made a formula for finding a phytagoras triple using rounding and other techniques usually considered useless

[@VicvicW]

Rickson Geometry Dash That sounds interesting! Care to elaborate?

[@lostfractal4801]

rounding is rather important in Chemistry. Use significant figures to prevent artificial look of high precision.