Golden Proof - Numberphile
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- Опубликовано: 21 сен 2014
- Part Two of Golden Ratio Trilogy
Proof that an infinite number of sequences have that "golden ratio" property - not just the Fibonacci Numbers.
More links & stuff in full description below ↓↓↓
Next in the trilogy: • Lucas Numbers - Number...
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Brady Numbers: • Brady Numbers - Number...
Proof: • Golden Proof - Numberp...
Lucas Numbers: • Lucas Numbers - Number...
Extra bit: • Golden Ratio and Fibon...
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Наука
"Why use whole numbers? Go bananas!"
Indeed.
+Eli Reid I'm obligated to agree. If we consider which is the wilder and more exotic of the two Fruits, it is the pomegranate.
Why use pomegranates? Go durians!
gghelis haha
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Why not start with π and e?
Curiously, it works even for complex numbers. Take any two complex numbers and build the sequence by adding the two last terms, and the ratio between two numbers in the sequence will start approaching phi+0i.
Oh wow, that really is interesting
Complex Fibonacci numbers
There is nothing curious about that. Yeah, of course, it will work for complex numbers too, because the reasoning in the video applies ...
@@samueldeandrade8535Indeed. A better word would have been "interestingly."
I lost it when he said, "I can multiply through by triangle..." lol
∆ makes me think of change....when he said ∆ i was like NOOOOOOOOOOOOOOOO! lmao.
Oh the days when delta used to make us think only of change
Only people who understand physics understand lol
just wait until 2021, Δ will make you think of something significantly worse than change
I want 0:42 as my rigtone
ughggh ughggh ughggh
I was hoping someone picked that up as well :D such a wonderful little gesture
"If I multiply through by triangle..."
WHEN DID MATH GET LIKE THIS!?!?!
(I know it's just a delta, but still)
Another interesting thing is that the elements of (1+sqrt5)/2 are the sides of the right angled triangle with sides 1 and 2,hypotenuse sqrt 5. This could be considered as the second most 'basic whole number ' right angled triangle' (the most basic would be with sides 1,1 and sqrt 2). Must have fascinated the ancient Greeks as being another of the 'Cosmic' connections between numbers and geometry
Ken Margo it's also the length of any diagonal of a unit pentagon
Delta = triangle = Brady is a Illuminati
voveve: He's one of the reptilians?!?!
_(Teacher Forming a question on board)_ : Gimme an example of whole number.
_Me_ : Why use whole numbers? Go Bananas!
_Teacher_ : Out!
Really important channel to follow. This is one of those channels that always makes you feel better about math, if you like math just a bit or just want to get a better grasp of it. This golden ratio stuff is particularly interesting to me, especially considering the
mathematician in this episode thinks it is overrated.
I like the way the RUclips subtitles call Matt 'Guy:'
0:43
2:19 I absolutely adore Matt for multiplying by Triangle
I would love to see you discuss the differences between the golden ratio and the super-golden ratio and why at all these differences are important in mathematics or where they crop up.
This is the solution to my first calculus homework in university. I was stuck until I noticed this video in my feed. Thank you Numberphile!!!
The golden ratio describes the characteristics of ordered accumulation in mathematics. There are a lot of ad hoc combinatorial models that ultimately demonstrate this.
It is interesting that the golden ratio squared is equal to the golden ratio + 1. This can be written as x^2=x+1. Solving for x, we get:
x+1-x^2=0
-x^2+x+1=0
(x^2-x-1=0)
Using the quadratic equation we get:
x = (1+sqrt(5))/2 which is equal to golden ratio.
This proof was far simpler than i expected.
But it wasn't completely clean, was it? I mean the first equation, where he put X_n/X_(n-1) = X_(n-1)/X_(n-2). Why should we be able to do that?
"If I multiply through by triangle" Best line
Really simple and really cool proof... this feels nice. Thanks.
Why is this private? This is very interesting, I didn't know where the (1 + sqrt 5)/2 came from.
Miguel Guerrero I have explained so many times why some videos start their lives unlisted and become listed later - I don't think I can do it any more.
Rest assured it will soon be public.
Ah, ok, sorry.
That's the quadratic formula.
Keppit Now I know
+Numberphile Sassy.
hey man, are you an actor in some monty phyton films? you looks so
He is not John Cleese. Unfortunately.
I always thought he looks like John Cleese.
The first time he said triangle I genuinely laughed.
Please do a video on a similar constant - the "silver ratio"! This is 1+sqrt(2)
Where the golden ratio is equivilant to (a+b)/a = a/b , the silver ratio is equivilant to (2a+b)/a = a/b
Interesting fact: if you remove the largest square possible from an A4 sheet of paper (ratio 1:sqrt2), the remaining rectangle is a "silver rectangle" in the ratio 1:1+sqrt2
Help he assumes the ratio must be constant, what is the proof for that?
That's an oversimplification in order to avoid speaking about limits. Just put a limit sign in front of both sides of the equation in your mind :)
Christian Kuhl And that still assumes that the ratio converges.
kilroy1964 Yes, indeed he does. But one can prove that this will always be the case except for the trivial one (0,0,,...) in which the ratio is never even defined.
Christian Kuhl Yes I assumed the sequence of ratios can actually be shown to converge, or else there would be no point in the rest. I was just pointing out that that proof is missing here. (But for obvious reasons.)
It's funny to watch these because sometimes I don't even understand what's happening but however I like to watch these videos. Maybe I'm learning something here. :)
i tried this on excel,(because i got curious,) taking the nth term and defining the ratio from there always receives an output value which is +/- from a number of decimal places no matter what are the 1st 2 numbers with the exception of about the first 5.
meaning on the 10th term, you will always have 1.61... (as i have tried)
Almost correct. If you begin exactly in the (1-sqrt(5))/2 subspace (starting with 1, (1-sqrt(5))/2, (3-sqrt(5))/2, ...), then the ratio is the negative root of delta.
@4:14-- don't just go bananas, go SINGING BANANA!!
This guy explains this really well. Good video.
how do we know that the ratio of 2 consecutive terms of fibaconni sequence converges
THIS IS THE GREATEST VIDEO EVER MADE!!!!
If only you could have posted this a week ago; it was one of my homework problems.
If you attach one simple one-dimensional harmonic oscillator to another (same mass, same spring constant), the appearing angular frequencies are the golden ratio or its inverse times the frequncy of a single simple one-dimensional harmonic oscillator - amazing :D
The Fibonacci Numbers have appeared in TV shows. For Example in the Criminal Minds episode Masterpiece, the serial killer used that sequence to pick out the number of victim as well as their hometown
Great video.
However, I think that it would be even better if you complete the video by also proving that the limit when n goes to infinity of x(n)/x(n-1) - x(n-1)/x(n-2) equals zero, therefore proving that indeed x(n)/x(n-1) tends to x(n-1)/x(n-2) when n goes to infinity.
Thanks for your wonderful work.
Or at least mention that he assumes the limit to exist…
To be fair, he mentions, en passant, that "it should be the same ratio give or take", but "it should be", certainly, is not "we assume".
Anyway, we should admit that this channel is great.
That formula for solving quadratic equations (ax^2+bx+c) seems unfamiliar to me.
We learned a similar one with p and q, where we set a=1 and then b=p and c=q.
(p/2)+-sqr( (p/2)^2- q )
I guess it's just the way we learned it, but it's surprising how you learn formulas differently in different places/countries. I would've expected some kind of standarisation.
What if you start with two complex numbers, does it still approach the golden ratio?
what happens if you use - instead of +? So: (1-√5)/2
Does it work with complex numbers? I assume so from the proof, but I would like to see it done.
Another property of all "Fibonacci-like" sequences is that the sum of the first ten terms of such a sequence is always divisible by 11; furthermore, that quotient equals the seventh term of the sequence. Would love to see a rigorous proof of this (don't think I can do it).
The Fibonacci numbers are the convergents of the continued fraction of the golden ratio, so they approximate the golden ratio extraordinarily well despite their small (positive integer) denominator. I guess that's some special property of them.
I like this new format where its cut up.
I love numbers, they make my day. Numbers are amazing.
That's just beautiful. Freaking magical
I didn't know you can do the pq-formula like he did it in this video o-o
THAT IS SO MUCH BETTER!
I gather that's another name for the quadratic formula. How in the world were they teaching it to you?
ZipplyZane
Well for the example of x^2+2x+1, then x = - (2/2) +- sqrt((2/2)^2 - 1) = - 1 +- sqrt(1-1) = - 1.
Or more generally:
x^2 + ax + b = - (a/2) +- sqrt((a/2)^2 - b)
It's how the swedish school teaches it :P
how did you say that ratio of the nth term by (n-1)th term is equal to the ratio of (n-1)th term by (n-2)th term?
He's taking a kind of limit... As the numbers in the Fibonacci sequence become larger and larger, the two ratios will approach a single value and will thus be almost equal
At that point he is assuming that the limit exists and continues to prove that, _if the limit does exist_ , it has to be equal to the number he produces. The step of actually proving that the sequence converges is omitted on the video.
thanks for the explanation. i can sleep now
Matt forgot to write "limit of x_n / x_n-1 as n approaches infinity". Technically what he wrote was wrong, so to correct it you need to put a "lim" or something next to "x_n / x_n-1".
if your starting conditions are 1, phi, the n-th term in the sequence is going to be phi^n-1:
1, phi, phi^2, phi^3, ...
phi is one of the roots of the polynomial x^n - x^n-1 - x^n-2 so you can clearly see whats happening. that is the most natural "Fibonacci" sequence. you can find the 1,1 Fib sequence if you add them instead of multipling by phi each time.
1, 1phi, 1phi + 1, 2phi + 1, 3phi + 2, 5phi + 3, 8phi + 5...
That explains why when you take the Fibonachi series: x(n) is the number of km an x(n-1) is the equivalent in miles. Because the golden ratio is similar to the convertion ratio between the two units. I.e 3 miles are 5km, 5mi are 8km, 8km are 13mi and so on.
W H A T HOW W H Y
How do we know if we pick 2 arbitrary starting numbers, whether we will arrive with the golden ratio or its negative reciprocal?
Great video!
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
Yay! I had been trying to remember this, and then I stumbled across this. Thanks Numberphile.
OMG THIS MADE SENSE!!!!
You just saved my assignment hahaha
What about sequence of complex numbers?
What about using 1 and -0,6180339887498950 (which is actually (1-root5)/2 ) for starting numbers?
Excel cannot calculate this effectively, but it tends to have a constant ratio of -0,6180339887498900 (except after about 36 numbers where it goes back to the golden number eventually).
(Credit to wowsaw0 who found that on the previous video).
Not sure, if this is completely correct, I claim that you have to prove that the sequence x_n/x_(n-1) actually converges and other stuff, for me it just seems unclean.
It probably should be easy to prove using the explicit form of the fibonacci sequence - or for any sequence like that, it's easy to find that form.
The Fibonacci sequence is defined in this video as X = X + X (making up some notation for this comment). But what what about a series like X = X + X? Or with X? Do these other series have the same aspect where they tend toward the same ratio regardless of starting numbers? And if they do is there some relationship between these ratios?
This was a surprisingly simple proof!
Amazing video
Just beautiful
Question :
1 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 ... What is the formula that will create this sequence beginning with the 3rd number ?It is not the fibonacci folrmula since the 3rd number is 1 not 2 .
If you take the difference between each number you get the fibonacci sequence again, just with a number of zeros at the start.
Here is a solution I like, using linear algebra (which actually fills a hole in the argument of the video). Consider pairs (x_n,x_{n+1}) in the sequence. How do we work out the next pair of terms (x_{n+1},x_{n+2})? Well, it will be given by multiplying the (column) vector (x_n,x_{n+1}) by the matrix
0 1
1 1
(try multiplying this out with a column vector with some terms of the Fibonacci sequence, if you like). This is because the first entry of the new pair should be: (x_{n+1} , x_n + x_{n+1}).
You can work out the eigenvectors and eigenvalues of this matrix easily, using the usual tricks. One of the eigenvalues has eigenvalue bigger than one, and the other perpendicular direction is less than one. So, by iterating this matrix, one can easily show that it "squishes" pairs to the line (1,golden ratio) with leading eigenvalue but pushes the point to infinity, unless the pair belongs to the other eigenline (-golden ratio,1). This means that, except in "exceptional cases" the ratio of terms converges to the golden ratio.
=======
What I like about this solution is the following:
1) As long as you understand the intuitive ideas about eigenvalues and eigenvectors, this proof is very "geometric". It's easy to picture all of your points being "squished" to the line of a certain slope. I know lots of people won't know anything about "eigenvectors" and so on, *but it's worth learning, and it's very intuitive!*
2) It generalises easily to rules other than x_{n+1}=x_n+x_{n-1}. Things can become more complicated when you start allowing sums of more terms, it's fun to investigate!
3) It shows you PRECISELY what pairs you get which tend towards the negative inverse of the golden ratio. They are precisely those points lying on the line spanned by (-golden ratio,1). And, just to point it out, if you stick in the pair (0,0), then they stay at (0,0) (the map is linear!), and so doesn't ever have a well-defined ratio.
4) It actually shows that the ratios CONVERGE to these values. The proof in the video above isn't quite rigorous, in that one was assuming that the ratios are "exactly" equal, term by term. So there is a hole in the argument of the video: one would need to show that the ratios must always converge, amongst a couple of other things. The proof using eigenvalues above intuitively explains why this is the case.
I should point out that my argument clarifies something incorrectly stated in the video: he claims that you "always" converge to the golden ratio (I'm not blaming him, just a slip of the tongue I imagine).
But this is not true, as the above shows. Indeed, plug in (-golden ratio,1) (or any multiple of it) as your first two terms. Then what will happen is that the terms flip-flop from positive to negative, but tend towards zero. Their ratios will always be the negative of the inverse of the golden ratio. This is where the other solution to the quadratic formula comes in from the video. But it is very delicate! If you even mess up the terms by a very small amount, eventually you will start to tend towards the golden ratio.
Hope this clears things up for people!
jamma246 A most interesting approach indeed! Thank you for sharing. :-)
This proof is really for the limit as n approaches infinity, right?
1:30 i thought you couldn't cancel out terms if they were added together, you could only do that if they are multiplied together. for example 4+3/4 wouldn't equal 3, but 4x3/4 would.
EDIT: I just understood what he did by adding one. Ignore previous comment
What happens if you make an Ulam spiral with the Fibonacci numbers?
+Sam C You don't get anything interesting, the Fibonacci numbers are spaced far too widely apart from one another for any visual patterns to form. Even at a 100x100 scale you can only see twenty of them (compare that to over a thousand if you're graphing the primes).
wow I had this interesting problem at my school which asked to calculate the number which added to its square is equal to its cube, and I found out it to be the golden ratio!!! came directly to this vid
Great Video! So if there are two solutions for the golden ratio, how come when you divided terms in both the fibonacci sequence and the 'brady sequence' both approached only one of the solutions (1.618... as opposed to -0.618...)
Divide the preceding number with the succeeding one instead.
***** Does that imply that the negative solution is meaningless and should be ignored? Or does it still have some inherent properties associated with the sequences?
Daniel D For starters: -0.618... is the negative inverse of the golden ratio, so it has some meaning of its own. And there are indeed sequences (for example starting with 1 and -0.618...) which end up with limit -0.618..., though these are the minority, in some sense.
So does it work with complex numbers if you take the ratio of the absolute values?
[about the proof at 0:58] - for example, 5/3 is not equal to 3/2...have I missed the point?
If Xn/Xn-1=Xn-1/Xn-2 then 5/3=3/2 but that is not true... Do you mean it is limit (it goes near there more and more)?
There was one step I didn't get: Why is he so sure the ratio converges to 1 number? That seems quite random to just assume that's the case. I can think of a lot of sequences that don't follow that rule, so why?
i think it fails for 10,50,60,110,170,280,450 series ..
and also it doesn't work if x1
I don't understand. At the first step, how do you know that those to ratios are equal?
I really like the thumbnails.
@ 2:00 i don't understand how did he go from
1 + x_n-2 / x_n-1 = delta
to
1 + 1 / delta = delta ???
But the Fibbonacci no. make the best rational approximations of phi, as we get from its continued simple infinite fraction and that is .Mathologer did a great video , search "the most irrational number".
Avatar should be a phi, it's not Numberpile
THE BRADY NUMBERS! 4:39
Freakin' hilarious... LOL
Wait, i don't get it...1/1 = 1 and 2/1 is, well..2, how is that the golden ratio? Do they mean when n is getting close to infinity or did i miss something?
This might be a dumb question but at 1:14 if lets say Xsubn=5 and Xsubn-1 = 3 wouldn't that be 5/3? If that is a the case wouldn't xsubn-1/xsubn-2=3/1=3? Those are obviously not equal to I'm not getting something. Maybe I'm not understanding the notation...
I think he means what delta approaches... But valid point
What about this sequence:
0.618
-0.382
0.236
-0.146
0.090
-0.056
...
the ratio doesn't tend to 1.618, but the terms satisfy the fibonacci property...
How do they come up with the fact that x(n)/x(n-1) = x(n-1)/x(n-2)?
Because even when the series reaches infinity, this still ist't true becaue phi has infinite decimals.
Lets take some of the first numbers for example: 8/5=1.6 and 5/3=1.667
So the outcomes are NOT the same.
When you get bigger numbers the outcomes are closer to each other but are never the same. So how and why can they directly assume that x(n)/x(n-1) = x(n-1)/x(n-2)?
Am i thinking wrong somewhere or do more people feel the same?
How come the proof equates exactly the golden ratio but in the video's before the actual calculation only approaches the golden ratio?
can you express the golden ratio with limit?
"if i multiply through by triangle" I literally guffawed at that!! hahahaha what a strange thing to catch you off guard.
This proof is so much more satisfying if you complete the square!
What happens to the ratio if you add one term? 1,1,1,3,5,9,17,31...
I can't edit my previous comment now, so I'll repost briefly then add the new stuff.
I used two complex numbers (1+2i) + (3+4i) then added them. The third and fourth members of the sequence are 4 + 6i and 7 + 10i respectively. Then if you take the ratio of the magnitudes: ( 7+10i)/(4+6i) -> sqrt(7^2+10^2)/sqrt(4^2+6^2) = 12.20656.../7.21110 = 1.692746
Now calculated the phase angles for the last two complex numbers in the sequence and divided them:
arctan(10/7) / arctan(6/4) = 0.598046 = 1 / 1.672113 ~ 1 / phi
does it work with complex numbers as well?
in fibonacci numbers there are no geometrical progresion, for example if n=5 x(n)=5 x(n-1)=3 and x(n-2)=2 and x(n)/x(n-1)=5/3 and x(n-1)/x(n-2)=3/2 and 5/3 isn't equal to 3/2
5/3=3/2 ?
i love how the caption writer just calls matt "Guy"
No one will answer, but I try anyway. If I had discovered a new property of the Golden ratio, and new family of numbers with the same property, what I have to do? I'm not a mathematician professionally speaking, and I did not write an appropriate formal paper (because I don't know how), but the property is real and experimentaly proved! thank you.
What about Complex numbers? Would this work out to?
(a+bi)+(a+bi)?
What happens when you use a positive number and a negative number to start off?
= 0.5 + 0.5 * (5)^0.5
magic 5 number, notable, also the diagonals of a pentagon intersect each other at goldenratio.
i guess our choice to represent the number in base 10 provides this beautiful coincidence
Why did he start with the ration of Xn/Xn-1 = Xn-1/Xn-2???????
8/5 is not equal to 5/3.
I don't understand why we can start with that when solving for delta.
***** same problem!
Next time choose the Greek letter Xi.
It's a good trap. Xi = Ξ or ξ
;-)
I don't know why but the idea of multiplying by triangle amuses me greatly.