# Golden Proof – Numberphile

1. July 13, 2016 at 10:22 pm

2. But why does the ratio of two numbers from that sequence tend to the golden number ? Shouldn't it be equal to it ?

3. i started dividing the numbers in fibonaccis sequence and their ratio wasnt exactly the golden ratio, instead it seemed to aproach more and more to that value as i divided numbers further into the sequence, but never quite reaching it, even though it should (or so i think) why does this happen despite the fact that this numbers are indeed the sum of its predecesors in the sequence? it really bugs me that their ratio is not exactly phi and not knowing why.

4. 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 .

5. What if the two numbers you start with are both 0?

6. 0:41 "hshskhmmhmh"

7. Does the Fibonacci sequence start with a 0 or a 1?

8. Actually he did NOT really prove that the Fibonacci ratio of two consecutive terms will tend towards the golden ratio.

Look at 0:47. He assumes that consecutive terms will tend towards a fixed ratio.
So what he actually proves here is "simply" that if a sequence tends towards a fixed ratio for two consecutive terms, then this ratio will be the golden ratio…

But he did not prove here that the Fibonacci sequence will tend towards a fixed ratio. This is a missing element in his proof.

Nice and instructive video anyway

9. 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?

10. Where's 4.. You wrote 1.2.3.5

11. This guy sounds like Wallace from Wallace & Gromit.

12. I think I saw this formula in Einsteins equation, but I am not quite sure… Its that dejavu effect.

13. What if you start with two complex numbers, does it still approach the golden ratio?

14. how do we know that the ratio of 2 consecutive terms of fibaconni sequence converges

15. If I take the equation x_n/x_n-1= x_n-1/x_n-2 and multiply each side by (x_n-1)(x_n-2) I get
(x_n)(x_n-2)=(x_n-1)^2 and that is not entirely true how does that work? Because for instance if we let x_n=5 then we get 5(2)=3^2 which is off by 1. If we let x_n=8 then 8(3)=5^2 which is also off by one. x_n=3 then 3(1)=2^2 and also off by one and so on. Could it be possible to demonstrate that by doing this we will always be off by one; meaning (x_n)(x_n-2)+1=(x_n-1)^2 and if so why does that happens, if the ratio is the same then so should be the equality without the +1.

17. Which program do you use to plot these formulars?

18. This isn't a proof. You have proven that if it converges to a ratio then that ratio is (root5+1)/2 but not that it converges.

19. But the Fibbonacci no. make the best rational approximations of phi, as we get from its continued simple infinite fraction and that is <1;1;1;1;1;1;1;1;1;1;1;…>.Mathologer did a great video , search "the most irrational number".

20. Avatar should be a phi, it's not Numberpile

21. Know what's cool?

The -0.618 golden ratio is for when you cross from negative to positive numbers or vice versa

22. 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.

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…

24. Is there a relationship between the golden ratio and the fractals?

25. does it work with complex numbers as well?

26. go bananas

27. your Side-burns are very cute.

28. March 12, 2017 at 1:04 pm

I don't understand. At the first step, how do you know that those to ratios are equal?

29. why the x/xn-1 =xn-1/xn-2

30. Turn on subtitles
1:04
I think it means "I love you" in some ancient language

31. Doesn't b = -1 in that quadratic formula ?

32. that makes 1/0 a golden ratio.. so basically infinity is a golden ratio?

33. If we start with 1 and Ф, the ratio stays constant

34. 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.

35. he was briefly possessed at 0:42

36. Those are ugly X's

37. mistake found. it's ∆2-∆+1= 0

38. "Add two terms, you get the next one . . . you always get the golden ratio"

That's not entirely true. I can think of two sequences – one trivial and one not so trivial – that do not converge to Φ.

There's the trivial sequence starting with 0,0 that doesn't work as others pointed out in the comments. The part that breaks down is assuming you can always take the ratio of consecutive terms since 0/0 is undefined.

However, there is a more interesting sequence which doesn't converge to the golden ratio but does converge to something else. There are two solutions to the quadratic equation, namely Φ and -1/Φ. The second solution is always in the shadow of its big brother but a solution nonetheless.

Take the sequence -1/Φ, 1/Φ^2, -1/Φ^3, 1/Φ^4, -1/Φ^5, . . ., (-1/Φ)^n , . . .
which is approx. -0.618.., 0.382.., -0.236.., 0.146.., -0.090..,….
The sum of two consecutive terms equals the term after that.
But the ratio between two consecutive numbers is -1/Φ.
You could also multiply the sequence by a constant and that too would have ratio -1/Φ.

39. April 14, 2017 at 4:11 pm

It seems that you are saying that any two numbers in a fib sequence are the golden ratio, which is kind of weird in itself, but why can't we apply this to geometry. A sequence of squares whose area is 1, 1, 2, 3, 5…. We can find the ratio of any two squares in that sequence by the golden ration and we may find something interesting since the golden ratio comes from geometry itself.

40. There is a lot more to this. F(n)+hF(n+1)=F(n+2)   =>  F(n)=[((h+sqr(h^2+4))/2)^n – ((h-sqr(h^2+4)/2)^n]/(sqr(h^2+4)). Also see how to find square roots with Fibonacci. See Maths Montage. There is more to come including gh-sequences.

41. What about tribonacci numbers? What is the ratio there

42. That's just beautiful. Freaking magical

43. Why is "0" never formally included at the beginning of the Fibonacci Sequence?

44. But that only works when n tends to infinity and we never said that in this explanation did we?

45. This proof is so much more satisfying if you complete the square!

46. This proof is so much more satisfying if you complete the square!

47. May 6, 2017 at 2:10 am

Seriously, what is with those "x's"? He is not even simplifying it, his "x" is bigger and takes more effort than a standard x. Maddening, I could barely follow this because of his x's.

48. Simply put, phi is one h of a lot cooler than pi.

49. what?

50. Ah…that moment when you get numbers from triangles!

51. Pretty x's.

52. Sniff the sharpie

53. can someone explain to me what he did at 1:33?

54. i think it fails for 10,50,60,110,170,280,450 series ..
and also it doesn't work if x1<<<<x2 , where x1 and x2 are the first two whole numbers in any series.

55. and I think π is overrated. GLORIA φ!

56. 1.61803398874989404586834365638… That's how far I have it memorized.

57. 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.

58. 2461088728020852650965837278824744888195501202033208602494809016150345898655881568529668632145075095684064475574129304979502076478845558119223703716529010199517881773147465912048047355958278354093324012611115410331140881757507234580768781634807813079555237687138396348552813749066168694119624173736122996979931367342080

59. I like the way the YouTube subtitles call Matt 'Guy:'

60. The best Fibonacci-like sequence is:
F(1) = 1
F(2) = 0
F(n) = F(n-1) + F(n-2)
because every Fibonacci-like sequence:
S(1) = x
S(2) = y
S(n)=S(n-1) + S(n-2)
is equivalent to S(n)=F(n)x +F(n+1)y

61. 0:43

62. That's not how you write an X! lol

63. i got it all intil the quadratic equations which melted my left brain

64. mate if you just show what a b and c where in tthe quadratic equation i could have got it

65. 8 + 9 = 17 17+9= 26 26 + 17= 43 43 + 26= 69 69/43= 1.60465116279. He was right

66. i still dont know ahat hes applying the quadratic equation too

67. mate just specificy what you mean a bit more

68. i must say, it's much more roundabout way of findig phi then, say, the normal equation

69. I'm confused. The proof given seems to imply that the ratio of any consecutive pair of numbers in a sequence, which obeys the property that the next number in the sequence is the sum of the previous two numbers, is equal to the golden ratio. However, this is not true; because only the limit of consecutive pairs in such a sequence, as n goes to infinity, is equal to the golden ratio. Why does the proof seem to imply the ratio of consecutive terms in the sequence is the golden ratio?

70. Is 1 equal to 0+(-1)?

71. can you express the golden ratio with limit?

72. 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 4×3/4 would.

EDIT: I just understood what he did by adding one. Ignore previous comment

73. 1.61803398874989…

74. Pi and golden ratio always show up everywhere. It's creepy.

75. 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

76. But why does this happen doe?

77. Golden ratio skeptic, lol 🙂

78. If you make the plus-minus negative, you get negative inverse Phi.

79. Let Me Do A Math Equation,
X = 2C back to back.
(:

80. R R R H H H H H

81. angry because delta is already used in math

82. The other number that you get, 1-sqrt5/2 also satiasfies as a ratio for sequences like that. So wouldn't be fair to say that that number could also fit a sequence. Yes the Fibonacci sequence is one of them if you go to the left instead those numbers look like this
…-8 5 -3 2 -1 1 0 1 1 2…. It's the same just back wards and every other number is negative. The ratios are -1/2,2/-3,-3/5,5/-8 which you can see is the ratios to right just flipped and negative, so it converges to -1/phi and 1/phi is phi -1 you can see this because phi^2 =phi +1 and divide by phi so it's -(phi-1) which is -phi +1 and you can do the math and see that that is 1-sqrt5/2

83. Very, Very Nice Explanation!.  What I wonder though, is this.  You have proven that if a limit exists here, IT IS phi.  How would we prove that THE LIMIT EXISTS?  That's a lumpy stumper for me at this point.  Or have we in principle proven it here?

84. 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.

85. smart is the new sexy

86. You said the -0.6180….
is also golden ratio but how can the ratio of two positive no. be negative???

88. Fibonacci numbers appear in nature (sunflower, pinecone).

You just saved my assignment hahaha

90. As others have pointed out, there's errors in the way this way explained. Led to misunderstanding for my younger sister who's in calculus now. You need to talk about limits to do this right. For those with more experience, it's fine, but for those who don't get math as well, I think it causes more problems than it solves to leave things out.

91. Doesn't work for sequence starting with 0, 0

92. What if u start with 0 and 0? Ha I win

93. can someone explain me how x(n)/x(n-1) = x(n-1)/x(n-2)?

94. How do we know if we pick 2 arbitrary starting numbers, whether we will arrive with the golden ratio or its negative reciprocal?

95. August 7, 2019 at 2:25 am

(Teacher Forming a question on board) : Gimme an example of whole number.

Me : Why use whole numbers? Go Bananas!

Teacher : Out!

96. The first time he said triangle I genuinely laughed.