The magic of Fibonacci numbers | Arthur Benjamin

The magic of Fibonacci numbers | Arthur Benjamin

So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration. Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics
that we learn in school is not effectively motivated, and when our students ask, “Why are we learning this?” then they often hear that they’ll need it in an upcoming math class or on a future test. But wouldn’t it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind? Now, I know many people have not had the opportunity to see how this can happen, so let me give you a quick example with my favorite collection of numbers, the Fibonacci numbers. (Applause) Yeah! I already have Fibonacci fans here. That’s great. Now these numbers can be appreciated in many different ways. From the standpoint of calculation, they’re as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on. Indeed, the person we call Fibonacci was actually named Leonardo of Pisa, and these numbers appear in his book “Liber Abaci,” which taught the Western world the methods of arithmetic that we use today. In terms of applications, Fibonacci numbers appear in nature surprisingly often. The number of petals on a flower is typically a Fibonacci number, or the number of spirals on a sunflower or a pineapple tends to be a Fibonacci number as well. In fact, there are many more
applications of Fibonacci numbers, but what I find most inspirational about them are the beautiful number patterns they display. Let me show you one of my favorites. Suppose you like to square numbers, and frankly, who doesn’t? (Laughter) Let’s look at the squares of the first few Fibonacci numbers. So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on. Now, it’s no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right? That’s how they’re created. But you wouldn’t expect anything special to happen when you add the squares together. But check this out. One plus one gives us two, and one plus four gives us five. And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues. In fact, here’s another one. Suppose you wanted to look at adding the squares of
the first few Fibonacci numbers. Let’s see what we get there. So one plus one plus four is six. Add nine to that, we get 15. Add 25, we get 40. Add 64, we get 104. Now look at those numbers. Those are not Fibonacci numbers, but if you look at them closely, you’ll see the Fibonacci numbers buried inside of them. Do you see it? I’ll show it to you. Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate? (Laughter) Fibonacci! Of course. Now, as much fun as it is to discover these patterns, it’s even more satisfying to understand why they are true. Let’s look at that last equation. Why should the squares of one, one,
two, three, five and eight add up to eight times 13? I’ll show you by drawing a simple picture. We’ll start with a one-by-one square and next to that put another one-by-one square. Together, they form a one-by-two rectangle. Beneath that, I’ll put a two-by-two square, and next to that, a three-by-three square, beneath that, a five-by-five square, and then an eight-by-eight square, creating one giant rectangle, right? Now let me ask you a simple question: what is the area of the rectangle? Well, on the one hand, it’s the sum of the areas of the squares inside it, right? Just as we created it. It’s one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? That’s the area. On the other hand, because it’s a rectangle, the area is equal to its height times its base, and the height is clearly eight, and the base is five plus eight, which is the next Fibonacci number, 13. Right? So the area is also eight times 13. Since we’ve correctly calculated the area two different ways, they have to be the same number, and that’s why the squares of one,
one, two, three, five and eight add up to eight times 13. Now, if we continue this process, we’ll generate rectangles of the form 13 by 21, 21 by 34, and so on. Now check this out. If you divide 13 by eight, you get 1.625. And if you divide the larger number
by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians, scientists and artists for centuries. Now, I show all this to you because, like so much of mathematics, there’s a beautiful side to it that I fear does not get enough attention in our schools. We spend lots of time learning about calculation, but let’s not forget about application, including, perhaps, the most
important application of all, learning how to think. If I could summarize this in one sentence, it would be this: Mathematics is not just solving for x, it’s also figuring out why. Thank you very much. (Applause)

Dereck Turner

100 thoughts on “The magic of Fibonacci numbers | Arthur Benjamin

  1. asdf ghjk says:

    Listens to Tool once

  2. Lisamarie says:

    living, application, inspiration, transportation./. I10…..=vehicle NOT cubed, it is extravehicular. LIFE is common like a hexagon it shows all sides. That is true LIGHT

  3. Mariana Veranyan18 says:

    Who has the text of this talk, send me, please👋🙏

  4. Angie Bradley says:

    n e v e r e n d i n g

  5. Eduardo Mendes says:


  6. Mike Ivy says:

    There were times and teachers that made math fun…and it was bc they made each of us a part of a whole…we were called to the Baird, worked the problem to its conclusion and how or why did we reach this answer…I remember a math teacher in HS,( and this was my 3rd try at algebra 1), and he told us we were going to have many problems to work at home and at school and sure enough we went to the board in 5’s and worked algebra problems and if u didn’t get the answer correctly , he didn’t chastise u, no , he guided u through the problem until u figured out ur mistake. And I made an A for my final and I left this class feeling I could do anything…wasn’t true…when I went to college and took my first college algebra I was lost and the teacher said they didn’t have time to help and so I withdrew from that class and eventually college…effed up teachers just don’t know they are poor teachers.

  7. Forex Life says:

    fibonacci is God's signature

  8. Chandni Vora says:

    The golden ratio is also called as phi, right?

  9. gunerguk says:

    "Everything We created is precisely measured." Quran 54:49

    "Praise the Name of your Lord, the Most High. He who creates and regulates. He who measures and guides." Quran 87:1-3

    "You see no discrepancy in the creation of the Compassionate." Quran 67:3

    "The sun and the moon move according to plan. And the stars and the trees prostrate themselves. And the sky, He raised; and He set up the balance. So do not transgress in the balance." Quran 55:5-8

  10. Nazeer Hussain says:

    Close your eyes and imagine Heath Ledger 's Joker when listening to his voice

  11. Dyên Nguyễn says:

    So great

  12. John says:

    I only know 25 letters of the alphabet and I don't know Y.

  13. Victoria DH says:

    Amazing, thank you. I wish my math teacher was like him, but mine was some selfish dude, and that's why I started to hate Math, but I always applaud to those folks who love math.

  14. Mikey Comicguy says:

    He should have walked into the room to the sound of Lateralus blaring over the speakers

  15. Alisher says:

    and the application is still secret?

  16. Elephant In The Room says:

    Fibonacchi introduced the formula to the "western world" its concerning that people are afraid to say that the muslim mastered it due to the quran. They got the world into the modern world and out of the darkness into the light. The entire quran is based around the Fibonacchi code. It should be called the quranic numbers and not the Fibonacchi if people are truthfull.

  17. seema verma says:

    Woh superb👍🏻🔥

  18. Abhiram Pratapa says:

    Good video useful

  19. Hai Nguyen Trong says:

    Arthur Benjamin, your lecture is very insightful for the new beginners, still this does not solve the string theory in space relativity. Fibonacci is really a genius in definition of a golden spiral to which we all agreed upon. We are waiting for welcoming of great you tube on math insight from you.

  20. Hai Nguyen Trong says:

    The Fibonacci numbers are the structural representation of the galaxies origins and its expansion in space and mouvements of planets. Thank you for this original video of high learning. Intelligence is the Alma Mata of all human kind that separate us from the animal world.

  21. Alefbet says:

    What a wonderful and brilliant Creator of all this and us. To God be all glory. Amen.

  22. Oleg K.O.A. says:

    Golden ratio melody

  23. It's Pompey Jomuad says:

    Simon where's Garfunkel?

  24. MARK EBRAHIM says:

    The sequence with dividing Fibonacci numbers works for every sequence but you get even closer to the golden ratio. Like if it works

  25. Divyanshu Gairola हरिदास says:

    Please make another videos on applied maths😍😘

  26. KnightFox18 says:

    1,3,6,9,18…..27,36,45/ 54 63 72 81 . so on so fort.out of all of an infinite amount of numbers, 18 , when you add its individual digits equal half its value.9. only number to do so.

  27. Σταμάτης Μισιρλής says:

    The golden ratio of 1,618, has first discovered by the Greek mathematician Pythagoras, 2,5 millenniums ago.

  28. Wingate8 7zero says:

    time 20 9 18 there is only one way

  29. Anurag Kumar says:

    Speak Hindi

  30. Luca Martin says:

    FIBONACCI was a genius.

  31. Manav Gupta says:


  32. sanjay gowda says:

    great video.. I could have watched it before the end of my college studies… 😕

  33. Rahul Sharma says:

    In the introduction to his book, Fibonacci (c. 13th century CE) makes the following revelations

    1) "I am the son of an official working in Bugia, Algeria".

    2) There was a colony of Indian Merchants in that city.

    3) "It was there that I was introduced to Indian Mathematics".

    Fibonacci further says-

    " I loved Indian Mathematics to such an extent above all others that I completely devoted myself to it"

    "I was also introduced to Greek, Arabic & Egyptian Math"

    "But I found ALL of them, EVEN Pythagoras, to be erroneous compared to Indian Mathematics"

    Fibonacci further says:

    "For this reason, basing my book COMPLETELY on Indian methods and applying myself with greatest attention to it, but not without adding something of my own thought, I forced myself to compose this book.

    I demonstrated everything with proof"

    Finally, Fibonacci says:

    " In my book, I have published the doctrine of Mathematics completely according to the Method of Indians.

    I have COMPLETELY adopted the (Mathematical) Method of Indians because it is the MOST effective"

    Thus, in his book, Fibonacci does NOT refer to #Fibonacci Series as "Fibonacci Series"

    Rather, he simply calls it "Indian Series".

    Unlike many other Europeans, Fibonacci was NOT a plagiarist.

    He clearly mentioned his source and acknowledged his credit to ancient Indians.

    Fibonacci's introduction makes it clear that he considered himself "Indian Mathematician" insomuch as he adhered to Indian Mathematical Methodology and contributed to it.

    The real name of the so called "Fibonacci Series" is "Indian Series".

    This comes from the horse's mouth !

    So far as the so called "Fibonacci Series" is concerned, Fibonacci was only TRANSLATING the Sutras of Pingala (c.3rd century CE) and his commentator Virahanka who derived "Fibonacci Series" several hundreds of years before Fibonacci was even born .

    I was very shocked reading Fibonacci's introduction. Why are these facts kept concealed?

    A more important question. Why should it be called 'Fibonacci series' when Fibonacci himself does not claim to have discovered it and simply acknowledges Indian Mathematics as his source?

    The precepts of Pythagoras and Euclid were forgotten in early middle ages and revived only later.

    Yet, the credit always goes to Pythagoras and Euclid. Never to the later day Mathematicians who revived their works. Why is Pingala never extended the same courtesy?

    I wonder why!

    Fibonacci was NOT a European Mathematician, except by flesh and blood.

    He explicitly rejected the European methodology of Mathematics. He denounced even the path of Pythagoras as "erroneous".

    He followed footsteps of exemplary Vaidika Mathematicians like Pingala and Virahanka

    Fibonacci does not describe his book as "European Mathematics".

    He explicitly describes his book as " treatise on Indian Mathematical methods".

    As such, it is hard to even consider him a "European Mathematician". He followed the footsteps of Vaidika Sanskritic Mathematicians

    Reference and the source
    i have used :
    English translation of introduction to Fibonacci's book "Liber Abaci". Published in the scholarly journal Reti Medievali Rivista by Giuseppe Germano (2013)

  34. Suraj Goud says:

    Its real name is Indian Series .

  35. Jishnu Jishnu says:

    I liked it

  36. Kate Kate says:

    its called ted talk but when is ted gonna show up

  37. Ben Chaggares says:

    *Enter Tool fans

  38. james mcguinness says:

    LSD numbers 😂

  39. urbanjunior says:

    nardwar is it you??? 0.0

  40. Hajdu Zsolt says:

    When you're a member of a neapolitan executioner family

  41. Lord of the Phantoms says:


  42. N8riz says:

    Give Jesus Christ credit for Him creating everything and we get to be at awe

  43. Malka Manuranga says:

    That is why I love Benjamin

  44. tonton inbas says:

    Foutage de gueule à l'encontre des non anglophones !

  45. Aatif Ahmed K says:

    "Mathematics is not just solving for x, its also figuring out y" Loved this quote

  46. Alex 3M says:

    Someone please help me with 3,6 and 9 the Tesla's legend.

  47. sharkawi jirim says:

    U make me sleepy

  48. Priscila Canaan Oliver says:

    still no fun. boring

  49. I am an egg says:

    “Why do we learn mathematics?”

    So I can spin a steel ball

  50. David Schwartzberg says:

    You want to hear a joke?

    Well, I took two others and mixed them but got something no so greater

  51. robson silva says:

    Incrible! This is a excelent presentation!

  52. Rashesh Patel says:

    Its adapted from Indian Theory of Sankhya, written 33000 years back in Sanskrit. Adopted even by Tesla.

  53. W Mobberley says:

    This is one example of a Ted Talk which just stops. It should have gone on for at least another hour!

  54. James Laid Ler says:

    "Over thinking over analysing separates the body from the mind"

  55. Faris Al says:

    you better also discuss how this fibbonacci learnt about the theory. then you probably find God

  56. Doruk Kavraz says:

    his speaking & pronounciation are very clear.

  57. Fai Lua says:

    I enjoyed every minute

  58. Krishnakumar M says:


  59. remconet says:

    This was like a book with just an intro… kinda weird.

  60. McBlcVar aka Varun says:

    It's actually 4: Graduation

  61. wtfMYRON says:

    I've seen better short videos on it on YouTube

  62. Duhck says:

    i’m sorry but in the thumbnail he looks like when dollar store bill gates tries to introduce a new iphone

  63. Mike Calz says:

    Who is trader here ?

  64. Nilmani Ray says:

    Happy New Year 2020 to all Pi (π) lovers. Each and every book may help you to learn something as you are seeking. Even a piece of papers laying on foot path may encourage you to success your dream as ATTENTION is the key to success of life. Guys I am going to say that as per Greek alphabet value of Pi (π) is 3.14 and its place in the alphabet is 16. But the origin of the Pi(π)= 22/7 is correct.
    Now I am planning to publish a book about π, 0(Zero) and numbers. But due to unavoidable circumstances, I can not do that. I have been researching on this topic since 2013. Due to lack of proper guidance and qualification it is not possible to research under an university. If some one willing to help in this regards I shall try my best to proof the origin of Pi (π) & 0 (Zero) and its relation to all.
    A sample example is given below.
    5 x 1 = 5
    5 x 2 = 10
    5 x 3 = 15
    5 x 4 = 20
    5 x 5 = 25
    5 x 6 = 30
    5 x 7 = 35
    5 x 8 = 40
    5 x 9 = 45
    5 x 10 = 50
    Total = 275

    275÷ 5 = 55 ÷ 70
    = 0.7857142857…… x 4
    = 3.142857142857….
    Note : The calculation has been done as per my research process.

    Nilmani Ray
    Guwahati, India
    Email : [email protected]

  65. Mounjia Abdeltif says:

    very interesting

  66. Srim3359 says:

    I mean, yeah, he explained it in a truly simple manner. But I still don't see the practical use of this number sequence…

  67. lob1952 says:

    The serie really start with zero, "0 + 1 = 1"

  68. MRTACPANS says:

    what is the real value of one since its located in the 5 position center to the 4

  69. aniello fontana says:

    Great final.

  70. Dafan says:

    Incrível. " Matemática não é só encontrar o X"

  71. jalal BENLAIDI says:


  72. jalal BENLAIDI says:

    Love me or hate me, both are in my favour. If you love me, I'll always be in your heart… If you hate me, I'll always be in your mind.”


    Very nice

  74. Rich Channel says:


  75. Àlan 237 says:


  76. Alby Corrin says:

    That was hella good

  77. Phú Hoàng says:

    Sure but why my nails are SPINNING ?

  78. Parker XX14 -opponent police says:


  79. stefan goldberg says:

    it should be called God's number! we did not invent it,just discovered it.

  80. Riovo Gaming says:

    The video is 6:25 which is 25 squared

  81. Don R. Mueller, Ph.D. says:

    This guy looks like the legal douche-bag from the OJ trial (the one that went bad) what's his name? Oh yeah, Barry Schmuck. Hopefully, this guy is better at math.

  82. Adel Merah says:

    Arab numbers not Fibonacciʼs

  83. EnergeticWaves says:

    He looks like David Brenner

  84. S says:

    How can you assemble the most, in the least amount of space, with the least amount of energy, but with the highest degree of efficiency. THAT is the philotaxis of the golden section, the golden proportionality.

    The Golden ratio goes way deeper than simple math. We see the pattern everywhere in nature, from the galactic to DNA, to the geometry of magnetism. And since all atoms are basically tiny magnets, think of the ramifications…

  85. ProfOmarMath says:

    Art is a wonderful colleague to have. He spreads the joy of mathematics in a fun and compelling way.

  86. Joe Blechl says:

    wow wow wow. this is a pic of a pyramid i built out of cardboard from a cropcircle of a hypercube. call me!!!

  87. Hashim Khan says:


  88. Komal Ghadigaonkar says:

    wonderfully explained. great animation. Its a need of time we should motivate younger ones to take math. There should be more such videos.

  89. The Gripmaster says:

    Nose: 1
    Eyes/ears/hands/legs: 2
    Fingers: 5

  90. Eunus Mondal says:

    WOW loved it.

  91. Mayra Calizaya says:

    Omg!! He's amazing

  92. James Allen jr. says:

    I knew there were teachers like this… I had nothing like this learning Calculus. Memorizing formulas was not my thing. So, my professor pretty much ruined my computer career by making math horrible.

  93. Reazul Hasan says:

    Nice lecture sir…i love your class

  94. Gabriel Braun Valle says:

    math does not inspire me

  95. Nasrin Khatun says:


  96. Ronney Rendon says:

    Why on earth is this only 6.25 min?! I was really getting into it then it ended! Aren’t ted talks usually 15 min?! So sad over here.

  97. Ricky Bhattacharya says:

    Fibonacci Tribonacci and n-bonacci numbers.

  98. Phillip Senn says:

    2, 3, 5, 8 who do we appreciate?

Leave a Reply

Your email address will not be published. Required fields are marked *