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)




Comments
  1. First discovered in India – they should give credit where credit is due – it's like saying Columbus discovered America – we have moved forward from that – but where dark skin is concerned we remain racist

  2. After reading 'being surprised' comments. I must say Western people are very poor at maths because this the very very very basic thing what he explained.

  3. I prefer to notice the beauty. The math is irrelevant to me. I was a hard science major. Then I died in a sailing accident.

    Sit down. Slow down. Notice the beauty. Nature will handle the math for you. Leonardo would agree.

  4. Also interesting…
    10 squared 100 (01 squared 001)
    11 squared 121 (11 squared 121)
    12 squared 144 (21 squared 441)
    13 squared 169 (31 squared 961)

  5. This isn't related to Fibonacci, but I just turned 53 and I figure I'm in the prime of my life. I wanted to tell this joke to math people.

  6. I'm an art teacher and I teach my students golden ratio and Fibonacci numbers all the time. It's conected of course with visuality also with biology and phisics. Amazing thing is, as wild the students behave, as soon as I start talking about it, they all calm down and listen.

  7. 2:27 whats interesting is that for each number listed, you can express either it or its square as a 1 2 or 3 if zeros dont matter. You do this by taking the square, multiplying each of the numbers in the square until the units cannot be multiplied to create something larger. This would either be a whole number 1 through 9, or a number like 10 20 30 ect. Once you get to 5 for example, 25=2×5=10=1. For number 13^2=169 1×6×9=54, then 5×4=20=2 so its like a second derivative only with the multiplying of digits within the square…until you get to 21, but wait a minute, 21 itself can be expressed as a 2 or a one, so the previous number 169 with 2 of these derivatives is followed by one with no derivatives (or whatever you call them) which is interesting because 5, which is the only fibonacci number which in sequence=itself, is also the first time a 1,2, or 3 is not used in its original form

  8. Before Fibonacci Hemichandra (Indian mathematician) of 1000 CE had already invented this. Indian poetry is based on Fibonacci. First Fibonacci series is described by Pingala(450BC). So give credits to Indians.

  9. Miracle of mathamatics,,!! Fine,
    Enlightened a lot,! Appreciated &
    Encouraged, Thanks to Arthur Benjamin & TED for uploading,,

  10. I often thought about different sums and i also thought about this numbers, but i have known about name of this. This video is very interesting! And some properties were surprise to me

  11. Here's something interesting I think:
    12th fibonacci number = 89 (btw 144 is next one, which is 12×12).

    89 x 89 = 7921 – diameter of earth in miles.(well – its 7920, but v close)

    7921 / 144 (next fib numb after 89 ) = 55 (fib number below 89 – 11th one)

    11 x 60 (secs/mins) = 660 = 1 furlong (660 feet)
    55 x 12 = 660

    7920 (diameter of earth) / months (12) = 660

    1 furlong (660) x 8 = 5280 (1 mile )
    1 mile = 5280 feet
    7920 / 11 = 720
    720 /5 = 144
    7x6x5x4x3x2x1 = 5040 (anti-prime – Platos best number for city inhabitants – has 60 divisors)
    5040/7 = 720
    720 gives us everything (sun,earth,moon dimensions and time / dates etc)..too much to list
    without coming across as a number nutter;)…but we get 2160 (moon),4320,8640(both sun) , 7920(earth) as well as exterior angle of heptagon (51.43) 720/14

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  13. I really liked this video and i like Arthur Benjamin. It would have been very cool and nice to be his student.

  14. I learned about Fibonacci numbers in 6th grade after we were done with all of the standards(and 7th grade standards)

  15. I am Japanese third Junior high school student. I’m learning English and this video is hard to understand for me. But someday I hope I can’understand this video

  16. 13*2=26-5=21*2=42-8=34*2=68-13=55*2=110-21=89*2=178-34=144*2=288-55=233*2=466-89=377*2=754-144=610*2=1220-233=987*2=1974-377=1597*2=3194-610=2584*2=5168-987=4181*2=8362-1597=6765*2=13530-2584=10946. This pattern just keeps going. Can someone please explain why this happens? PLEASE?

  17. 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

  18. 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.

  19. "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

  20. 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 and know how in a deep way. I think Math is very important in our life, it changes the way you think.

  21. 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.

  22. 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.

  23. 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.

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

  25. 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.

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

  27. 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)

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