The magic of Fibonacci numbers | Arthur Benjamin

Translator: Elian Muftiu Reviewer: Helena Bedalli Why do we learn mathematics? Firstly, for three reasons: calculation, execution and, lastly, unfortunately less important in terms of the time we dedicate to it, inspiration. Mathematics is the science of motivation, and we study it to learn to think logically, critically, and creatively, but much of the mathematics we learn in school is not usefully motivated, and when our students ask, "Why?" What are we learning this? "thing?" We often respond that they will need it in a lesson or in a later exam. But wouldn't it be great if we sometimes learned math just because it's fun, nice, or just to stimulate the mind?

I know a lot of people haven't had a chance to see how this can happen, so let me give you a quick example through my favorite collection of

numbers

, the Fibonacci

numbers

. (Applause) Yes! We have Fibonacci fans here. Now, these figures can be estimated in many different ways, from the point of view of calculations, it is easy to understand how one and one are two, then one and two are three, two and three are five, three and five are. eight, and so on. In fact, the person we know as Fibonacci was actually named Leonardo Pisano, and these numbers appear in his book "Liber Abaci", which taught the West the arithmetic methods we use today.

In terms of application, Fibonacci numbers surprisingly often appear in nature. The number of petals on a flower is usually a Fibonacci number, or the number of spirals on a sunflower or pineapple is also usually a Fibonacci number. There are actually more applications of Fibonacci numbers, but what I find most inspiring about them are the beautiful numerical motifs they show us. Let me show you one of my favorites. It seems crazy to me that you like to balance numbers, honestly, who doesn't? (Laughter) Let's take a look at the squares of the Fibonacci primes. So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on.

Now, it is not surprising that when you add the Fibonacci numbers in a row, you find the next Fibonacci number. Or not? This is how they are formed. But you wouldn't expect anything special to happen when you put the squares together. Watch this. One and one give us two, one and four give us five. And four plus nine is 13, nine plus 25 is 34, and so the motif continues. In fact, here we have another one. Suppose you want to see how the squares of some of the Fibonacci primes add up. Let's see what we get. So one plus one plus four is six.

We add nine, we get 15. We add 25, we get 40. We add 64, we get 104. Now we look at these numbers. These are not Fibonacci numbers, but if you look closely, you will find Fibonacci numbers hidden inside them. You see it? I'm telling you. Six is ​​two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we “grade”? (pun intended) (Laughs) Fibonacci! Of course. Now, as fun as it is to discover these reasons, it is even more fun to understand why they are true. Let's take the last equation. Why must the squares of one, one, two, three, five, and eight added together be eight times 13?

I'll show you by making a simple drawing. We start with a square one by one and next to it we put another square one by one. Together they form a one by two rectangle. Underneath I'll put a two-by-two square, and then a three-by-three square, under it a five-by-five square, and then an eight-by-eight square, making a giant rectangle, right? Now let me ask you a simple question: what is the area of ​​the rectangle? On the one hand, it is the sum of the areas of the squares inside, right? Exactly how we created it. It's one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. correct?

That's the surface. On the other hand, since it's a rectangle, the area is the height times the base, and the height is obviously eight, and the base is five plus eight, or the next Fibonacci number, 13. Right? So the area is also eight times 13. Since we calculate the area in exactly two ways, these must be the same number, and that is why the squares of one, one, two, three, five, and eight added together are eight times 13. If we continue with this process, we will produce rectangles of the shape 13 by 21, 21 by 34, etc. Look now. If you divide 13 by eight, you get 1.625.

And if you divide the larger number by the smaller number, the ratio gets closer and closer to the number 1,618, known to many as the golden ratio, a number that has fascinated mathematicians, scientists, and artists for centuries. Now, I tell you all this because, as with much mathematics, there is a beautiful part that I fear does not receive enough attention in our schools. We spend a lot of time learning about calculations, but let's not forget about the application, including, perhaps most important of all, learning to think. If I could sum this up in one sentence, it would be: Mathematics is not just about solving x, but also about finding why.

Thank you so much. (Applause)