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.
Zašto učimo matematiku? U suštini, iz tri razloga: računanje, primena i poslednje i nažalost najmanje bitno, što se tiče vremena koje mu posvećujemo, nadahnuće.
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)
Matematika je nauka o šablonima i proučavamo je kako bismo saznali kako da mislimo logički, kritički i kreativno, ali previše matematike koju učimo u školama nije valjano motivisano i kada naši učenici pitaju: "Zašto učimo ovo?", često čuju da će im to biti potrebno na nekom budućem testu ili času matematike. Ali zar ne bi bilo sjajno kad bismo se, s vremena na vreme, bavili matematikom jednostavno jer je zabavna ili predivna ili zato što je uzbudljiva za um? Znam da dosta ljudi nije imalo priliku da vidi kako ovo može da se desi, zato hajde da vam dam brz primer sa mojim omiljenim skupom brojeva, Fibonačijevim brojevima. (Aplauz)
Yeah! I already have Fibonacci fans here. That's great.
Da! Ovde već imam Fibonačijeve fanove. To je sjajno.
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.
Ove brojeve možete razumeti na mnogo različitih načina. Sa stanovišta računanja, jednako ih je lako razumeti kao 1 + 1, što je 2. 1 + 2 je onda 3, 2 + 3 je 5, 3 + 5 je 8, i tako dalje. Zaista, osoba koju nazivamo Fibonači se zapravo zvala Leonardo od Pize i ovi brojevi se pojavljuju u njegovoj knjizi "Liber Abaci", koja je Zapadni svet naučila aritmetičkim metodama koje danas koristimo. Što se tiče primene, Fibonačijevi brojevi se u prirodi pojavljuju iznenađujuće često. Broj latica na cvetu je tipično Fibonačijev broj, ili broj spirala na suncokretu ili ananasu, to su takođe uglavnom Fibonačijevi brojevi.
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)
Zapravo postoji još dosta primena Fibonačijevih brojeva, ali ono što je za mene najinspirativnije u vezi sa njima su predivni šabloni brojeva koje oni prikazuju. Dozvolite da vam pokažem jedan od meni omiljenih. Recimo da volite da kvadrirate brojeve, a iskreno, ko to ne voli? (Smeh)
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.
Hajde da pogledamo kvadrate prvih nekoliko Fibonačijevih brojeva. 1 na kvadrat je 1, 2 na kvadrat je 4. 3 na kvadrat je 9, 5 na kvadrat je 25, i tako dalje. Ne iznenađuje činjenica da kada dodate uzastopne Fibonačijeve brojeve, dobijate sledeći Fibonačijev broj. Zar ne? Tako oni nastaju. Ali ne biste očekivali da se desi ništa posebno kada saberete kvadratne vrednosti. Ali pogledajte ovo. 1 + 1 nam daje 2, i 1 + 4 daje 5. 4 + 9 je 13, 9 + 25 je 34, i da, šablon se nastavlja.
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.
Zapravo, evo još jednog. Recimo da želite da pogledate sabiranje kvadratnih vrednosti prvih nekoliko Fibonačijevih brojeva. Da vidimo šta tu dobijamo. 1 + 1 + 4 je 6. Tome dodajte 9, to je 15. Dodajte 25, to je 40. Dodajte 64 i to je 104. Sada pogledajte te brojke. To nisu Fibonačijevi brojevi, ali ako ih pogledate pažljivo, videćete Fibonačijeve brojeve sakrivene unutar njih.
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?
Vidite li ih? Pokazaću vam. 6 je 2 puta 3, 15 je 3 puta 5, 40 je 5 puta 8, 2, 3, 5, 8. Kome odajemo priznanje?
(Laughter)
(Smeh)
Fibonacci! Of course.
Fibonačiju! Naravno.
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?
Koliko god da je zabavno otkrivati ove šablone, još je veće zadovoljstvo razumeti zašto su tačni. Hajde da pogledamo poslednju jednačinu. Zašto bi kvadratne vrednosti brojeva 1, 1, 2, 3, 5 i 8 sabrane, dale 8 puta 13? Pokazaću vam uz pomoć jednostavnog crteža. Počećemo sa kvadratom dimenzija 1x1 i pored ćemo dodati još jedan kvadrat dimenzija 1x1. Zajedno daju pravougaonik dimenzija 1x2. Ispod toga, dodaću kvadrat dimenzija 2x2, a pored toga, kvadrat dimenzija 3x3, ispod toga, kvadrat dimenzija 5x5 i onda kvadrat dimenzija 8x8, stvarajući jedan ogromni pravougaonik, zar ne?
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.
Dozvolite da vam postavim jednostavno pitanje: koja je površina pravouganika? Sa jedne strane, to je zbir površina kvadrata unutar njega, zar ne? Baš kao što smo ga napravili. To je 1 na kvadrat + 1 na kvadrat + 2 na kvadrat + 3 na kvadrat + 5 na kvadrat + 8 na kvadrat. Zar ne? To je površina. Sa druge strane, zbog toga što je to pravougonik, površinu dobijemo kada pomnožimo visinu i osnovu, a visina je očigledno 8 dok je osnova 5 + 8, što je sledeći Fibonačijev broj, 13. Zar ne? Površina je takođe 8 puta 13. Pošto smo tačno izračunali površinu na dva različita načina, to mora da bude isti broj i zbog toga kvadradne vrednosti brojeva 1, 1, 2, 3, 5 i 8 sabrane daju 8 puta 13.
Now, if we continue this process, we'll generate rectangles of the form 13 by 21, 21 by 34, and so on.
Ako nastavimo ovaj proces dobićemo pravouganike formata 13x21, 21x34 i tako dalje.
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.
Pogledajte sada ovo. Ako podelite 13 sa 8 dobijate 1,625. A ako veći broj podelite manjim brojem, ove srazmere se sve više približavaju vrednosti oko 1,618, što je mnogima poznato kao Zlatni presek, broj koji vekovima fascinira matematičare, naučnike i umetnike.
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.
Ovo sve vam pokazujem zato što, baš kao u dobrom delu matematike, postoji prelepa strana toga za koju se bojim da ne dobija dovoljno pažnje u našim školama. Puno vremena provodimo učeći o računanju, ali ne zaboravimo na primenu, uključujući možda i najbitniju primenu od svih, učenje kako se misli.
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.
Kada bih ovo mogao da sažmem u jednu rečenicu, to bi bilo sledeće: matematika ne znači samo pronaći vrednost x, već takođe i otkriti zašto.
Thank you very much.
Hvala vam mnogo.
(Applause)
(Aplauz)