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.
Torej, zakaj se učimo matematike? V glavnem imamo tri razloge: računanje uporaba in na koncu še razlog, ki je žal daleč zadaj, kar se tiče časa, ki mu ga namenimo, navdih.
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 znanost vzorcev in učimo se je, da se naučimo razmišljati logično, kritično in ustvarjalno. Ampak prevečkrat za matematiko, ki jo učijo v šoli, ni učinkovite motivacije in ko nas učenci vprašajo: "Zakaj se to učimo?" pogosto slišijo, da bodo znanje potrebovali pri pouku matematike ali pri naslednjem testu. Ampak, ali ne bi bilo krasno, če bi kdaj pa kdaj uporabljali matematiko preprosto zato, ker je zabavna ali lepa ali pa ker spodbuja razmišljanje? Veliko ljudi ni imelo priložnosti, da bi videli, kako se to lahko zgodi, zato vam bom na hitro pokazal primer s svojo najljubšo zbirko številk, Fibonaccijevimi števili. (Aplavz)
Yeah! I already have Fibonacci fans here. That's great.
To! Tu je nekaj Fibonaccijevih oboževalcev. Odlično.
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.
Torej, ta števila so krasna na veliko različnih načinov. Z vidika računanja so tako lahko razumljiva kot ena plus ena, kar je dva. Potem imamo ena plus dva je tri, dva plus tri je pet, tri plus pet je osem in tako naprej. V resnici se je oseba, ki ji pravimo Fibonacci, imenovala Leonardo Pisano in ta števila so zapisana v njegovi knjigi "Liber Abaci", ki je zahodni svet naučila aritmetičnih metod, ki jih uporabljamo danes. Kar se tiče uporabe, se Fibonaccijeva števila v naravi pojavljajo presenetljivo pogosto. Število cvetnih listov na roži je ponavadi Fibonaccijevo število, pa tudi število spiral na sončnici ali ananasu je pogosto Fibonaccijevo število.
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)
Pravzaprav je možnosti uporabe Fibonaccijevih števil veliko več, a sam mislim, da so pri njih najbolj navdušujoči lepi številski vzorci, ki jih ustvarjajo. Pokazal vam bom enega od svojih najljubših. Recimo, da radi kvadrirate števila, konec koncev, kdo jih pa ne? (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.
Poglejmo kvadrate prvih nekaj Fibonaccijevih števil. Torej, ena na kvadrat je ena, dva na kvadrat je štiri, tri na kvadrat je devet, pet na kvadrat je 25 in tako naprej. No, ni prav presenetljivo, da, ko seštejemo zaporedna Fibonaccijeva števila, dobimo naslednje Fibonaccijevo število. Drži? Tako nastanejo. Ne bi pa pričakovali, da se zgodi kaj posebnega, ko seštejemo njihove kvadrate. Pa poglejte zdaj tole. Ena plus ena je dva in ena plus štiri je pet. In štiri plus devet je 13, devet plus 25 je 34, in ja, vzorec se nadaljuje.
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.
V bistvu imamo še en vzorec. Recimo, da bi hoteli pogledati seštevek kvadratov prvih nekaj Fibonaccijevih števil. Pa poglejmo, kaj dobimo. Torej, ena plus ena plus štiri je šest. Dodajmo še devet in dobimo 15. Dodamo 25 in dobimo 40. Dodamo 64, dobimo 104. Zdaj pa poglejmo ta števila. To niso Fibonaccijeva števila, ampak, če jih pogledate od blizu, boste videli, da se Fibonaccijeva števila skrivajo v 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?
Jih vidite? Vam bom pokazal. Šest je dva krat tri, 15 je tri krat pet, 40 je pet krat osem, dva, tri, pet, osem, koga občudujemo?
(Laughter)
(Smeh)
Fibonacci! Of course.
Fibonaccija! Jasno.
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?
Zelo zabavno je odkrivati vzorce, a v še večje zadovoljstvo je razumeti zakaj držijo. Poglejmo zadnjo enačbo. Zakaj mora seštevek kvadratov od ena, ena, dva, tri, pet in osem znašati osem krat 13? To vam bom pokazal s preprosto sliko. Začeli bomo s kvadratom ena krat ena in zraven njega narisali še en kvadrat ena krat ena. Skupaj sestavljata pravokotnik ena krat dva. Pod njega bom narisal kvadrat dva krat dva, zraven njega pa kvadrat tri krat tri, pod njega kvadrat pet krat pet in nato kvadrat osem krat osem, in tako sem sestavil ogromen pravokotnik.
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.
Zdaj vam bom postavil preprosto vprašanje: Kolikšna je ploščina pravokotnika? No, po svoje je vsota ploščin vseh kvadratov v njem, drži? Kot smo ga naredili. Ena na kvadrat plus ena na kvadrat plus dva na kvadrat plus tri na kvadrat plus pet na kvadrat plus osem na kvadrat. To je ploščina. Po drugi strani pa, ker je pravokotnik, je ploščina enaka višini krat širini in višina je očitno osem, širina pa pet plus osem, kar je naslednje Fibonaccijevo število, 13. Je tako? Tako imamo ploščino osem krat 13. Ker smo pravilno izračunali ploščino na dva različna načina, moramo dobiti enako številko in zato je seštevek kvadratov od ena, ena, dva, tri, pet in osem skupaj osem krat 13.
Now, if we continue this process, we'll generate rectangles of the form 13 by 21, 21 by 34, and so on.
Če nadaljujemo s tem postopkom, bomo ustvarili pravokotnike s stranicami 13 krat 21, 21 krat 34 in tako naprej.
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.
Zdaj pa poglejte tole. Če 13 delimo z osem, dobimo 1,625. In če delimo večje število z manjšim številom, se razmerje vedno bolj približuje okoli 1,618, kar veliko ljudi pozna kot zlati rez, število, ki je stoletja navduševalo matematike, znanstvenike in 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.
To vam kažem, ker, kot toliko matematike, v sebi skriva nekaj lepega, čemur mislim, da v naših šolah žal ne posvečamo dovolj pozornosti. Veliko časa se učimo o računanju, ampak ne smemo pozabiti na uporabo, vključno z morda najpomembnejšo uporabo, da se naučimo, kako razmišljati.
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.
Če bi lahko to zajel v enem stavku, bi rekel tole: Matematika ni samo iskanje x-a, ampak tudi smisla.
Thank you very much.
Najlepša hvala.
(Applause)
(Aplavz)