The magic of Fibonacci numbers | Arthur Benjamin | TED

Prevodilac: Mile Živković Lektor: Miloš Milosavljević

So why do we learn mathematics?

Zašto učimo matematiku?

Essentially, for three reasons:

U suštini, iz tri razloga:

calculation,

računanje,

application,

primena

and last, and unfortunately least

i poslednje i nažalost najmanje bitno,

in terms of the time we give it,

što se tiče vremena koje mu posvećujemo,

inspiration.

nadahnuće.

Mathematics is the science of patterns,

Matematika je nauka o šablonima

and we study it to learn how to think logically,

i proučavamo je kako bismo saznali kako da mislimo logički,

critically and creatively,

kritički i kreativno,

but too much of the mathematics that we learn in school

ali previše matematike koju učimo u školama

is not effectively motivated,

nije valjano motivisano

and when our students ask,

i kada naši učenici pitaju:

"Why are we learning this?"

"Zašto učimo ovo?",

then they often hear that they'll need it

često čuju da će im to biti potrebno

in an upcoming math class or on a future test.

na nekom budućem testu ili času matematike.

But wouldn't it be great

Ali zar ne bi bilo sjajno

if every once in a while we did mathematics

kad bismo se, s vremena na vreme, bavili matematikom

simply because it was fun or beautiful

jednostavno jer je zabavna ili predivna

or because it excited the mind?

ili zato što je uzbudljiva za um?

Now, I know many people have not

Znam da dosta ljudi nije imalo

had the opportunity to see how this can happen,

priliku da vidi kako ovo može da se desi,

so let me give you a quick example

zato hajde da vam dam brz primer

with my favorite collection of numbers,

sa mojim omiljenim skupom brojeva,

the Fibonacci numbers. (Applause)

Fibonačijevim brojevima. (Aplauz)

Yeah! I already have Fibonacci fans here.

Da! Ovde već imam Fibonačijeve fanove.

That's great.

To je sjajno.

Now these numbers can be appreciated

Ove brojeve možete razumeti

in many different ways.

na mnogo različitih načina.

From the standpoint of calculation,

Sa stanovišta računanja,

they're as easy to understand

jednako ih je lako razumeti

as one plus one, which is two.

kao 1 + 1, što je 2.

Then one plus two is three,

1 + 2 je onda 3,

two plus three is five, three plus five is eight,

2 + 3 je 5, 3 + 5 je 8,

and so on.

i tako dalje.

Indeed, the person we call Fibonacci

Zaista, osoba koju nazivamo Fibonači

was actually named Leonardo of Pisa,

se zapravo zvala Leonardo od Pize

and these numbers appear in his book "Liber Abaci,"

i ovi brojevi se pojavljuju u njegovoj knjizi "Liber Abaci",

which taught the Western world

koja je Zapadni svet naučila

the methods of arithmetic that we use today.

aritmetičkim metodama koje danas koristimo.

In terms of applications,

Što se tiče primene,

Fibonacci numbers appear in nature

Fibonačijevi brojevi se u prirodi

surprisingly often.

pojavljuju iznenađujuće često.

The number of petals on a flower

Broj latica na cvetu

is typically a Fibonacci number,

je tipično Fibonačijev broj,

or the number of spirals on a sunflower

ili broj spirala na suncokretu

or a pineapple

ili ananasu,

tends to be a Fibonacci number as well.

to su takođe uglavnom Fibonačijevi brojevi.

In fact, there are many more applications of Fibonacci numbers,

Zapravo postoji još dosta primena Fibonačijevih brojeva,

but what I find most inspirational about them

ali ono što je za mene najinspirativnije u vezi sa njima

are the beautiful number patterns they display.

su predivni šabloni brojeva koje oni prikazuju.

Let me show you one of my favorites.

Dozvolite da vam pokažem jedan od meni omiljenih.

Suppose you like to square numbers,

Recimo da volite da kvadrirate brojeve,

and frankly, who doesn't? (Laughter)

a iskreno, ko to ne voli? (Smeh)

Let's look at the squares

Hajde da pogledamo kvadrate

of the first few Fibonacci numbers.

prvih nekoliko Fibonačijevih brojeva.

So one squared is one,

1 na kvadrat je 1,

two squared is four, three squared is nine,

2 na kvadrat je 4. 3 na kvadrat je 9,

five squared is 25, and so on.

5 na kvadrat je 25, i tako dalje.

Now, it's no surprise

Ne iznenađuje činjenica

that when you add consecutive Fibonacci numbers,

da kada dodate uzastopne Fibonačijeve brojeve,

you get the next Fibonacci number. Right?

dobijate sledeći Fibonačijev broj. Zar ne?

That's how they're created.

Tako oni nastaju.

But you wouldn't expect anything special

Ali ne biste očekivali da se desi ništa posebno

to happen when you add the squares together.

kada saberete kvadratne vrednosti.

But check this out.

Ali pogledajte ovo.

One plus one gives us two,

1 + 1 nam daje 2,

and one plus four gives us five.

i 1 + 4 daje 5.

And four plus nine is 13,

4 + 9 je 13,

nine plus 25 is 34,

9 + 25 je 34,

and yes, the pattern continues.

i da, šablon se nastavlja.

In fact, here's another one.

Zapravo, evo još jednog.

Suppose you wanted to look at

Recimo da želite da pogledate

adding the squares of the first few Fibonacci numbers.

sabiranje kvadratnih vrednosti prvih nekoliko Fibonačijevih brojeva.

Let's see what we get there.

Da vidimo šta tu dobijamo.

So one plus one plus four is six.

1 + 1 + 4 je 6.

Add nine to that, we get 15.

Tome dodajte 9, to je 15.

Add 25, we get 40.

Dodajte 25, to je 40.

Add 64, we get 104.

Dodajte 64 i to je 104.

Now look at those numbers.

Sada pogledajte te brojke.

Those are not Fibonacci numbers,

To nisu Fibonačijevi brojevi,

but if you look at them closely,

ali ako ih pogledate pažljivo,

you'll see the Fibonacci numbers

videćete Fibonačijeve brojeve

buried inside of them.

sakrivene unutar njih.

Do you see it? I'll show it to you.

Vidite li ih? Pokazaću vam.

Six is two times three, 15 is three times five,

6 je 2 puta 3, 15 je 3 puta 5,

40 is five times eight,

40 je 5 puta 8,

two, three, five, eight, who do we appreciate?

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,

Koliko god da je zabavno otkrivati ove šablone,

it's even more satisfying to understand

još je veće zadovoljstvo razumeti

why they are true.

zašto su tačni.

Let's look at that last equation.

Hajde da pogledamo poslednju jednačinu.

Why should the squares of one, one, two, three, five and eight

Zašto bi kvadratne vrednosti brojeva 1, 1, 2, 3, 5 i 8

add up to eight times 13?

sabrane, dale 8 puta 13?

I'll show you by drawing a simple picture.

Pokazaću vam uz pomoć jednostavnog crteža.

We'll start with a one-by-one square

Počećemo sa kvadratom dimenzija 1x1

and next to that put another one-by-one square.

i pored ćemo dodati još jedan kvadrat dimenzija 1x1.

Together, they form a one-by-two rectangle.

Zajedno daju pravougaonik dimenzija 1x2.

Beneath that, I'll put a two-by-two square,

Ispod toga, dodaću kvadrat dimenzija 2x2,

and next to that, a three-by-three square,

a pored toga, kvadrat dimenzija 3x3,

beneath that, a five-by-five square,

ispod toga, kvadrat dimenzija 5x5

and then an eight-by-eight square,

i onda kvadrat dimenzija 8x8,

creating one giant rectangle, right?

stvarajući jedan ogromni pravougaonik, zar ne?

Now let me ask you a simple question:

Dozvolite da vam postavim jednostavno pitanje:

what is the area of the rectangle?

koja je površina pravouganika?

Well, on the one hand,

Sa jedne strane,

it's the sum of the areas

to je zbir površina

of the squares inside it, right?

kvadrata unutar njega, zar ne?

Just as we created it.

Baš kao što smo ga napravili.

It's one squared plus one squared

To je 1 na kvadrat + 1 na kvadrat

plus two squared plus three squared

+ 2 na kvadrat + 3 na kvadrat

plus five squared plus eight squared. Right?

+ 5 na kvadrat + 8 na kvadrat. Zar ne?

That's the area.

To je površina.

On the other hand, because it's a rectangle,

Sa druge strane, zbog toga što je to pravougonik,

the area is equal to its height times its base,

površinu dobijemo kada pomnožimo visinu i osnovu,

and the height is clearly eight,

a visina je očigledno 8

and the base is five plus eight,

dok je osnova 5 + 8,

which is the next Fibonacci number, 13. Right?

što je sledeći Fibonačijev broj, 13. Zar ne?

So the area is also eight times 13.

Površina je takođe 8 puta 13.

Since we've correctly calculated the area

Pošto smo tačno izračunali površinu

two different ways,

na dva različita načina,

they have to be the same number,

to mora da bude isti broj

and that's why the squares of one, one, two, three, five and eight

i zbog toga kvadradne vrednosti brojeva 1, 1, 2, 3, 5 i 8

add up to eight times 13.

sabrane daju 8 puta 13.

Now, if we continue this process,

Ako nastavimo ovaj proces

we'll generate rectangles of the form 13 by 21,

dobićemo pravouganike formata 13x21,

21 by 34, and so on.

21x34 i tako dalje.

Now check this out.

Pogledajte sada ovo.

If you divide 13 by eight,

Ako podelite 13 sa 8

you get 1.625.

dobijate 1,625.

And if you divide the larger number by the smaller number,

A ako veći broj podelite manjim brojem,

then these ratios get closer and closer

ove srazmere se sve više približavaju

to about 1.618,

vrednosti oko 1,618,

known to many people as the Golden Ratio,

što je mnogima poznato kao Zlatni presek,

a number which has fascinated mathematicians,

broj koji vekovima fascinira

scientists and artists for centuries.

matematičare, naučnike i umetnike.

Now, I show all this to you because,

Ovo sve vam pokazujem zato što,

like so much of mathematics,

baš kao u dobrom delu matematike,

there's a beautiful side to it

postoji prelepa strana toga

that I fear does not get enough attention

za koju se bojim da ne dobija dovoljno pažnje

in our schools.

u našim školama.

We spend lots of time learning about calculation,

Puno vremena provodimo učeći o računanju,

but let's not forget about application,

ali ne zaboravimo na primenu,

including, perhaps, the most important application of all,

uključujući možda i najbitniju primenu od svih,

learning how to think.

učenje kako se misli.

If I could summarize this in one sentence,

Kada bih ovo mogao da sažmem u jednu rečenicu,

it would be this:

to bi bilo sledeće:

Mathematics is not just solving for x,

matematika ne znači samo pronaći vrednost x,

it's also figuring out why.

već takođe i otkriti zašto.

Thank you very much.

Hvala vam mnogo.

(Applause)

(Aplauz)