The magic of Fibonacci numbers | Arthur Benjamin | TED

Translator: Petra Zajc Reviewer: Nika Kotnik

So why do we learn mathematics?

Torej, zakaj se učimo matematike?

Essentially, for three reasons:

V glavnem imamo tri razloge:

calculation,

računanje

application,

uporaba

and last, and unfortunately least

in na koncu še razlog, ki je žal daleč zadaj,

in terms of the time we give it,

kar se tiče časa, ki mu ga namenimo,

inspiration.

navdih.

Mathematics is the science of patterns,

Matematika je znanost vzorcev

and we study it to learn how to think logically,

in učimo se je, da se naučimo razmišljati logično,

critically and creatively,

kritično in ustvarjalno.

but too much of the mathematics that we learn in school

Ampak prevečkrat za matematiko, ki jo učijo v šoli,

is not effectively motivated,

ni učinkovite motivacije

and when our students ask,

in ko nas učenci vprašajo:

"Why are we learning this?"

"Zakaj se to učimo?"

then they often hear that they'll need it

pogosto slišijo, da bodo znanje potrebovali

in an upcoming math class or on a future test.

pri pouku matematike ali pri naslednjem testu.

But wouldn't it be great

Ampak, ali ne bi bilo krasno,

if every once in a while we did mathematics

če bi kdaj pa kdaj uporabljali matematiko

simply because it was fun or beautiful

preprosto zato, ker je zabavna ali lepa

or because it excited the mind?

ali pa ker spodbuja razmišljanje?

Now, I know many people have not

Veliko ljudi ni imelo priložnosti,

had the opportunity to see how this can happen,

da bi videli, kako se to lahko zgodi,

so let me give you a quick example

zato vam bom na hitro pokazal primer

with my favorite collection of numbers,

s svojo najljubšo zbirko številk,

the Fibonacci numbers. (Applause)

Fibonaccijevimi števili. (Aplavz)

Yeah! I already have Fibonacci fans here.

To! Tu je nekaj Fibonaccijevih oboževalcev.

That's great.

Odlično.

Now these numbers can be appreciated

Torej, ta števila so krasna

in many different ways.

na veliko različnih načinov.

From the standpoint of calculation,

Z vidika računanja

they're as easy to understand

so tako lahko razumljiva

as one plus one, which is two.

kot ena plus ena, kar je dva.

Then one plus two is three,

Potem imamo ena plus dva je tri,

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

dva plus tri je pet, tri plus pet je osem

and so on.

in tako naprej.

Indeed, the person we call Fibonacci

V resnici se je oseba, ki ji pravimo Fibonacci,

was actually named Leonardo of Pisa,

imenovala Leonardo Pisano

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

in ta števila so zapisana v njegovi knjigi "Liber Abaci",

which taught the Western world

ki je zahodni svet naučila

the methods of arithmetic that we use today.

aritmetičnih metod, ki jih uporabljamo danes.

In terms of applications,

Kar se tiče uporabe,

Fibonacci numbers appear in nature

se Fibonaccijeva števila v naravi pojavljajo

surprisingly often.

presenetljivo pogosto.

The number of petals on a flower

Število cvetnih listov na roži

is typically a Fibonacci number,

je ponavadi Fibonaccijevo število,

or the number of spirals on a sunflower

pa tudi število spiral na sončnici

or a pineapple

ali ananasu

tends to be a Fibonacci number as well.

je pogosto Fibonaccijevo število.

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

Pravzaprav je možnosti uporabe Fibonaccijevih števil veliko več,

but what I find most inspirational about them

a sam mislim, da so pri njih najbolj navdušujoči

are the beautiful number patterns they display.

lepi številski vzorci, ki jih ustvarjajo.

Let me show you one of my favorites.

Pokazal vam bom enega od svojih najljubših.

Suppose you like to square numbers,

Recimo, da radi kvadrirate števila,

and frankly, who doesn't? (Laughter)

konec koncev, kdo jih pa ne? (Smeh)

Let's look at the squares

Poglejmo kvadrate

of the first few Fibonacci numbers.

prvih nekaj Fibonaccijevih števil.

So one squared is one,

Torej, ena na kvadrat je ena,

two squared is four, three squared is nine,

dva na kvadrat je štiri, tri na kvadrat je devet,

five squared is 25, and so on.

pet na kvadrat je 25 in tako naprej.

Now, it's no surprise

No, ni prav presenetljivo,

that when you add consecutive Fibonacci numbers,

da, ko seštejemo zaporedna Fibonaccijeva števila,

you get the next Fibonacci number. Right?

dobimo naslednje Fibonaccijevo število. Drži?

That's how they're created.

Tako nastanejo.

But you wouldn't expect anything special

Ne bi pa pričakovali, da se zgodi

to happen when you add the squares together.

kaj posebnega, ko seštejemo njihove kvadrate.

But check this out.

Pa poglejte zdaj tole.

One plus one gives us two,

Ena plus ena je dva

and one plus four gives us five.

in ena plus štiri je pet.

And four plus nine is 13,

In štiri plus devet je 13,

nine plus 25 is 34,

devet plus 25 je 34,

and yes, the pattern continues.

in ja, vzorec se nadaljuje.

In fact, here's another one.

V bistvu imamo še en vzorec.

Suppose you wanted to look at

Recimo, da bi hoteli pogledati

adding the squares of the first few Fibonacci numbers.

seštevek kvadratov prvih nekaj Fibonaccijevih števil.

Let's see what we get there.

Pa poglejmo, kaj dobimo.

So one plus one plus four is six.

Torej, ena plus ena plus štiri je šest.

Add nine to that, we get 15.

Dodajmo še devet in dobimo 15.

Add 25, we get 40.

Dodamo 25 in dobimo 40.

Add 64, we get 104.

Dodamo 64, dobimo 104.

Now look at those numbers.

Zdaj pa poglejmo ta števila.

Those are not Fibonacci numbers,

To niso Fibonaccijeva števila,

but if you look at them closely,

ampak, če jih pogledate od blizu,

you'll see the Fibonacci numbers

boste videli, da se Fibonaccijeva števila

buried inside of them.

skrivajo v njih.

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

Jih vidite? Vam bom pokazal.

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

Šest je dva krat tri, 15 je tri krat pet,

40 is five times eight,

40 je pet krat osem,

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

dva, tri, pet, osem, koga občudujemo?

(Laughter)

(Smeh)

Fibonacci! Of course.

Fibonaccija! Jasno.

Now, as much fun as it is to discover these patterns,

Zelo zabavno je odkrivati vzorce,

it's even more satisfying to understand

a v še večje zadovoljstvo je razumeti

why they are true.

zakaj držijo.

Let's look at that last equation.

Poglejmo zadnjo enačbo.

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

Zakaj mora seštevek kvadratov od ena, ena, dva, tri, pet in osem

add up to eight times 13?

znašati osem krat 13?

I'll show you by drawing a simple picture.

To vam bom pokazal s preprosto sliko.

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

Začeli bomo s kvadratom ena krat ena

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

in zraven njega narisali še en kvadrat ena krat ena.

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

Skupaj sestavljata pravokotnik ena krat dva.

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

Pod njega bom narisal kvadrat dva krat dva,

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

zraven njega pa kvadrat tri krat tri,

beneath that, a five-by-five square,

pod njega kvadrat pet krat pet

and then an eight-by-eight square,

in nato kvadrat osem krat osem,

creating one giant rectangle, right?

in tako sem sestavil ogromen pravokotnik.

Now let me ask you a simple question:

Zdaj vam bom postavil preprosto vprašanje:

what is the area of the rectangle?

Kolikšna je ploščina pravokotnika?

Well, on the one hand,

No, po svoje

it's the sum of the areas

je vsota ploščin

of the squares inside it, right?

vseh kvadratov v njem, drži?

Just as we created it.

Kot smo ga naredili.

It's one squared plus one squared

Ena na kvadrat plus ena na kvadrat

plus two squared plus three squared

plus dva na kvadrat plus tri na kvadrat

plus five squared plus eight squared. Right?

plus pet na kvadrat plus osem na kvadrat.

That's the area.

To je ploščina.

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

Po drugi strani pa, ker je pravokotnik,

the area is equal to its height times its base,

je ploščina enaka višini krat širini

and the height is clearly eight,

in višina je očitno osem,

and the base is five plus eight,

širina pa pet plus osem,

which is the next Fibonacci number, 13. Right?

kar je naslednje Fibonaccijevo število, 13. Je tako?

So the area is also eight times 13.

Tako imamo ploščino osem krat 13.

Since we've correctly calculated the area

Ker smo pravilno izračunali ploščino

two different ways,

na dva različna načina,

they have to be the same number,

moramo dobiti enako številko

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

in zato je seštevek kvadratov od ena, ena, dva, tri, pet in osem

add up to eight times 13.

skupaj osem krat 13.

Now, if we continue this process,

Če nadaljujemo s tem postopkom,

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

bomo ustvarili pravokotnike s stranicami 13 krat 21,

21 by 34, and so on.

21 krat 34 in tako naprej.

Now check this out.

Zdaj pa poglejte tole.

If you divide 13 by eight,

Če 13 delimo z osem,

you get 1.625.

dobimo 1,625.

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

In če delimo večje število z manjšim številom,

then these ratios get closer and closer

se razmerje vedno bolj približuje

to about 1.618,

okoli 1,618,

known to many people as the Golden Ratio,

kar veliko ljudi pozna kot zlati rez,

a number which has fascinated mathematicians,

število, ki je stoletja navduševalo

scientists and artists for centuries.

matematike, znanstvenike in umetnike.

Now, I show all this to you because,

To vam kažem, ker,

like so much of mathematics,

kot toliko matematike,

there's a beautiful side to it

v sebi skriva nekaj lepega,

that I fear does not get enough attention

čemur mislim, da v naših šolah žal

in our schools.

ne posvečamo dovolj pozornosti.

We spend lots of time learning about calculation,

Veliko časa se učimo o računanju,

but let's not forget about application,

ampak ne smemo pozabiti na uporabo,

including, perhaps, the most important application of all,

vključno z morda najpomembnejšo uporabo,

learning how to think.

da se naučimo, kako razmišljati.

If I could summarize this in one sentence,

Če bi lahko to zajel v enem stavku,

it would be this:

bi rekel tole:

Mathematics is not just solving for x,

Matematika ni samo iskanje x-a,

it's also figuring out why.

ampak tudi smisla.

Thank you very much.

Najlepša hvala.

(Applause)

(Aplavz)