TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Arthur Benjamin: Čarobnost Fibonaccijevih števil
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Matematika je logična, uporabna in preprosto ... krasna. Matemag Arthur Benjamin raziskuje skrite lastnosti čudnega in čudovitega niza števil, ki se imenuje Fibonaccijevo zaporedje. (In nas opomni, da je tudi matematika lahko vir navdiha!)
Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Double-click the English transcript below to play the video.
00:12
So why do we learn mathematics?
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Torej, zakaj se učimo matematike?
00:15
Essentially, for three reasons:
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V glavnem imamo tri razloge:
00:18
calculation,
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računanje
00:19
application,
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uporaba
00:21
and last, and unfortunately least
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in na koncu še razlog,
ki je žal daleč zadaj,
ki je žal daleč zadaj,
00:24
in terms of the time we give it,
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kar se tiče časa, ki mu ga namenimo,
00:26
inspiration.
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navdih.
00:28
Mathematics is the science of patterns,
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Matematika je znanost vzorcev
00:30
and we study it to learn how to think logically,
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in učimo se je, da se naučimo
razmišljati logično,
razmišljati logično,
00:34
critically and creatively,
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kritično in ustvarjalno.
00:36
but too much of the mathematics
that we learn in school
that we learn in school
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Ampak prevečkrat za matematiko,
ki jo učijo v šoli,
ki jo učijo v šoli,
00:39
is not effectively motivated,
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ni učinkovite motivacije
00:41
and when our students ask,
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in ko nas učenci vprašajo:
00:43
"Why are we learning this?"
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"Zakaj se to učimo?"
00:44
then they often hear that they'll need it
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pogosto slišijo,
da bodo znanje potrebovali
da bodo znanje potrebovali
00:46
in an upcoming math class or on a future test.
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pri pouku matematike
ali pri naslednjem testu.
ali pri naslednjem testu.
00:50
But wouldn't it be great
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Ampak, ali ne bi bilo krasno,
00:51
if every once in a while we did mathematics
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če bi kdaj pa kdaj uporabljali matematiko
00:54
simply because it was fun or beautiful
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preprosto zato, ker je zabavna ali lepa
00:57
or because it excited the mind?
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ali pa ker spodbuja razmišljanje?
00:59
Now, I know many people have not
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Veliko ljudi ni imelo priložnosti,
01:01
had the opportunity to see how this can happen,
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da bi videli, kako se to lahko zgodi,
01:03
so let me give you a quick example
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zato vam bom na hitro pokazal primer
01:05
with my favorite collection of numbers,
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s svojo najljubšo zbirko številk,
01:07
the Fibonacci numbers. (Applause)
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Fibonaccijevimi števili. (Aplavz)
01:10
Yeah! I already have Fibonacci fans here.
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To! Tu je nekaj
Fibonaccijevih oboževalcev.
Fibonaccijevih oboževalcev.
01:12
That's great.
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Odlično.
01:13
Now these numbers can be appreciated
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Torej, ta števila so krasna
01:15
in many different ways.
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na veliko različnih načinov.
01:17
From the standpoint of calculation,
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Z vidika računanja
01:20
they're as easy to understand
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so tako lahko razumljiva
01:22
as one plus one, which is two.
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kot ena plus ena, kar je dva.
01:24
Then one plus two is three,
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Potem imamo ena plus dva je tri,
01:26
two plus three is five, three plus five is eight,
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dva plus tri je pet, tri plus pet je osem
01:29
and so on.
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in tako naprej.
01:31
Indeed, the person we call Fibonacci
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V resnici se je oseba,
ki ji pravimo Fibonacci,
ki ji pravimo Fibonacci,
01:33
was actually named Leonardo of Pisa,
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imenovala Leonardo Pisano
01:36
and these numbers appear in his book "Liber Abaci,"
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in ta števila so zapisana
v njegovi knjigi "Liber Abaci",
v njegovi knjigi "Liber Abaci",
01:39
which taught the Western world
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ki je zahodni svet naučila
01:41
the methods of arithmetic that we use today.
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aritmetičnih metod,
ki jih uporabljamo danes.
ki jih uporabljamo danes.
01:44
In terms of applications,
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Kar se tiče uporabe,
01:45
Fibonacci numbers appear in nature
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se Fibonaccijeva števila
v naravi pojavljajo
v naravi pojavljajo
01:48
surprisingly often.
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presenetljivo pogosto.
01:49
The number of petals on a flower
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Število cvetnih listov na roži
01:51
is typically a Fibonacci number,
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je ponavadi Fibonaccijevo število,
01:53
or the number of spirals on a sunflower
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pa tudi število spiral na sončnici
01:56
or a pineapple
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ali ananasu
01:57
tends to be a Fibonacci number as well.
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je pogosto Fibonaccijevo število.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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Pravzaprav je možnosti uporabe
Fibonaccijevih števil veliko več,
Fibonaccijevih števil veliko več,
02:03
but what I find most inspirational about them
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a sam mislim, da so pri njih
najbolj navdušujoči
najbolj navdušujoči
02:06
are the beautiful number patterns they display.
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lepi številski vzorci, ki jih ustvarjajo.
02:08
Let me show you one of my favorites.
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Pokazal vam bom enega
od svojih najljubših.
od svojih najljubših.
02:11
Suppose you like to square numbers,
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Recimo, da radi kvadrirate števila,
02:13
and frankly, who doesn't? (Laughter)
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konec koncev, kdo jih pa ne? (Smeh)
02:16
Let's look at the squares
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Poglejmo kvadrate
02:18
of the first few Fibonacci numbers.
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prvih nekaj Fibonaccijevih števil.
02:20
So one squared is one,
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Torej, ena na kvadrat je ena,
02:22
two squared is four, three squared is nine,
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dva na kvadrat je štiri,
tri na kvadrat je devet,
tri na kvadrat je devet,
02:24
five squared is 25, and so on.
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pet na kvadrat je 25 in tako naprej.
02:27
Now, it's no surprise
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No, ni prav presenetljivo,
02:29
that when you add consecutive Fibonacci numbers,
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da, ko seštejemo
zaporedna Fibonaccijeva števila,
zaporedna Fibonaccijeva števila,
02:32
you get the next Fibonacci number. Right?
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dobimo naslednje
Fibonaccijevo število. Drži?
Fibonaccijevo število. Drži?
02:34
That's how they're created.
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Tako nastanejo.
02:35
But you wouldn't expect anything special
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Ne bi pa pričakovali, da se zgodi
02:37
to happen when you add the squares together.
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kaj posebnega,
ko seštejemo njihove kvadrate.
ko seštejemo njihove kvadrate.
02:40
But check this out.
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Pa poglejte zdaj tole.
02:42
One plus one gives us two,
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Ena plus ena je dva
02:44
and one plus four gives us five.
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in ena plus štiri je pet.
02:46
And four plus nine is 13,
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In štiri plus devet je 13,
02:48
nine plus 25 is 34,
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devet plus 25 je 34,
02:52
and yes, the pattern continues.
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in ja, vzorec se nadaljuje.
02:54
In fact, here's another one.
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V bistvu imamo še en vzorec.
02:56
Suppose you wanted to look at
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Recimo, da bi hoteli pogledati
seštevek kvadratov
prvih nekaj Fibonaccijevih števil.
prvih nekaj Fibonaccijevih števil.
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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03:00
Let's see what we get there.
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Pa poglejmo, kaj dobimo.
03:02
So one plus one plus four is six.
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Torej, ena plus ena plus štiri je šest.
03:04
Add nine to that, we get 15.
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Dodajmo še devet in dobimo 15.
03:07
Add 25, we get 40.
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Dodamo 25 in dobimo 40.
03:09
Add 64, we get 104.
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Dodamo 64, dobimo 104.
03:12
Now look at those numbers.
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Zdaj pa poglejmo ta števila.
03:14
Those are not Fibonacci numbers,
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To niso Fibonaccijeva števila,
03:16
but if you look at them closely,
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ampak, če jih pogledate od blizu,
03:18
you'll see the Fibonacci numbers
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boste videli, da se Fibonaccijeva števila
03:20
buried inside of them.
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skrivajo v njih.
03:22
Do you see it? I'll show it to you.
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Jih vidite? Vam bom pokazal.
03:24
Six is two times three, 15 is three times five,
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Šest je dva krat tri, 15 je tri krat pet,
03:28
40 is five times eight,
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40 je pet krat osem,
03:30
two, three, five, eight, who do we appreciate?
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dva, tri, pet, osem, koga občudujemo?
03:33
(Laughter)
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(Smeh)
03:34
Fibonacci! Of course.
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Fibonaccija! Jasno.
03:36
Now, as much fun as it is to discover these patterns,
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Zelo zabavno je odkrivati vzorce,
03:40
it's even more satisfying to understand
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a v še večje zadovoljstvo je razumeti
03:42
why they are true.
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zakaj držijo.
03:44
Let's look at that last equation.
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Poglejmo zadnjo enačbo.
03:46
Why should the squares of one, one,
two, three, five and eight
two, three, five and eight
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Zakaj mora seštevek kvadratov od
ena, ena, dva, tri, pet in osem
ena, ena, dva, tri, pet in osem
03:50
add up to eight times 13?
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znašati osem krat 13?
03:53
I'll show you by drawing a simple picture.
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To vam bom pokazal s preprosto sliko.
03:56
We'll start with a one-by-one square
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Začeli bomo s kvadratom ena krat ena
03:58
and next to that put another one-by-one square.
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in zraven njega narisali
še en kvadrat ena krat ena.
še en kvadrat ena krat ena.
04:02
Together, they form a one-by-two rectangle.
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Skupaj sestavljata
pravokotnik ena krat dva.
pravokotnik ena krat dva.
04:06
Beneath that, I'll put a two-by-two square,
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Pod njega bom narisal
kvadrat dva krat dva,
kvadrat dva krat dva,
04:08
and next to that, a three-by-three square,
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zraven njega pa kvadrat tri krat tri,
04:11
beneath that, a five-by-five square,
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pod njega kvadrat pet krat pet
04:13
and then an eight-by-eight square,
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in nato kvadrat osem krat osem,
04:15
creating one giant rectangle, right?
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in tako sem sestavil ogromen pravokotnik.
04:18
Now let me ask you a simple question:
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Zdaj vam bom postavil preprosto vprašanje:
04:20
what is the area of the rectangle?
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Kolikšna je ploščina pravokotnika?
04:23
Well, on the one hand,
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No, po svoje
04:25
it's the sum of the areas
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je vsota ploščin
04:28
of the squares inside it, right?
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vseh kvadratov v njem, drži?
04:30
Just as we created it.
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Kot smo ga naredili.
04:31
It's one squared plus one squared
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Ena na kvadrat plus ena na kvadrat
04:33
plus two squared plus three squared
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plus dva na kvadrat plus tri na kvadrat
04:35
plus five squared plus eight squared. Right?
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plus pet na kvadrat plus osem na kvadrat.
04:38
That's the area.
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To je ploščina.
04:40
On the other hand, because it's a rectangle,
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Po drugi strani pa, ker je pravokotnik,
04:42
the area is equal to its height times its base,
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je ploščina enaka višini krat širini
04:46
and the height is clearly eight,
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in višina je očitno osem,
04:48
and the base is five plus eight,
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širina pa pet plus osem,
04:51
which is the next Fibonacci number, 13. Right?
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kar je naslednje
Fibonaccijevo število, 13. Je tako?
Fibonaccijevo število, 13. Je tako?
04:55
So the area is also eight times 13.
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Tako imamo ploščino osem krat 13.
04:58
Since we've correctly calculated the area
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Ker smo pravilno izračunali ploščino
05:00
two different ways,
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na dva različna načina,
05:02
they have to be the same number,
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moramo dobiti enako številko
05:04
and that's why the squares of one,
one, two, three, five and eight
one, two, three, five and eight
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in zato je seštevek kvadratov od
ena, ena, dva, tri, pet in osem
ena, ena, dva, tri, pet in osem
05:08
add up to eight times 13.
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skupaj osem krat 13.
05:10
Now, if we continue this process,
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Če nadaljujemo s tem postopkom,
05:12
we'll generate rectangles of the form 13 by 21,
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bomo ustvarili pravokotnike
s stranicami 13 krat 21,
s stranicami 13 krat 21,
05:16
21 by 34, and so on.
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21 krat 34 in tako naprej.
05:19
Now check this out.
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Zdaj pa poglejte tole.
05:20
If you divide 13 by eight,
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Če 13 delimo z osem,
05:22
you get 1.625.
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dobimo 1,625.
05:24
And if you divide the larger number
by the smaller number,
by the smaller number,
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In če delimo večje število
z manjšim številom,
z manjšim številom,
05:28
then these ratios get closer and closer
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se razmerje vedno bolj približuje
05:31
to about 1.618,
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okoli 1,618,
05:33
known to many people as the Golden Ratio,
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kar veliko ljudi pozna kot zlati rez,
05:37
a number which has fascinated mathematicians,
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število, ki je stoletja navduševalo
05:39
scientists and artists for centuries.
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matematike, znanstvenike in umetnike.
05:42
Now, I show all this to you because,
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To vam kažem, ker,
05:45
like so much of mathematics,
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kot toliko matematike,
05:47
there's a beautiful side to it
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v sebi skriva nekaj lepega,
05:49
that I fear does not get enough attention
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čemur mislim, da v naših šolah žal
05:51
in our schools.
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ne posvečamo dovolj pozornosti.
05:52
We spend lots of time learning about calculation,
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Veliko časa se učimo o računanju,
05:55
but let's not forget about application,
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ampak ne smemo pozabiti na uporabo,
05:58
including, perhaps, the most
important application of all,
important application of all,
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vključno z morda najpomembnejšo uporabo,
06:01
learning how to think.
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da se naučimo, kako razmišljati.
06:03
If I could summarize this in one sentence,
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Če bi lahko to zajel v enem stavku,
06:05
it would be this:
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bi rekel tole:
06:07
Mathematics is not just solving for x,
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Matematika ni samo iskanje x-a,
06:10
it's also figuring out why.
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ampak tudi smisla.
06:13
Thank you very much.
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Najlepša hvala.
06:15
(Applause)
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(Aplavz)
ABOUT THE SPEAKER
Arthur Benjamin - MathemagicianUsing daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty.
Why you should listen
Arthur Benjamin makes numbers dance. In his day job, he's a professor of math at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage in his tuxedo to perform high-speed mental calculations, memorizations and other astounding math stunts. It's part of his drive to teach math and mental agility in interesting ways, following in the footsteps of such heroes as Martin Gardner.
Benjamin is the co-author, with Michael Shermer, of Secrets of Mental Math (which shares his secrets for rapid mental calculation), as well as the co-author of the MAA award-winning Proofs That Really Count: The Art of Combinatorial Proof. For a glimpse of his broad approach to math, see the list of research talks on his website, which seesaws between high-level math (such as his "Vandermonde's Determinant and Fibonacci SAWs," presented at MIT in 2004) and engaging math talks for the rest of us ("An Amazing Mathematical Card Trick").
Arthur Benjamin | Speaker | TED.com