ABOUT THE SPEAKER
Arthur Benjamin - Mathemagician
Using 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").

More profile about the speaker
Arthur Benjamin | Speaker | TED.com
TEDGlobal 2013

Arthur Benjamin: The magic of Fibonacci numbers

Arthur Benjamin: Čarobnost Fibonaccijevih števil

Filmed:
7,057,274 views

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!)
- Mathemagician
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 learnučiti se mathematicsmatematika?
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Torej, zakaj se učimo matematike?
00:15
EssentiallyV bistvu, for threetri reasonsrazlogov:
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V glavnem imamo tri razloge:
00:18
calculationizračun,
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računanje
00:19
applicationaplikacija,
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uporaba
00:21
and last, and unfortunatelyna žalost leastvsaj
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in na koncu še razlog,
ki je žal daleč zadaj,
00:24
in termspogoji of the time we give it,
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kar se tiče časa, ki mu ga namenimo,
00:26
inspirationnavdih.
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navdih.
00:28
MathematicsMatematika is the scienceznanost of patternsvzorce,
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Matematika je znanost vzorcev
00:30
and we studyštudija it to learnučiti se how to think logicallylogično,
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in učimo se je, da se naučimo
razmišljati logično,
00:34
criticallykritično and creativelyustvarjalno,
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kritično in ustvarjalno.
00:36
but too much of the mathematicsmatematika
that we learnučiti se in schoolšola
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Ampak prevečkrat za matematiko,
ki jo učijo v šoli,
00:39
is not effectivelyučinkovito motivatedmotivirani,
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ni učinkovite motivacije
00:41
and when our studentsštudenti askvprašajte,
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in ko nas učenci vprašajo:
00:43
"Why are we learningučenje this?"
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"Zakaj se to učimo?"
00:44
then they oftenpogosto hearslišite that they'lloni bodo need it
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pogosto slišijo,
da bodo znanje potrebovali
00:46
in an upcomingprihajajoči mathmatematika classrazred or on a futureprihodnost testtest.
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pri pouku matematike
ali pri naslednjem testu.
00:50
But wouldn'tne bi it be great
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Ampak, ali ne bi bilo krasno,
00:51
if everyvsak onceenkrat in a while we did mathematicsmatematika
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če bi kdaj pa kdaj uporabljali matematiko
00:54
simplypreprosto because it was funzabavno or beautifullepo
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preprosto zato, ker je zabavna ali lepa
00:57
or because it excitednavdušen the mindum?
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ali pa ker spodbuja razmišljanje?
00:59
Now, I know manyveliko people have not
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Veliko ljudi ni imelo priložnosti,
01:01
had the opportunitypriložnost to see how this can happense zgodi,
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da bi videli, kako se to lahko zgodi,
01:03
so let me give you a quickhitro exampleprimer
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zato vam bom na hitro pokazal primer
01:05
with my favoritenajljubši collectionzbirka of numbersštevilke,
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s svojo najljubšo zbirko številk,
01:07
the FibonacciFibonacci numbersštevilke. (ApplauseAplavz)
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Fibonaccijevimi števili. (Aplavz)
01:10
Yeah! I alreadyže have FibonacciFibonacci fansnavijači here.
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To! Tu je nekaj
Fibonaccijevih oboževalcev.
01:12
That's great.
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Odlično.
01:13
Now these numbersštevilke can be appreciatedcenijo
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Torej, ta števila so krasna
01:15
in manyveliko differentdrugačen waysnačinov.
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na veliko različnih načinov.
01:17
From the standpointstališče of calculationizračun,
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Z vidika računanja
01:20
they're as easyenostavno to understandrazumeti
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so tako lahko razumljiva
01:22
as one plusplus one, whichki is two.
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kot ena plus ena, kar je dva.
01:24
Then one plusplus two is threetri,
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Potem imamo ena plus dva je tri,
01:26
two plusplus threetri is fivepet, threetri plusplus fivepet is eightosem,
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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
IndeedDejansko, the personoseba we call FibonacciFibonacci
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V resnici se je oseba,
ki ji pravimo Fibonacci,
01:33
was actuallydejansko namedimenovan LeonardoLeonardo of PisaPisa,
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imenovala Leonardo Pisano
01:36
and these numbersštevilke appearPojavi se in his bookknjigo "LiberLiber AbaciAbaci,"
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in ta števila so zapisana
v njegovi knjigi "Liber Abaci",
01:39
whichki taughtučil the WesternWestern worldsvet
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ki je zahodni svet naučila
01:41
the methodsmetode of arithmeticaritmetično that we use todaydanes.
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aritmetičnih metod,
ki jih uporabljamo danes.
01:44
In termspogoji of applicationsaplikacije,
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Kar se tiče uporabe,
01:45
FibonacciFibonacci numbersštevilke appearPojavi se in naturenarava
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se Fibonaccijeva števila
v naravi pojavljajo
01:48
surprisinglypresenetljivo oftenpogosto.
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presenetljivo pogosto.
01:49
The numberštevilka of petalscvetnih listov on a flowercvet
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Število cvetnih listov na roži
01:51
is typicallyobičajno a FibonacciFibonacci numberštevilka,
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je ponavadi Fibonaccijevo število,
01:53
or the numberštevilka of spiralsspirale on a sunflowerSončnica
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pa tudi število spiral na sončnici
01:56
or a pineappleananas
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ali ananasu
01:57
tendstežava to be a FibonacciFibonacci numberštevilka as well.
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je pogosto Fibonaccijevo število.
02:00
In factdejstvo, there are manyveliko more
applicationsaplikacije of FibonacciFibonacci numbersštevilke,
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Pravzaprav je možnosti uporabe
Fibonaccijevih števil veliko več,
02:03
but what I find mostnajbolj inspirationalinspirativno about them
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a sam mislim, da so pri njih
najbolj navdušujoči
02:06
are the beautifullepo numberštevilka patternsvzorce they displayprikaz.
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lepi številski vzorci, ki jih ustvarjajo.
02:08
Let me showshow you one of my favoritespriljubljene.
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Pokazal vam bom enega
od svojih najljubših.
02:11
SupposeRecimo, da you like to squarekvadrat numbersštevilke,
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Recimo, da radi kvadrirate števila,
02:13
and franklyodkrito, who doesn't? (LaughterSmeh)
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konec koncev, kdo jih pa ne? (Smeh)
02:16
Let's look at the squareskvadratov
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Poglejmo kvadrate
02:18
of the first fewmalo FibonacciFibonacci numbersštevilke.
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prvih nekaj Fibonaccijevih števil.
02:20
So one squaredkvadrat is one,
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Torej, ena na kvadrat je ena,
02:22
two squaredkvadrat is fourštiri, threetri squaredkvadrat is ninedevet,
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dva na kvadrat je štiri,
tri na kvadrat je devet,
02:24
fivepet squaredkvadrat is 25, and so on.
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pet na kvadrat je 25 in tako naprej.
02:27
Now, it's no surprisepresenečenje
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No, ni prav presenetljivo,
02:29
that when you adddodaj consecutivezaporednih FibonacciFibonacci numbersštevilke,
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da, ko seštejemo
zaporedna Fibonaccijeva števila,
02:32
you get the nextNaslednji FibonacciFibonacci numberštevilka. Right?
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dobimo naslednje
Fibonaccijevo število. Drži?
02:34
That's how they're createdustvarjeno.
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Tako nastanejo.
02:35
But you wouldn'tne bi expectpričakovati anything specialposeben
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Ne bi pa pričakovali, da se zgodi
02:37
to happense zgodi when you adddodaj the squareskvadratov togetherskupaj.
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kaj posebnega,
ko seštejemo njihove kvadrate.
02:40
But checkpreveri this out.
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Pa poglejte zdaj tole.
02:42
One plusplus one givesdaje us two,
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Ena plus ena je dva
02:44
and one plusplus fourštiri givesdaje us fivepet.
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in ena plus štiri je pet.
02:46
And fourštiri plusplus ninedevet is 13,
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In štiri plus devet je 13,
02:48
ninedevet plusplus 25 is 34,
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devet plus 25 je 34,
02:52
and yes, the patternvzorec continuesse nadaljuje.
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in ja, vzorec se nadaljuje.
02:54
In factdejstvo, here'sTukaj je anotherdrugo one.
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V bistvu imamo še en vzorec.
02:56
SupposeRecimo, da you wanted to look at
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Recimo, da bi hoteli pogledati
seštevek kvadratov
prvih nekaj Fibonaccijevih števil.
02:58
addingdodajanje the squareskvadratov of
the first fewmalo FibonacciFibonacci numbersštevilke.
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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 plusplus one plusplus fourštiri is sixšest.
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Torej, ena plus ena plus štiri je šest.
03:04
AddDodaj ninedevet to that, we get 15.
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Dodajmo še devet in dobimo 15.
03:07
AddDodaj 25, we get 40.
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Dodamo 25 in dobimo 40.
03:09
AddDodaj 64, we get 104.
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Dodamo 64, dobimo 104.
03:12
Now look at those numbersštevilke.
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Zdaj pa poglejmo ta števila.
03:14
Those are not FibonacciFibonacci numbersštevilke,
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To niso Fibonaccijeva števila,
03:16
but if you look at them closelytesno,
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ampak, če jih pogledate od blizu,
03:18
you'llboš see the FibonacciFibonacci numbersštevilke
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boste videli, da se Fibonaccijeva števila
03:20
buriedpokopan insideznotraj of them.
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skrivajo v njih.
03:22
Do you see it? I'll showshow it to you.
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Jih vidite? Vam bom pokazal.
03:24
SixŠest is two timeskrat threetri, 15 is threetri timeskrat fivepet,
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Šest je dva krat tri, 15 je tri krat pet,
03:28
40 is fivepet timeskrat eightosem,
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40 je pet krat osem,
03:30
two, threetri, fivepet, eightosem, who do we appreciatecenite?
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dva, tri, pet, osem, koga občudujemo?
03:33
(LaughterSmeh)
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(Smeh)
03:34
FibonacciFibonacci! Of courseseveda.
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Fibonaccija! Jasno.
03:36
Now, as much funzabavno as it is to discoverodkrijte these patternsvzorce,
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Zelo zabavno je odkrivati vzorce,
03:40
it's even more satisfyingki izpolnjuje to understandrazumeti
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a v še večje zadovoljstvo je razumeti
03:42
why they are trueresnično.
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zakaj držijo.
03:44
Let's look at that last equationenačba.
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Poglejmo zadnjo enačbo.
03:46
Why should the squareskvadratov of one, one,
two, threetri, fivepet and eightosem
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Zakaj mora seštevek kvadratov od
ena, ena, dva, tri, pet in osem
03:50
adddodaj up to eightosem timeskrat 13?
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znašati osem krat 13?
03:53
I'll showshow you by drawingrisanje a simplepreprosto pictureslika.
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To vam bom pokazal s preprosto sliko.
03:56
We'llBomo startZačni with a one-by-oneenega squarekvadrat
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Začeli bomo s kvadratom ena krat ena
03:58
and nextNaslednji to that put anotherdrugo one-by-oneenega squarekvadrat.
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in zraven njega narisali
še en kvadrat ena krat ena.
04:02
TogetherSkupaj, they formobrazec a one-by-twoena dva rectanglepravokotnik.
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Skupaj sestavljata
pravokotnik ena krat dva.
04:06
BeneathPod that, I'll put a two-by-twodva po dva squarekvadrat,
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Pod njega bom narisal
kvadrat dva krat dva,
04:08
and nextNaslednji to that, a three-by-threetri za tri squarekvadrat,
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zraven njega pa kvadrat tri krat tri,
04:11
beneathspodaj that, a five-by-fivepet pet squarekvadrat,
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pod njega kvadrat pet krat pet
04:13
and then an eight-by-eightosem jih osem squarekvadrat,
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in nato kvadrat osem krat osem,
04:15
creatingustvarjanje one giantvelikan rectanglepravokotnik, right?
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in tako sem sestavil ogromen pravokotnik.
04:18
Now let me askvprašajte you a simplepreprosto questionvprašanje:
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Zdaj vam bom postavil preprosto vprašanje:
04:20
what is the areaobmočje of the rectanglepravokotnik?
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Kolikšna je ploščina pravokotnika?
04:23
Well, on the one handroka,
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No, po svoje
04:25
it's the sumvsota of the areasobmočja
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je vsota ploščin
04:28
of the squareskvadratov insideznotraj it, right?
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vseh kvadratov v njem, drži?
04:30
Just as we createdustvarjeno it.
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Kot smo ga naredili.
04:31
It's one squaredkvadrat plusplus one squaredkvadrat
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Ena na kvadrat plus ena na kvadrat
04:33
plusplus two squaredkvadrat plusplus threetri squaredkvadrat
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plus dva na kvadrat plus tri na kvadrat
04:35
plusplus fivepet squaredkvadrat plusplus eightosem squaredkvadrat. Right?
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plus pet na kvadrat plus osem na kvadrat.
04:38
That's the areaobmočje.
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To je ploščina.
04:40
On the other handroka, because it's a rectanglepravokotnik,
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Po drugi strani pa, ker je pravokotnik,
04:42
the areaobmočje is equalenako to its heightvišina timeskrat its basebazo,
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je ploščina enaka višini krat širini
04:46
and the heightvišina is clearlyjasno eightosem,
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in višina je očitno osem,
04:48
and the basebazo is fivepet plusplus eightosem,
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širina pa pet plus osem,
04:51
whichki is the nextNaslednji FibonacciFibonacci numberštevilka, 13. Right?
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kar je naslednje
Fibonaccijevo število, 13. Je tako?
04:55
So the areaobmočje is alsotudi eightosem timeskrat 13.
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Tako imamo ploščino osem krat 13.
04:58
SinceOd we'vesmo correctlypravilno calculatedizračuna the areaobmočje
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Ker smo pravilno izračunali ploščino
05:00
two differentdrugačen waysnačinov,
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na dva različna načina,
05:02
they have to be the sameenako numberštevilka,
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moramo dobiti enako številko
05:04
and that's why the squareskvadratov of one,
one, two, threetri, fivepet and eightosem
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in zato je seštevek kvadratov od
ena, ena, dva, tri, pet in osem
05:08
adddodaj up to eightosem timeskrat 13.
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skupaj osem krat 13.
05:10
Now, if we continuenadaljuj this processproces,
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Če nadaljujemo s tem postopkom,
05:12
we'llbomo generateustvarjati rectanglespravokotniki of the formobrazec 13 by 21,
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bomo ustvarili pravokotnike
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 checkpreveri this out.
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Zdaj pa poglejte tole.
05:20
If you divideRazdelite 13 by eightosem,
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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 divideRazdelite the largervečje numberštevilka
by the smallermanjši numberštevilka,
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In če delimo večje število
z manjšim številom,
05:28
then these ratiosrazmerja get closerbližje and closerbližje
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se razmerje vedno bolj približuje
05:31
to about 1.618,
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okoli 1,618,
05:33
knownznano to manyveliko people as the GoldenZlati RatioRazmerje,
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kar veliko ljudi pozna kot zlati rez,
05:37
a numberštevilka whichki has fascinatedfasciniran mathematiciansmatematiki,
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število, ki je stoletja navduševalo
05:39
scientistsznanstveniki and artistsumetniki for centuriesstoletja.
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matematike, znanstvenike in umetnike.
05:42
Now, I showshow all this to you because,
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To vam kažem, ker,
05:45
like so much of mathematicsmatematika,
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kot toliko matematike,
05:47
there's a beautifullepo sidestran to it
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v sebi skriva nekaj lepega,
05:49
that I fearstrah does not get enoughdovolj attentionpozornost
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čemur mislim, da v naših šolah žal
05:51
in our schoolsšole.
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ne posvečamo dovolj pozornosti.
05:52
We spendporabiti lots of time learningučenje about calculationizračun,
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Veliko časa se učimo o računanju,
05:55
but let's not forgetpozabi about applicationaplikacija,
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ampak ne smemo pozabiti na uporabo,
05:58
includingvključno z, perhapsmorda, the mostnajbolj
importantpomembno applicationaplikacija of all,
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vključno z morda najpomembnejšo uporabo,
06:01
learningučenje how to think.
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da se naučimo, kako razmišljati.
06:03
If I could summarizepovzamemo this in one sentencestavek,
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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
MathematicsMatematika is not just solvingreševanje for x,
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Matematika ni samo iskanje x-a,
06:10
it's alsotudi figuringfiguring out why.
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ampak tudi smisla.
06:13
Thank you very much.
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Najlepša hvala.
06:15
(ApplauseAplavz)
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(Aplavz)
Translated by Petra Zajc
Reviewed by Nika Kotnik

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ABOUT THE SPEAKER
Arthur Benjamin - Mathemagician
Using 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").

More profile about the speaker
Arthur Benjamin | Speaker | TED.com