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: Keajaiban nombor Fibonacci

Filmed:
7,057,274 views

Matematik ialah sesuatu yang logik, berfungsi dan.. hebat. Ahli Mate-magik, Arthur Benjamin, meneroka sifat-sifat tersembunyi dalam satu set nombor yang pelik dan menarik, iaitu siri Fibonacci. (Dan mengingatkan anda bahawa matematik juga memberikan inspirasi!)
- 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 learnbelajar mathematicsmatematik?
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Kenapa kita belajar matematik?
00:15
EssentiallyPada asasnya, for threetiga reasonssebab:
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Asasnya, kerana tiga sebab:
00:18
calculationpengiraan,
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pengiraan, aplikasi,
00:19
applicationpermohonan,
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pengiraan, aplikasi,
00:21
and last, and unfortunatelymalangnya leastpaling kurang
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dan yang terakhir, yang malangnya
kurang diberikan perhatian,
00:24
in termsterma of the time we give it,
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dan yang terakhir, yang malangnya
kurang diberikan perhatian,
00:26
inspirationinspirasi.
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inspirasi.
00:28
MathematicsMatematik is the sciencesains of patternscorak,
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Matematik merupakan sains corak.
00:30
and we studybelajar it to learnbelajar how to think logicallysecara logik,
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Kita belajar berfikir secara logik, kritikal dan kreatif,
00:34
criticallykritikal and creativelysecara kreatif,
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Kita belajar berfikir secara logik, kritikal dan kreatif,
00:36
but too much of the mathematicsmatematik
that we learnbelajar in schoolsekolah
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tetapi yang diajar di sekolah,
00:39
is not effectivelyberkesan motivatedbermotivasi,
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tidak memberikan rangsangan yang baik.
00:41
and when our studentspelajar asktanya,
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Apabila pelajar kita bertanya,
00:43
"Why are we learningpembelajaran this?"
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"Kenapa kita belajar ni?"
00:44
then they oftenselalunya heardengar that they'llmereka akan need it
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Ini penting untuk kelas yang berikutnya
00:46
in an upcomingakan datang mathmatematik classkelas or on a futuremasa depan testujian.
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atau ujian yang akan datang.
00:50
But wouldn'ttidak akan it be great
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Bukankah lebih bagus
00:51
if everysetiap oncesekali in a while we did mathematicsmatematik
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kalau kadangkala kita membuat matematik
00:54
simplysemata-mata because it was funkeseronokan or beautifulcantik
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kerana ia menyeronokkan, mengasyikkan
00:57
or because it excitedteruja the mindfikiran?
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atau kerana ia merangsang minda?
00:59
Now, I know manyramai people have not
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Ramai yang tak berpeluang untuk
01:01
had the opportunitypeluang to see how this can happenberlaku,
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memahami bagaimana ini boleh berlaku,
01:03
so let me give you a quickcepat examplecontohnya
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jadi biar saya berikan contoh
01:05
with my favoritekegemaran collectionkoleksi of numbersnombor,
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dengan koleksi nombor kegemaran saya,
01:07
the FibonacciFibonacci numbersnombor. (ApplauseTepukan)
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nombor Fibonacci. (Tepukan)
01:10
Yeah! I alreadysudah have FibonacciFibonacci fanspeminat here.
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Ya, ada peminat Fibonacci di sini. Bagus.
01:12
That's great.
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Ya, ada peminat Fibonacci di sini. Bagus.
01:13
Now these numbersnombor can be appreciateddihargai
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Nombor-nombor ini boleh dihargai
01:15
in manyramai differentberbeza wayscara.
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dalam berbagai-bagai cara.
01:17
From the standpointsudut pandangan of calculationpengiraan,
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Dari sudut pengiraan,
01:20
they're as easymudah to understandfaham
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ia sangat senang difahami
01:22
as one plusditambah one, whichyang mana is two.
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seperti 1 + 1 = 2,
01:24
Then one plusditambah two is threetiga,
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1 + 2 = 3,
01:26
two plusditambah threetiga is fivelima, threetiga plusditambah fivelima is eightlapan,
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2 + 3 = 5,
3 + 5 = 8,
01:29
and so on.
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dan begitulah seterusnya.
01:31
IndeedSesungguhnya, the personorang we call FibonacciFibonacci
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Orang yang dikenali sebagai Fibonacci
01:33
was actuallysebenarnya namedbernama LeonardoLeonardo of PisaPisa,
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sebenarnya bernama Leonardo of Pisa,
01:36
and these numbersnombor appearmuncul in his bookbuku "LiberSejarah AbaciSepua (Abacus),"
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nombor-nombor ini diterangkan
dalam buku "Liber Abaci",
01:39
whichyang mana taughtdiajar the WesternWestern worlddunia
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di mana dunia Barat telah diajar
01:41
the methodskaedah of arithmeticaritmetik that we use todayhari ini.
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kaedah aritmetik yang digunakan sekarang.
01:44
In termsterma of applicationsaplikasi,
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Dari segi aplikasi, nombor Fibonacci
01:45
FibonacciFibonacci numbersnombor appearmuncul in naturesifat
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selalu muncul dalam alam semula jadi.
01:48
surprisinglymengejutkan oftenselalunya.
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selalu muncul dalam alam semula jadi.
01:49
The numbernombor of petalskelopak on a flowerbunga
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Bilangan kelopak bunga
01:51
is typicallybiasanya a FibonacciFibonacci numbernombor,
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selalunya ialah nombor Fibonacci,
01:53
or the numbernombor of spiralsbentuk on a sunflowerSunflower
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lingkaran bunga matahari atau nenas,
01:56
or a pineapplenenas
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lingkaran bunga matahari atau nenas,
01:57
tendscenderung to be a FibonacciFibonacci numbernombor as well.
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biasanya merupakan nombor Fibonacci.
02:00
In factfakta, there are manyramai more
applicationsaplikasi of FibonacciFibonacci numbersnombor,
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Banyak lagi aplikasi nombor Fibonacci,
02:03
but what I find mostpaling banyak inspirationalmenjadi sumber inspirasi about them
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yang paling memberikan inspirasi
02:06
are the beautifulcantik numbernombor patternscorak they displaypaparan.
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ialah corak nombor yang dipaparkan.
02:08
Let me showtunjukkan you one of my favoriteskegemaran.
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Ini salah satu kegemaran saya.
02:11
SupposeAndaikan you like to squarepersegi numbersnombor,
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Katakan anda suka nombor kuasa dua,
02:13
and franklyterus terang, who doesn't? (LaughterGelak ketawa)
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siapa yang tak suka, kan? (Gelak ketawa)
02:16
Let's look at the squaresdataran
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Mari kita lihat nombor kuasa dua
02:18
of the first fewbeberapa FibonacciFibonacci numbersnombor.
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bagi nombor-nombor Fibonacci.
02:20
So one squaredkuasa dua is one,
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1 kuasa dua = 1,
02:22
two squaredkuasa dua is fourempat, threetiga squaredkuasa dua is ninesembilan,
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2 kuasa dua = 4,
3 kuasa dua = 9,
02:24
fivelima squaredkuasa dua is 25, and so on.
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5 kuasa dua = 25,
dan seterusnya.
02:27
Now, it's no surprisekejutan
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Jadi, tak hairanlah apabila
02:29
that when you addTambah consecutiveberturut-turut FibonacciFibonacci numbersnombor,
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jumlah dua nombor Fibonacci yang berturut
02:32
you get the nextseterusnya FibonacciFibonacci numbernombor. Right?
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menghasilkan nombor Fibonacci yang berikutnya.
02:34
That's how they're createddicipta.
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Itu merupakan cara ia dicipta.
02:35
But you wouldn'ttidak akan expectmenjangkakan anything specialistimewa
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Anda tak akan menjangkakan apa-apa jika
02:37
to happenberlaku when you addTambah the squaresdataran togetherbersama-sama.
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nombor-nombor kuasa dua tersebut ditambah.
02:40
But checksemak this out.
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Cuba tengok ni.
02:42
One plusditambah one givesmemberi us two,
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1 + 1 = 2,
02:44
and one plusditambah fourempat givesmemberi us fivelima.
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1 + 4 = 5,
02:46
And fourempat plusditambah ninesembilan is 13,
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4 + 9 = 13,
02:48
ninesembilan plusditambah 25 is 34,
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9 + 25 = 34,
02:52
and yes, the patterncorak continuesterus.
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dan corak itu berterusan.
02:54
In factfakta, here'sdi sini anotherlain one.
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Ini satu lagi contoh.
02:56
SupposeAndaikan you wanted to look at
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Katakan anda tambah beberapa
02:58
addingtambahnya the squaresdataran of
the first fewbeberapa FibonacciFibonacci numbersnombor.
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nombor kuasa dua Fibonacci yang awal.
03:00
Let's see what we get there.
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Mari kita lihat apa hasilnya.
03:02
So one plusditambah one plusditambah fourempat is sixenam.
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1 + 1 + 4 = 6.
03:04
AddTambah ninesembilan to that, we get 15.
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6 + 9 = 15.
03:07
AddTambah 25, we get 40.
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15 + 25 = 40.
03:09
AddTambah 64, we get 104.
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40 + 64 = 104.
03:12
Now look at those numbersnombor.
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Tengok nombor-nombor ini.
03:14
Those are not FibonacciFibonacci numbersnombor,
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Ia bukan nombor-nombor Fibonacci.
03:16
but if you look at them closelyrapat,
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Tetapi jika anda lihat dengan teliti,
03:18
you'llawak akan see the FibonacciFibonacci numbersnombor
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ada nombor Fibonacci
03:20
burieddikebumikan insidedalam of them.
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yang tersembunyi di dalamnya.
03:22
Do you see it? I'll showtunjukkan it to you.
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Nampak tak? Saya akan tunjukkan.
03:24
SixEnam is two timeskali threetiga, 15 is threetiga timeskali fivelima,
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6 = 2 x 3,
15 = 3 x 5,
03:28
40 is fivelima timeskali eightlapan,
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40 = 5 x 8,
03:30
two, threetiga, fivelima, eightlapan, who do we appreciatemenghargai?
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2, 3, 5, 8, terima kasih kepada siapa?
03:33
(LaughterGelak ketawa)
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(Gelak ketawa)
03:34
FibonacciFibonacci! Of coursekursus.
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Semestinya, Fibonacci!
03:36
Now, as much funkeseronokan as it is to discovertemui these patternscorak,
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Corak ini memang menyeronokkan,
03:40
it's even more satisfyingmemuaskan to understandfaham
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tapi lebih memuaskan jika kita faham
03:42
why they are truebenar.
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kenapa ia begitu.
03:44
Let's look at that last equationpersamaan.
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Cuba lihat persamaan yang terakhir.
03:46
Why should the squaresdataran of one, one,
two, threetiga, fivelima and eightlapan
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Kenapa kuasa dua kepada 1, 1, 2, 3, 5 dan 8
03:50
addTambah up to eightlapan timeskali 13?
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jumlahnya sama dengan 8 x 13?
03:53
I'll showtunjukkan you by drawinglukisan a simplemudah picturegambar.
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Saya akan lukiskan satu gambar.
03:56
We'llKami akan startmulakan with a one-by-onesatu demi satu squarepersegi
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Ada satu segi empat 1 x 1,
03:58
and nextseterusnya to that put anotherlain one-by-onesatu demi satu squarepersegi.
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dan satu lagi segi empat 1 x 1.
04:02
TogetherBersama-sama, they formborang a one-by-twosatu-dua rectanglesegi empat tepat.
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Hasilnya segi empat tepat 1 x 2.
04:06
BeneathDi bawah that, I'll put a two-by-twodua-dua squarepersegi,
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Letakkan segi empat 2 x 2 di bawah,
04:08
and nextseterusnya to that, a three-by-threetiga oleh tiga squarepersegi,
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dan segi empat 3 x 3 di sebelah,
04:11
beneathdi bawah that, a five-by-fivelima oleh lima squarepersegi,
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segi empat 5 x 5 di bawah,
04:13
and then an eight-by-eightlapan oleh lapan squarepersegi,
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dan satu lagi segi empat 8 x 8,
04:15
creatingmencipta one giantgergasi rectanglesegi empat tepat, right?
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membentuk segi empat tepat yang besar, kan?
04:18
Now let me asktanya you a simplemudah questionsoalan:
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Izinkan saya bertanya,
04:20
what is the areakawasan of the rectanglesegi empat tepat?
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berapakah luas segi empat tepat itu?
04:23
Well, on the one handtangan,
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Yang pertama, ia merupakan jumlah luas
04:25
it's the sumjumlah of the areaskawasan-kawasan
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Yang pertama, ia merupakan jumlah luas
04:28
of the squaresdataran insidedalam it, right?
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semua segi empat di dalamnya, kan?
04:30
Just as we createddicipta it.
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Sama seperti yang kita buat tadi.
04:31
It's one squaredkuasa dua plusditambah one squaredkuasa dua
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1 kuasa dua + 1 kuasa dua,
04:33
plusditambah two squaredkuasa dua plusditambah threetiga squaredkuasa dua
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+ 2 kuasa dua, + 3 kuasa dua,
04:35
plusditambah fivelima squaredkuasa dua plusditambah eightlapan squaredkuasa dua. Right?
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+ 5 kuasa dua, + 8 kuasa dua.
04:38
That's the areakawasan.
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Itu merupakan luasnya.
04:40
On the other handtangan, because it's a rectanglesegi empat tepat,
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Yang kedua, luas sebuah segi empat tepat,
04:42
the areakawasan is equalsama to its heightketinggian timeskali its baseasas,
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ialah tinggi x tapak,
04:46
and the heightketinggian is clearlyjelas eightlapan,
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tinggi = 8,
04:48
and the baseasas is fivelima plusditambah eightlapan,
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tapak = 5 + 8,
04:51
whichyang mana is the nextseterusnya FibonacciFibonacci numbernombor, 13. Right?
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iaitu 13, nombor Fibonacci yang berikutnya, kan?
04:55
So the areakawasan is alsojuga eightlapan timeskali 13.
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Jadi luasnya ialah 8 x 13 juga.
04:58
SinceSejak we'vekami sudah correctlybetul calculateddikira the areakawasan
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Kita telah mengira luas
05:00
two differentberbeza wayscara,
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dengan dua cara yang berbeza,
05:02
they have to be the samesama numbernombor,
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hasilnya mesti sama,
05:04
and that's why the squaresdataran of one,
one, two, threetiga, fivelima and eightlapan
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sebab itu kuasa dua kepada 1, 1, 2, 3, 5 dan 8,
05:08
addTambah up to eightlapan timeskali 13.
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jumlahnya sama dengan 8 x 13.
05:10
Now, if we continueteruskan this processproses,
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Jika kita teruskan proses ini,
05:12
we'llkita akan generatemenjana rectanglessegi empat tepat of the formborang 13 by 21,
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hasilnya ialah segi empat tepat 13 x 21,
05:16
21 by 34, and so on.
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21 x 34, dan seterusnya.
05:19
Now checksemak this out.
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Sekarang tengok ni.
05:20
If you dividemembahagi 13 by eightlapan,
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Jika anda bahagi 13 dengan 8,
05:22
you get 1.625.
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anda dapat 1.625. Bahagikan nombor
05:24
And if you dividemembahagi the largerlebih besar numbernombor
by the smallerlebih kecil numbernombor,
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yang lebih besar dengan yang sebelumnya
05:28
then these ratiosnisbah get closerlebih dekat and closerlebih dekat
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nisbahnya akan semakin hampir
05:31
to about 1.618,
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dengan kira-kira 1.618,
05:33
knowndiketahui to manyramai people as the GoldenEmas RatioNisbah,
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juga dikenali sebagai Nisbah Keemasan,
05:37
a numbernombor whichyang mana has fascinatedterpesona mathematiciansahli matematik,
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nombor yang mempesonakan ahli matematik,
05:39
scientistssaintis and artistsartis for centuriesberabad-abad lamanya.
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saintis dan seniman sejak dulu.
05:42
Now, I showtunjukkan all this to you because,
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Saya bentangkan semua ini kerana,
05:45
like so much of mathematicsmatematik,
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seperti kebanyakan matematik,
05:47
there's a beautifulcantik sidesebelah to it
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ia mempunyai aspek yang menakjubkan
05:49
that I fearketakutan does not get enoughcukup attentionperhatian
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yang sayangnya tak mendapat perhatian
05:51
in our schoolssekolah.
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di sekolah-sekolah kita.
05:52
We spendbelanja lots of time learningpembelajaran about calculationpengiraan,
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Banyak masa dihabiskan untuk belajar mengira,
05:55
but let's not forgetlupa about applicationpermohonan,
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tetapi jangan lupa tentang aplikasinya
05:58
includingtermasuk, perhapsmungkin, the mostpaling banyak
importantpenting applicationpermohonan of all,
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termasuk aplikasi yang paling penting,
06:01
learningpembelajaran how to think.
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belajar cara berfikir.
06:03
If I could summarizemeringkaskan this in one sentencehukuman,
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Saya simpulkan dalam satu ayat:
06:05
it would be this:
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Saya simpulkan dalam satu ayat:
06:07
MathematicsMatematik is not just solvingmenyelesaikan for x,
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Matematik bukan hanya untuk mencari x,
06:10
it's alsojuga figuringmemikirkan out why.
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tapi juga untuk mengetahui kenapa (why).
06:13
Thank you very much.
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Terima kasih.
06:15
(ApplauseTepukan)
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(Tepukan)
Translated by Ros Rosli
Reviewed by PF Ng

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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