TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Arthur Benjamin: Keajaiban nombor Fibonacci
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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!)
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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Kenapa kita belajar matematik?
00:15
Essentially, for three reasons:
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Asasnya, kerana tiga sebab:
00:18
calculation,
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pengiraan, aplikasi,
00:19
application,
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pengiraan, aplikasi,
00:21
and last, and unfortunately least
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dan yang terakhir, yang malangnya
kurang diberikan perhatian,
kurang diberikan perhatian,
00:24
in terms of the time we give it,
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dan yang terakhir, yang malangnya
kurang diberikan perhatian,
kurang diberikan perhatian,
00:26
inspiration.
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inspirasi.
00:28
Mathematics is the science of patterns,
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Matematik merupakan sains corak.
00:30
and we study it to learn how to think logically,
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Kita belajar berfikir secara logik, kritikal dan kreatif,
00:34
critically and creatively,
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Kita belajar berfikir secara logik, kritikal dan kreatif,
00:36
but too much of the mathematics
that we learn in school
that we learn in school
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tetapi yang diajar di sekolah,
00:39
is not effectively motivated,
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tidak memberikan rangsangan yang baik.
00:41
and when our students ask,
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Apabila pelajar kita bertanya,
00:43
"Why are we learning this?"
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"Kenapa kita belajar ni?"
00:44
then they often hear that they'll need it
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Ini penting untuk kelas yang berikutnya
00:46
in an upcoming math class or on a future test.
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atau ujian yang akan datang.
00:50
But wouldn't it be great
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Bukankah lebih bagus
00:51
if every once in a while we did mathematics
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kalau kadangkala kita membuat matematik
00:54
simply because it was fun or beautiful
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kerana ia menyeronokkan, mengasyikkan
00:57
or because it excited the mind?
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atau kerana ia merangsang minda?
00:59
Now, I know many people have not
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Ramai yang tak berpeluang untuk
01:01
had the opportunity to see how this can happen,
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memahami bagaimana ini boleh berlaku,
01:03
so let me give you a quick example
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jadi biar saya berikan contoh
01:05
with my favorite collection of numbers,
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dengan koleksi nombor kegemaran saya,
01:07
the Fibonacci numbers. (Applause)
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nombor Fibonacci. (Tepukan)
01:10
Yeah! I already have Fibonacci fans 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 numbers can be appreciated
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Nombor-nombor ini boleh dihargai
01:15
in many different ways.
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dalam berbagai-bagai cara.
01:17
From the standpoint of calculation,
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Dari sudut pengiraan,
01:20
they're as easy to understand
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ia sangat senang difahami
01:22
as one plus one, which is two.
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seperti 1 + 1 = 2,
01:24
Then one plus two is three,
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1 + 2 = 3,
01:26
two plus three is five, three plus five is eight,
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2 + 3 = 5,
3 + 5 = 8,
3 + 5 = 8,
01:29
and so on.
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dan begitulah seterusnya.
01:31
Indeed, the person we call Fibonacci
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Orang yang dikenali sebagai Fibonacci
01:33
was actually named Leonardo of Pisa,
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sebenarnya bernama Leonardo of Pisa,
01:36
and these numbers appear in his book "Liber Abaci,"
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nombor-nombor ini diterangkan
dalam buku "Liber Abaci",
dalam buku "Liber Abaci",
01:39
which taught the Western world
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di mana dunia Barat telah diajar
01:41
the methods of arithmetic that we use today.
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kaedah aritmetik yang digunakan sekarang.
01:44
In terms of applications,
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Dari segi aplikasi, nombor Fibonacci
01:45
Fibonacci numbers appear in nature
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selalu muncul dalam alam semula jadi.
01:48
surprisingly often.
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selalu muncul dalam alam semula jadi.
01:49
The number of petals on a flower
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Bilangan kelopak bunga
01:51
is typically a Fibonacci number,
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selalunya ialah nombor Fibonacci,
01:53
or the number of spirals on a sunflower
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lingkaran bunga matahari atau nenas,
01:56
or a pineapple
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lingkaran bunga matahari atau nenas,
01:57
tends to be a Fibonacci number as well.
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biasanya merupakan nombor Fibonacci.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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Banyak lagi aplikasi nombor Fibonacci,
02:03
but what I find most inspirational about them
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yang paling memberikan inspirasi
02:06
are the beautiful number patterns they display.
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ialah corak nombor yang dipaparkan.
02:08
Let me show you one of my favorites.
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Ini salah satu kegemaran saya.
02:11
Suppose you like to square numbers,
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Katakan anda suka nombor kuasa dua,
02:13
and frankly, who doesn't? (Laughter)
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siapa yang tak suka, kan? (Gelak ketawa)
02:16
Let's look at the squares
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Mari kita lihat nombor kuasa dua
02:18
of the first few Fibonacci numbers.
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bagi nombor-nombor Fibonacci.
02:20
So one squared is one,
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1 kuasa dua = 1,
02:22
two squared is four, three squared is nine,
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2 kuasa dua = 4,
3 kuasa dua = 9,
3 kuasa dua = 9,
02:24
five squared is 25, and so on.
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5 kuasa dua = 25,
dan seterusnya.
dan seterusnya.
02:27
Now, it's no surprise
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Jadi, tak hairanlah apabila
02:29
that when you add consecutive Fibonacci numbers,
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jumlah dua nombor Fibonacci yang berturut
02:32
you get the next Fibonacci number. Right?
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menghasilkan nombor Fibonacci yang berikutnya.
02:34
That's how they're created.
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Itu merupakan cara ia dicipta.
02:35
But you wouldn't expect anything special
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Anda tak akan menjangkakan apa-apa jika
02:37
to happen when you add the squares together.
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nombor-nombor kuasa dua tersebut ditambah.
02:40
But check this out.
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Cuba tengok ni.
02:42
One plus one gives us two,
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1 + 1 = 2,
02:44
and one plus four gives us five.
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1 + 4 = 5,
02:46
And four plus nine is 13,
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4 + 9 = 13,
02:48
nine plus 25 is 34,
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9 + 25 = 34,
02:52
and yes, the pattern continues.
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dan corak itu berterusan.
02:54
In fact, here's another one.
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Ini satu lagi contoh.
02:56
Suppose you wanted to look at
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Katakan anda tambah beberapa
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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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 plus one plus four is six.
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1 + 1 + 4 = 6.
03:04
Add nine to that, we get 15.
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6 + 9 = 15.
03:07
Add 25, we get 40.
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15 + 25 = 40.
03:09
Add 64, we get 104.
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40 + 64 = 104.
03:12
Now look at those numbers.
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Tengok nombor-nombor ini.
03:14
Those are not Fibonacci numbers,
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Ia bukan nombor-nombor Fibonacci.
03:16
but if you look at them closely,
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Tetapi jika anda lihat dengan teliti,
03:18
you'll see the Fibonacci numbers
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ada nombor Fibonacci
03:20
buried inside of them.
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yang tersembunyi di dalamnya.
03:22
Do you see it? I'll show it to you.
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Nampak tak? Saya akan tunjukkan.
03:24
Six is two times three, 15 is three times five,
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6 = 2 x 3,
15 = 3 x 5,
15 = 3 x 5,
03:28
40 is five times eight,
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40 = 5 x 8,
03:30
two, three, five, eight, who do we appreciate?
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2, 3, 5, 8, terima kasih kepada siapa?
03:33
(Laughter)
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(Gelak ketawa)
03:34
Fibonacci! Of course.
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Semestinya, Fibonacci!
03:36
Now, as much fun as it is to discover these patterns,
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Corak ini memang menyeronokkan,
03:40
it's even more satisfying to understand
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tapi lebih memuaskan jika kita faham
03:42
why they are true.
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kenapa ia begitu.
03:44
Let's look at that last equation.
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Cuba lihat persamaan yang terakhir.
03:46
Why should the squares of one, one,
two, three, five and eight
two, three, five and eight
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Kenapa kuasa dua kepada 1, 1, 2, 3, 5 dan 8
03:50
add up to eight times 13?
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jumlahnya sama dengan 8 x 13?
03:53
I'll show you by drawing a simple picture.
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Saya akan lukiskan satu gambar.
03:56
We'll start with a one-by-one square
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Ada satu segi empat 1 x 1,
03:58
and next to that put another one-by-one square.
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dan satu lagi segi empat 1 x 1.
04:02
Together, they form a one-by-two rectangle.
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Hasilnya segi empat tepat 1 x 2.
04:06
Beneath that, I'll put a two-by-two square,
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Letakkan segi empat 2 x 2 di bawah,
04:08
and next to that, a three-by-three square,
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dan segi empat 3 x 3 di sebelah,
04:11
beneath that, a five-by-five square,
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segi empat 5 x 5 di bawah,
04:13
and then an eight-by-eight square,
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dan satu lagi segi empat 8 x 8,
04:15
creating one giant rectangle, right?
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membentuk segi empat tepat yang besar, kan?
04:18
Now let me ask you a simple question:
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Izinkan saya bertanya,
04:20
what is the area of the rectangle?
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berapakah luas segi empat tepat itu?
04:23
Well, on the one hand,
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Yang pertama, ia merupakan jumlah luas
04:25
it's the sum of the areas
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Yang pertama, ia merupakan jumlah luas
04:28
of the squares inside it, right?
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semua segi empat di dalamnya, kan?
04:30
Just as we created it.
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Sama seperti yang kita buat tadi.
04:31
It's one squared plus one squared
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1 kuasa dua + 1 kuasa dua,
04:33
plus two squared plus three squared
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+ 2 kuasa dua, + 3 kuasa dua,
04:35
plus five squared plus eight squared. Right?
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+ 5 kuasa dua, + 8 kuasa dua.
04:38
That's the area.
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Itu merupakan luasnya.
04:40
On the other hand, because it's a rectangle,
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Yang kedua, luas sebuah segi empat tepat,
04:42
the area is equal to its height times its base,
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ialah tinggi x tapak,
04:46
and the height is clearly eight,
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tinggi = 8,
04:48
and the base is five plus eight,
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tapak = 5 + 8,
04:51
which is the next Fibonacci number, 13. Right?
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iaitu 13, nombor Fibonacci yang berikutnya, kan?
04:55
So the area is also eight times 13.
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Jadi luasnya ialah 8 x 13 juga.
04:58
Since we've correctly calculated the area
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Kita telah mengira luas
05:00
two different ways,
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dengan dua cara yang berbeza,
05:02
they have to be the same number,
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hasilnya mesti sama,
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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sebab itu kuasa dua kepada 1, 1, 2, 3, 5 dan 8,
05:08
add up to eight times 13.
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jumlahnya sama dengan 8 x 13.
05:10
Now, if we continue this process,
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Jika kita teruskan proses ini,
05:12
we'll generate rectangles of the form 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 check this out.
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Sekarang tengok ni.
05:20
If you divide 13 by eight,
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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 divide the larger number
by the smaller number,
by the smaller number,
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yang lebih besar dengan yang sebelumnya
05:28
then these ratios get closer and closer
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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
known to many people as the Golden Ratio,
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juga dikenali sebagai Nisbah Keemasan,
05:37
a number which has fascinated mathematicians,
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nombor yang mempesonakan ahli matematik,
05:39
scientists and artists for centuries.
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saintis dan seniman sejak dulu.
05:42
Now, I show all this to you because,
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Saya bentangkan semua ini kerana,
05:45
like so much of mathematics,
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seperti kebanyakan matematik,
05:47
there's a beautiful side to it
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ia mempunyai aspek yang menakjubkan
05:49
that I fear does not get enough attention
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yang sayangnya tak mendapat perhatian
05:51
in our schools.
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di sekolah-sekolah kita.
05:52
We spend lots of time learning about calculation,
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Banyak masa dihabiskan untuk belajar mengira,
05:55
but let's not forget about application,
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tetapi jangan lupa tentang aplikasinya
05:58
including, perhaps, the most
important application of all,
important application of all,
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termasuk aplikasi yang paling penting,
06:01
learning how to think.
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belajar cara berfikir.
06:03
If I could summarize this in one sentence,
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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
Mathematics is not just solving for x,
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Matematik bukan hanya untuk mencari x,
06:10
it's also figuring 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
(Applause)
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(Tepukan)
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