So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration.
Kenapa kita belajar matematik? Asasnya, kerana tiga sebab: pengiraan, aplikasi, pengiraan, aplikasi, dan yang terakhir, yang malangnya kurang diberikan perhatian, dan yang terakhir, yang malangnya kurang diberikan perhatian, inspirasi.
Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics that we learn in school is not effectively motivated, and when our students ask, "Why are we learning this?" then they often hear that they'll need it in an upcoming math class or on a future test. But wouldn't it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind? Now, I know many people have not had the opportunity to see how this can happen, so let me give you a quick example with my favorite collection of numbers, the Fibonacci numbers. (Applause)
Matematik merupakan sains corak. Kita belajar berfikir secara logik, kritikal dan kreatif, Kita belajar berfikir secara logik, kritikal dan kreatif, tetapi yang diajar di sekolah, tidak memberikan rangsangan yang baik. Apabila pelajar kita bertanya, "Kenapa kita belajar ni?" Ini penting untuk kelas yang berikutnya atau ujian yang akan datang. Bukankah lebih bagus kalau kadangkala kita membuat matematik kerana ia menyeronokkan, mengasyikkan atau kerana ia merangsang minda? Ramai yang tak berpeluang untuk memahami bagaimana ini boleh berlaku, jadi biar saya berikan contoh dengan koleksi nombor kegemaran saya, nombor Fibonacci. (Tepukan)
Yeah! I already have Fibonacci fans here. That's great.
Ya, ada peminat Fibonacci di sini. Bagus. Ya, ada peminat Fibonacci di sini. Bagus.
Now these numbers can be appreciated in many different ways. From the standpoint of calculation, they're as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on. Indeed, the person we call Fibonacci was actually named Leonardo of Pisa, and these numbers appear in his book "Liber Abaci," which taught the Western world the methods of arithmetic that we use today. In terms of applications, Fibonacci numbers appear in nature surprisingly often. The number of petals on a flower is typically a Fibonacci number, or the number of spirals on a sunflower or a pineapple tends to be a Fibonacci number as well.
Nombor-nombor ini boleh dihargai dalam berbagai-bagai cara. Dari sudut pengiraan, ia sangat senang difahami seperti 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, dan begitulah seterusnya. Orang yang dikenali sebagai Fibonacci sebenarnya bernama Leonardo of Pisa, nombor-nombor ini diterangkan dalam buku "Liber Abaci", di mana dunia Barat telah diajar kaedah aritmetik yang digunakan sekarang. Dari segi aplikasi, nombor Fibonacci selalu muncul dalam alam semula jadi. selalu muncul dalam alam semula jadi. Bilangan kelopak bunga selalunya ialah nombor Fibonacci, lingkaran bunga matahari atau nenas, lingkaran bunga matahari atau nenas, biasanya merupakan nombor Fibonacci.
In fact, there are many more applications of Fibonacci numbers, but what I find most inspirational about them are the beautiful number patterns they display. Let me show you one of my favorites. Suppose you like to square numbers, and frankly, who doesn't? (Laughter)
Banyak lagi aplikasi nombor Fibonacci, yang paling memberikan inspirasi ialah corak nombor yang dipaparkan. Ini salah satu kegemaran saya. Katakan anda suka nombor kuasa dua, siapa yang tak suka, kan? (Gelak ketawa)
Let's look at the squares of the first few Fibonacci numbers. So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on. Now, it's no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right? That's how they're created. But you wouldn't expect anything special to happen when you add the squares together. But check this out. One plus one gives us two, and one plus four gives us five. And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues.
Mari kita lihat nombor kuasa dua bagi nombor-nombor Fibonacci. 1 kuasa dua = 1, 2 kuasa dua = 4, 3 kuasa dua = 9, 5 kuasa dua = 25, dan seterusnya. Jadi, tak hairanlah apabila jumlah dua nombor Fibonacci yang berturut menghasilkan nombor Fibonacci yang berikutnya. Itu merupakan cara ia dicipta. Anda tak akan menjangkakan apa-apa jika nombor-nombor kuasa dua tersebut ditambah. Cuba tengok ni. 1 + 1 = 2, 1 + 4 = 5, 4 + 9 = 13, 9 + 25 = 34, dan corak itu berterusan.
In fact, here's another one. Suppose you wanted to look at adding the squares of the first few Fibonacci numbers. Let's see what we get there. So one plus one plus four is six. Add nine to that, we get 15. Add 25, we get 40. Add 64, we get 104. Now look at those numbers. Those are not Fibonacci numbers, but if you look at them closely, you'll see the Fibonacci numbers buried inside of them.
Ini satu lagi contoh. Katakan anda tambah beberapa nombor kuasa dua Fibonacci yang awal. Mari kita lihat apa hasilnya. 1 + 1 + 4 = 6. 6 + 9 = 15. 15 + 25 = 40. 40 + 64 = 104. Tengok nombor-nombor ini. Ia bukan nombor-nombor Fibonacci. Tetapi jika anda lihat dengan teliti, ada nombor Fibonacci yang tersembunyi di dalamnya.
Do you see it? I'll show it to you. Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate?
Nampak tak? Saya akan tunjukkan. 6 = 2 x 3, 15 = 3 x 5, 40 = 5 x 8, 2, 3, 5, 8, terima kasih kepada siapa?
(Laughter)
(Gelak ketawa)
Fibonacci! Of course.
Semestinya, Fibonacci!
Now, as much fun as it is to discover these patterns, it's even more satisfying to understand why they are true. Let's look at that last equation. Why should the squares of one, one, two, three, five and eight add up to eight times 13? I'll show you by drawing a simple picture. We'll start with a one-by-one square and next to that put another one-by-one square. Together, they form a one-by-two rectangle. Beneath that, I'll put a two-by-two square, and next to that, a three-by-three square, beneath that, a five-by-five square, and then an eight-by-eight square, creating one giant rectangle, right?
Corak ini memang menyeronokkan, tapi lebih memuaskan jika kita faham kenapa ia begitu. Cuba lihat persamaan yang terakhir. Kenapa kuasa dua kepada 1, 1, 2, 3, 5 dan 8 jumlahnya sama dengan 8 x 13? Saya akan lukiskan satu gambar. Ada satu segi empat 1 x 1, dan satu lagi segi empat 1 x 1. Hasilnya segi empat tepat 1 x 2. Letakkan segi empat 2 x 2 di bawah, dan segi empat 3 x 3 di sebelah, segi empat 5 x 5 di bawah, dan satu lagi segi empat 8 x 8, membentuk segi empat tepat yang besar, kan?
Now let me ask you a simple question: what is the area of the rectangle? Well, on the one hand, it's the sum of the areas of the squares inside it, right? Just as we created it. It's one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? That's the area. On the other hand, because it's a rectangle, the area is equal to its height times its base, and the height is clearly eight, and the base is five plus eight, which is the next Fibonacci number, 13. Right? So the area is also eight times 13. Since we've correctly calculated the area two different ways, they have to be the same number, and that's why the squares of one, one, two, three, five and eight add up to eight times 13.
Izinkan saya bertanya, berapakah luas segi empat tepat itu? Yang pertama, ia merupakan jumlah luas Yang pertama, ia merupakan jumlah luas semua segi empat di dalamnya, kan? Sama seperti yang kita buat tadi. 1 kuasa dua + 1 kuasa dua, + 2 kuasa dua, + 3 kuasa dua, + 5 kuasa dua, + 8 kuasa dua. Itu merupakan luasnya. Yang kedua, luas sebuah segi empat tepat, ialah tinggi x tapak, tinggi = 8, tapak = 5 + 8, iaitu 13, nombor Fibonacci yang berikutnya, kan? Jadi luasnya ialah 8 x 13 juga. Kita telah mengira luas dengan dua cara yang berbeza, hasilnya mesti sama, sebab itu kuasa dua kepada 1, 1, 2, 3, 5 dan 8, jumlahnya sama dengan 8 x 13.
Now, if we continue this process, we'll generate rectangles of the form 13 by 21, 21 by 34, and so on.
Jika kita teruskan proses ini, hasilnya ialah segi empat tepat 13 x 21, 21 x 34, dan seterusnya.
Now check this out. If you divide 13 by eight, you get 1.625. And if you divide the larger number by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians, scientists and artists for centuries.
Sekarang tengok ni. Jika anda bahagi 13 dengan 8, anda dapat 1.625. Bahagikan nombor yang lebih besar dengan yang sebelumnya nisbahnya akan semakin hampir dengan kira-kira 1.618, juga dikenali sebagai Nisbah Keemasan, nombor yang mempesonakan ahli matematik, saintis dan seniman sejak dulu.
Now, I show all this to you because, like so much of mathematics, there's a beautiful side to it that I fear does not get enough attention in our schools. We spend lots of time learning about calculation, but let's not forget about application, including, perhaps, the most important application of all, learning how to think.
Saya bentangkan semua ini kerana, seperti kebanyakan matematik, ia mempunyai aspek yang menakjubkan yang sayangnya tak mendapat perhatian di sekolah-sekolah kita. Banyak masa dihabiskan untuk belajar mengira, tetapi jangan lupa tentang aplikasinya termasuk aplikasi yang paling penting, belajar cara berfikir.
If I could summarize this in one sentence, it would be this: Mathematics is not just solving for x, it's also figuring out why.
Saya simpulkan dalam satu ayat: Saya simpulkan dalam satu ayat: Matematik bukan hanya untuk mencari x, tapi juga untuk mengetahui kenapa (why).
Thank you very much.
Terima kasih.
(Applause)
(Tepukan)