# My first book

On one rainy morning in November,
just a few days shy of your birthday,
you came to see me.

We have only known each other for several months,
but it seemed like we have been friends
for the longest time.

a book that was partially soaking wet,
and then placed it in my hand.

You told me to read it.
I laughed.
You knew I couldn’t read well.

I gave it back to you.

I couldn’t remember what you said to persuade me,
it was million years ago,
but you won. I promised you to read it.

You smiled ever so beautifully,
my attempt not to smile back failed miserably.
You told me you’d come back to make sure I read it.

You never did.

I never opened the book.
I buried it deep inside my closet,
because the sight of it reminded me of you.

But I made you a promise.
It takes exactly two years, three months, and seventeen days
to finally let you go.

Today, I want you to know,
although I believe you were watching me the whole time
from up there,

that I have finally finished reading my first book.

# The Power of Simple Things

Well it’s been ages since the last time I updated this blog. To be honest, I did want to write something, only if I haven’t been too busy sleeping. I am currently just doing research, reading literatures, and singing in the choir. Apparently doing research is really time-consuming, since I need to read many literatures and understand them before proceeding with the real research. Also the material I am currently studying is different with what I learned back in ITB, so I need to give more of my time to do the self-learning.

Speaking about reading literatures, I am currently self-learning one particular structure in mathematics called semisimple algebra. I won’t bother you with all the details, but it is an interesting object to study with, though. In short, an algebra $A$ is a ring which is also a vector space (so we can multiply vector with vector as well as scalar with vector). Now consider a module $U$ over $A$. Modules are basically just like vector space, where now the scalar set is no longer a field but a ring$. We call $U$ as an $A-$module. Now $U$ is called simple if its only submodules are the zero submodule and itself. We call $U$ semisimple if $U$ can be written as a direct sum of simple modules. Finally, we called an algebra $A$ to be semisimple if every non-zero module over $A$ is semisimple. What I am currently investigating is the special case when $A$ is actually a group algebra $\mathbb{C}G$ of $\mathbb{C}-$linear combination of elements in $G$, where $G$ is any finite group. Phew. That was long enough. Algebra is not really my forte, though. (I do miss integral and its friends!!) However, after spending so much time reading this particular thing, I admit I become more fascinated with algebra. To get the idea of this subject, I must have a strong understanding about group theory, but in a fancy level. At first I was quite skeptical about the idea of ‘fancy’ group theory. I didn’t have much experience dealing with groups. I did once learn it back then in my Algebra course in ITB, but what could be made complicated from just a set equipped with one single well-defined operation satisfying associativity, existence of the identity element, and existence of inverse elements? Turns out I was wrong. Like, totally wrong. Up until now I am still amazed by how rich group theory is. I read so many new things I haven’t even heard before, like derived groups, composition series, chief series, solvability; not to mention things I have already heard but I have no idea what they are, e.g. Sylow theorems, general linear groups, group action, representation theory (which has a deep connection with semisimple algebras and module theory). That is just group theory. I haven’t talked about another more complex structures which I believe to possess a richer theory. Before you stop reading because of what you just read, bear with me a little bit longer. What I really want to say from that rather abstract concepts which I believe not all of you would bother to understand is that we shouldn’t underestimate the power of simple things. Simple things can turn out to be not-so-simple and in fact they can probably involve a very serious and deeper understanding to master. Simple things are also the basic building blocks to grasp the more complicated ideas, so speaking that way, they are important. Why do I use such a nonconcrete analogue just to explain that? Well probably because I do feel impressed with all this learning process and I don’t want to wait any longer to write something LOL. Hopefully it won’t take too long time for me to update this blog after this post. Anyway, thanks for reading! # Tentang Kedewasaan Momen ulang tahun kemarin membuat saya berpikir cukup panjang mengenai kedewasaan. Saya sudah 22 tahun sekarang dan seperti harapan-harapan pada umumnya, saya diharapkan untuk makin dewasa. Nah, ini yang membuat saya berpikir. Kedewasaan itu apa ya? Apa yang saya lakukan setahun kemarin belum dewasa? Kalau begitu, kadar kedewasaan itu apa? Jujur, saya tidak tahu apa itu kedewasaan. Bersikap seperti orang dewasa? Sikap yang bagaimana? Orang dewasa juga bisa berbuat salah. Lha, banyak kasus kriminal kan juga rata-rata pelakunya orang dewasa. Jelas kalau berbuat kriminal itu bukan sesuatu yang baik, jadi tidak mungkin dijadikan teladan. Lalu sikap yang seperti apa? Ini membuat saya bingung. Saya cukup lama merenung sampai akhirnya sampai di suatu kesimpulan. Saya tidak perlu capek-capek memikirkan apa itu kedewasaan. Saya tidak perlu tahu juga apa kadar kedewasaan, karena itu bukan sesuatu yang saklek. Yang saya lakukan adalah berusaha menjalani hari ini lebih baik daripada kemarin. No thinking too much about the future. Ya merencanakan masa depan itu boleh, bagus malah. Tapi terlalu terbenam dalamnya, saya rasa tidak baik. Saya berprinsip, lakukan pekerjaan hari ini dengan baik dari kemarin. Kesusahan sehari cukup untuk sehari kok, jadi sebaiknya tidak dibawa-bawa lagi besoknya. Terdengar mudah, namun sulit. Kenapa? Karena.. You know, human. They like to overcomplicate things. Saya pun begitu. Ini yang jadi salah satu resolusi saya untuk tahun ini. Kalau dipikir-pikir lagi.. Saya malah harusnya meniru anak kecil. Mereka selalu bermain, penuh rasa ingin tahu, dan iman mereka sangatlah murni. Tuhan saja berkata kalau Kerajaan Allah itu milik anak-anak kecil (Markus 10:13-16). Ya, kadang dalam beberapa hal, menjadi ‘dewasa’ itu berarti menjadi seperti anak kecil. Membingungkan, bukan? Makanya saya putuskan tidak usah dipikir. Cukup jalani saja hidup seperti hari ini adalah hari terakhir kita hidup. Nah, menjalaninya tentu tidak asal-asalan. Jalani seperti yang Tuhan kehendaki. Sepertinya itu yang memang harus kita lakukan, bukan? Nah, kalau sekarang saya diminta mendefinisikan kedewasaan, saya akan jawab begini. Kedewasaan adalah suatu momen hidup dimana kita bisa menikmati Tuhan seutuhnya. Menjadi dewasa tidak sama dengan menjadi tua. Semakin tua tidak menjamin makin dewasa, meskipun semakin kita dewasa, pastilah kita makin tua 😛 Have a nice day, all! # Farewell, Bandung. For me, farewell is scary. When someone leaves, there is a slight possibility that you can never see them again. When you leave and somehow you make it to come back, no guarantee everything will stay the same as before. A little fact: it won’t. I am not a pro at handling farewell. Here’s a little story for you: just last week I left Bandung. My precious four and a half years life in Bandung sure has been very memorable. I met inspiring people there. I met new families there. I gained my precious best friends there. I learned a lot about myself there. I learned a lot about useful life skills there. I graduated there. I did what I like there. So many things to tell, I am getting confused which one I should mention. I was so sad when my train finally departed. I was so sad when I had to say goodbye to my remaining friends in Bandung. The moment I realised that I will leave Bandung I knew that I will miss this city very much. No, more importantly, I will miss those who have made me miss this city so much. When I’m back later in the future, I know I shouldn’t hope to meet my friends there. They will probably be scattered all around the world by then. Bandung will still be lovable, but it certainly will be different. Well, I guess it is the people in it that are more valuable to me. I may not be a member of HIMATIKA ITB anymore, but I still come to it when I have spare time. I may not be a member of PSM-ITB anymore, but I still come to FPS-IICC last week. It was because of the people in it. I might forget what I did back then when I was still an active member (in fact, I already forget it by now) but the moment I spent with my friends certainly cannot be forgotten. In fact, I still can laugh if I remember those good moments. How do you handle farewell? For me.. I think time will heal. I am still in the process of making myself occupied. I do hope that I can accept the fact that I have left Bandung to embrace my awaiting new life ahead. I will make new friends there, I will create memorable moments with them there, i will learn a lot about my calling there, so when my time there has ended, I have many good moments to be cherished. I think I have learned a new lesson: appreciate everything when you have time to do it. You cannot stay in your current place forever. There must be a time when you have to move, out of your comfort zone. When everything seems to be taken away from you, you will know that the moment you create will still remain in your heart. Your friends may leave to pursue their own dreams, but the moments spent with them will always be in your mind. Yes, farewell is still scary for me. Some people can really be good at handling farewell. But I am not one of them. Life is somehow fair and unfair. A meeting shall be followed by a farewell. Balance. Action – reaction. Newton’s third law. Well, the unfair part is that I have to go through a painful way to deal with it. I am still coping with that, though. But that doesn’t mean that I won’t meet anyone else ever. I am still going to meet new friends and hopefully when the moment to say Adieu comes again, I won’t be as scared as I am now 🙂 To everyone who knows me and live (or used to live) in Bandung, thank you for the moment I can spend with you. # What’s New? I’ve been away for such long time. Everytime I want to write something or when a bunch of ideas popped up, either the timing is not good or I have been procrastinating :p I have been offered to study at UNSW starting from next March. I will take mathematics again (well since I am so faithful), only this time I will undergo a full research instead of taking courses. Sometimes it is called master by research. For two years I will do a research about harmonic analysis on Lie group. For me, that is interesting! This semester has been quite memorable and different than previous semesters. As a fresh graduate, it means that I do not have to attend class anymore, which is.. quite weird. Sometimes I really want to attend classes, take the exam, and get graded. Although as a lecturer assistant I could get a chance to attend class informally, it still felt different. Next difference is that I rarely see my friends now. Most of my friends have been graduated and now they have chosen their own path. Most of them are working now. Some of them choose to continue to master study like me. I still meet some of my friends from my batch, sometimes. It is kind of relieving. Being surrounded by juniors makes you feel so old, you know :p Another difference is that this time I have a chance to be a tutor. You know, in ITB there are Calculus course for first year students. Since they are so many (around 3000 of them), almost all of the lecturer from Department of Mathematics have to teach Calculus. Well, there is 2-hour session per week for tutorial. My job was to assist those first year students to understand the concepts by doing some practices. I even had a chance to teach as a substitute. I have been accustomed to teach informally, but to teach formally in front of 60-70 students is of course something different. Luckily I managed to do it although there may be some minor flaws. Well, since this semester almost ends, it means that I have to move out from Bandung nearly soon. Maybe at the end of this month, maybe the month after. Since I have to start my master study on March, it means I have to depart to Sydney on February. Time does fly. I have to take care of my student visa, prepare the ticket to Sydney, and of course search for apartments in Sydney. Sydney is a very expensive city. The standard rate for student apartment in Sydney is around$250-300 per week, which is around 2.5-3 million rupiahs. It is per week, note that. I have to learn to spend my money wisely :p

That’s about it. Now if you’ll excuse me, I have a Doraemon movie to be watched with my friends at 9.30 pm. Cannot wait to watch it! 😀

# Hampir Tidak Ada Bilangan Rasional, Tapi…

Sebelum mulai, saya mau minta kalian pilih satu buah bilangan apa saja yang terletak di antara 0 dan 1 (ini berarti 0 dan 1 tidak boleh dipilih ya). Kalau sudah, diingat-ingat saja. Gak akan saya apa-apain kok.

Kenapa saya minta seperti itu? Kalau saya minta sebutkan satu bilangan di antara 0 dan 1, pasti kebanyakan akan jawab $\frac{1}{2}$. Mungkin ada beberapa yang kreatif jawabnya $\frac{5}{6}$. Lebih kreatif lagi kalau jawabnya $\frac{354566}{583948}$. Tapi jauh lebih kreatif orang yang jawab $\frac{1}{2}\sqrt{2}$ atau $\frac{\pi}{4}$ atau $\log(2)$. Nah, seberapa kreatifkah kalian? 😛

Faktanya, hampir setiap orang yang diminta untuk berikan satu angka di antara 0 dan 1, pasti akan memberikan jawaban berupa bilangan rasional. Mungkin karena bilangan-bilangan itu yang dekat dengan mereka. Mungkin cuma anak matematika yang akan jawab $\frac{\pi}{4}$ atau $\ln(2)$, karena mereka lebih sering berurusan dengan bilangan-bilangan semacam itu ketimbang anak-anak lainnya.

Sekarang, saya mau berikan suatu fakta yang mungkin agak ironis: di kumpulan bilangan real, hampir tidak ada bilangan rasional. Artinya, jumlah bilangan irasional jauh, jauh, jauh lebih banyak daripada bilangan rasional. Lebih mengejutkan lagi, dengan menganggap semua bilangan di antara 0 dan 1 terdistribusi secara merata, kalau kita harus mengambil satu angka dari antara 0 dan 1, katakanlah $a \in (0,1)$, maka peluang bahwa $a$ bilangan rasional adalah 0. Dengan kata lain, peluang bahwa $a$ bilangan irasional adalah 1.

Buktinya tidaklah sederhana. Bahkan tampak aneh dan tidak wajar ya? Jelas-jelas dari interval $(0,1)$ bilangan rasional ada banyak. Ambil saja $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots$ yang belum lagi ditambah $\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots$. Itu baru yang bentuknya $\frac{1}{n}$, belum ditambah dengan yang bentuknya lebih ‘tidak karuan’ seperti $\frac{11}{5476}$ dan lain sebagainya. Mau ditulis semua juga tidak mungkin saking banyaknya, kan? Bagaimana mungkin kalau kita mengambil satu bilangan dari antara 0 dan 1, peluang terambil bilangan rasional adalah 0?

Penjelasan di bawah ini sudah masuk ke ranah yang lebih matematis. Coba dibaca saja, tidak harus terlalu dipahami. Buat yang memang tidak bisa paham, percaya sajalah :D. Mungkin bisa langsung lompat ke dua paragraf terakhir.

Untuk menjelaskan ini, kita butuh suatu konsep yang disebut ukuran (measure). Secara kasar, namanya juga ukuran, pasti ia dibutuhkan untuk memberikan gambaran seberapa besar suatu objek. Nah dalam konteks ini, predikat ukuran melekat pada himpunan. Jadi, kita bisa mengukur seberapa besar suatu himpunan.

Secara intuitif, kalau kita punya selang $I = (0,1)$, ukuran himpunan ini adalah panjangnya kan? Jadi ukurannya 1. Sekarang, kalau kita punya persegi panjang di bidang berdimensi dua $R = \{(x,y) \in \mathbb{R}^2 \mid 0 \le x \le 2, 0 \le y \le 3\}$, ukuran himpunan ini secara intuitif adalah luasnya, yaitu 6. Kalau kita punya kubus di ruang berdimensi tiga $Q = \{(x,y,z) \in \mathbb{R}^3 \mid 0 \le x \le 2, 0 \le y \le 2, 0 \le z \le 2\}$, ukuran himpunan ini secara intuitif adalah volumenya, yaitu 8. Ukuran memang dipakai untuk memperumum konsep panjang (di ruang berdimensi satu), luas (di ruang berdimensi dua), dan volume (di ruang berdimensi tiga). Memperumum di sini maksudnya apa? Maksudnya adalah kalau kita ingin bekerja di ruang berdimensi $n$, kita tetap bisa punya suatu alat untuk mengukur seberapa besar suatu himpunan.

Ada banyak jenis ukuran, namun yang populer salah satunya adalah ukuran Lebesgue. Sekarang, sedikit lebih formal. Misalkan $m$ adalah ukuran Lebesgue, maka $m$ adalah suatu pemetaan yang membawa himpunan tak kosong ke suatu bilangan real non-negatif yang diperluas $\mathbb{R}_+^{\star} = \mathbb{R} \cup \{\infty\}$, yang kelak dinamakan ukuran himpunan ini. Dari penjelasan sebelumnya, kita memperbolehkan ukuran suatu himpunan tak berhingga.

Sekarang, mari bicara dalam domain bilangan real $\mathbb{R}$ saja. Jika $I = (a,b)$, definisikan $m(I) = b-a$. Sekarang, bagaimana untuk kelas himpunan yang lebih besar? Sekarang, coba pandang himpunan $(0,1) \cup (2,3)$. Masuk akal kan kalau kita mendefinisikan $m((0,1) \cup (2,3)) = m((0,1)) + m((2,3)) = 1 + 1 = 2$? Ini sudah membuat lebih banyak himpunan bisa diukur. Sekarang, berhingga banyaknya himpunan-himpunan yang saling disjoin (yaitu tidak memiliki irisan) bisa dihitung ukurannya.

Secara umum, jika $a_1 < b_1 < a_2 < b_2 < \ldots < a_n < b_n$, maka

$m\left(\displaystyle\bigcup_{i = 1}^n (a_i, b_i)\right) = \displaystyle\sum_{i = 1}^n m((a_i, b_i))$.

Bahkan, ini kita bisa perumum lagi untuk kasus terhitung banyaknya himpunan-himpunan yang saling disjoin, yaitu jika $a_1 < b_1 < a_2 < b_2 < \ldots < a_n < b_n < \ldots$, maka

$m\left(\displaystyle\bigcup_{i = 1}^{\infty} (a_i, b_i)\right) = \displaystyle\sum_{i = 1}^{\infty} m((a_i, b_i))$.

Ekspresi terakhir tidaklah masalah karena ukuran suatu himpunan boleh tak berhingga.

Sekarang, coba kita formalkan. Jika $E$ suatu himpunan terukur (yaitu yang ukurannya terdefinisi dengan baik), maka

1. $m(E) \ge 0$.
2. Jika $E_1, E_2, \ldots$ adalah subhimpunan dari $E$ yang saling disjoin, maka $m\left(\displaystyle\bigcup_{i = 1}^{\infty} E_i\right) = \displaystyle\sum_{i = 1}^{\infty} m(E_i)$.

Dari dua sifat di atas kita bisa dapatkan fakta ini.

Fakta: Jika $A$ dan $B$ adalah dua himpunan terukur dan $A \subset B$, maka $m(A) \le m(B)$. Ini bisa didapat dengan menggunakan fakta bahwa $B = A \cup (B\setminus A)$, dan karena $A$ dan $B\setminus A$ adalah dua himpunan yang saling disjoin, maka $m(B) = m(A) + m(B\setminus A) \ge m(A)$.

Sekarang, kita ingin menghitung ukuran suatu singleton, yaitu himpunan yang terdiri dari satu anggota. Misalkan $A = \{a\}$ dan $B = (a - \frac{\epsilon}{2}, a + \frac{\epsilon}{2})$, untuk suatu $\epsilon > 0$. Jelas $A \subset B$ dan kita tahu bahwa $m(B) = \epsilon$.

Karenanya, $0 \le m(A) \le m(B) = \epsilon$. Tapi kita bisa pilih $\epsilon$ yang sekecil-kecilnya, sehingga ini memberikan $m(A) = 0$. Jadi ukuran dari suatu himpunan singleton adalah 0.

Sekarang, perhatikan himpunan bilangan rasional $\mathbb{Q}$. Karena $\mathbb{Q}$ terhitung, maka kita bisa enumerasi $\mathbb{Q} = \{q_i \mid i \in \mathbb{N}\}$. Karenanya,

$m(\mathbb{Q}) = m\left(\displaystyle\bigcup_{i = 1}^{\infty} \{q_i\}\right) = \displaystyle\sum_{i = 1}^{\infty} m(\{q_i\}) = \displaystyle\sum_{i=1}^{\infty} 0 = 0$.

Jadi, ukuran bilangan rasional adalah nol! Kalau kita batasi diri bekerja pada bilangan rasional yang terletak di antara 0 dan 1, yaitu $\mathbb{Q} \cap (0,1)$, maka dengan menggunakan fakta di atas bisa didapatkan $m(\mathbb{Q} \cap (0,1)) = 0$.

Apa artinya ukuran suatu himpunan 0? Mengingat bahwa ukuran adalah perumuman konsep panjang, luas, atau volume, kita bisa bilang kalau himpunan yang berukuran 0 pada dasarnya dapat diabaikan; hampir tidak ada. Kenapa hampir? Ya karena sebenarnya mereka ada. Kita bekerja dengan bilangan rasional dari SD, kok. 😛

Kembali ke permasalahan di atas. Dari penjelasan secara matematis di atas (semoga tidak terlalu sulit untuk dipahami ya 😦 ) dapat disimpulkan bahwa bilangan rasional hampir tidak ada. Kalau kita kaitkan dengan peluang (yang sebenarnya juga adalah ukuran), kita bisa bilang bahwa peluang mengambil bilangan rasional dari antara 0 dan 1 adalah 0, karena ukuran bilangan rasional adalah 0. Artinya, kalau kita harus mengambil satu bilangan yang terdistribusi merata dari antara 0 dan 1, maka pasti bilangan tersebut adalah bilangan irasional.

Mengingat hampir setiap orang yang diminta untuk mengambil suatu bilangan dari antara 0 dan 1 akan menjawab dengan bilangan rasional, maka sebenarnya ini sangat ironis, kan?

# Solusi Kuis 1 MA1101 K01

Untuk adik-adik peserta MA1101 Kelas 01, ini solusi Kuis 1 kalian. Untuk adik-adik peserta MA1101 lainnya, ini bisa digunakan sebagai bahan belajar. Enjoy! 😀

Solusi Kuis 1