Archive for the ‘Math’ Category

Kenapa aku (dulu) tidak mau jadi dokter

2007 May 5

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Laborous questions in a test

2007 March 13

Why must instructors give a very “long” problem which doesn’t test understanding any better than a “shorter” problem?

Here’s an example problem to test the understanding of shift cipher:

Encrypt the plaintext “example” using the shift cipher with key B.

That problem should suffice. However here’s what some instructors like to give:

Encrypt the plaintext “iliketoseemystudentssufferhahahaiamevil” using the shift cipher with key P.

The second problem isn’t intellectually harder, it’s just more laborous!

I can forsee a similar agony in a microbiology test:

The nucleotide sequence of one DNA strand of a DNA double helix is:
-GGAGATCGCATGCATGCACAGCTGACGATGCA-
(dunno whether it is realistic, I just typed the ATGCs randomly)
What is the sequence of the complementary strand?

Isn’t a strand of -ATGC- enough?

PS: Oh and about that second example, it’s actually quite nice considering that my instructor gave a LONGER ciphertext to encrypt… Unbelievable…

Which would you choose? Learning a language in 1 day or 10 years?

2007 March 3

There are many random books here in Purwokerto, and some days ago I found two books with absurd titles.

First up is a book on my cousin’s bookcase: Bahasa JEPANG Hanya 1 Hari (Japanese in only 1 day) by Yan Tirtobisono. I was like “Heh!!!” when reading the title. I’ve learned this language on my spare time for around 2 years and still can’t do nothing :)… (well, yeah, probably because I’m not working hard enough)

The book’s 256 pages contain set phrases in roomaji without any grammar explanation. More like a travel pamphlet for me. I know that the Indonesian mentality is wanting results instantly without much effort, but, 1 day?!? Give me a break…

The other one is from my brother’s room: Korea dalam 10 x 365 hari (Korean in 10 x 365 days) by HS. Maru Lis. We all know that 10 x 365 days means 3650 days which means around 10 years!!! I was like “What the?!? This is a complete opposite of the other book!”.

I knew the author must meant something else, so I peeked at the contents. It turned out to be a 365-days lesson, with each lesson teaching 10 words. More like a dictionary for me. Anyway, the title is one of the most serious abuse of mathematical notation that I’ve encountered.

Welcome to WordPress, LaTeX…

2007 February 19

It is now possible to use LaTeX in WordPress! Two small tests:

\frac{dy}{dx} = \sin{x}

\sum_{k=1}^{n} k = \frac{(n)(n+1)}{2}

For more info about using it, read this post.

Substandard translations

2007 February 10

When lecturers give students writing homework, most students just Google for the topic and then translate some (English) web pages they found. However, ignorance of the topic or underutilization of common sense usually creates a translation that’s silly.

Last semester, I took “Introduction to the history and philosophy of mathematics”. The class is divided into groups and each group must make a paper about the topic they choose. Before the test, I managed to get all the group’s paper. One paper particularly stood out, so let’s try to reverse engineer it!

  • “Ini dapat dibuktikan dengan PENGENALAN” -> “This can be proved using INTRODUCTION”. Great, a new proving method has just been invented. It obviously should be “Ini dapat dibuktikan dengan INDUKSI” which in English is “This can be proved using INDUCTION“.
  • “Teorema PENGINGAT Cina” -> “Chinese REMINDER theorem”. As in, “the theorem that reminds you to eat and take a bath”. The correct translation should be “Teorema SISA Cina” which in English is “Chinese REMAINDER theorem“. A famous theorem in number theory.
  • “Seekor laba-laba memanjat DENGAN beberapa kaki di dinding…” -> “A spider climbs the wall USING some feet…”. Because the problem doesn’t ask for the number of feet the spider has (or uses), the translation that makes sense is “Seekor laba-laba memanjat SEJAUH beberapa kaki di dinding…” (“A spider climbs the wall FOR some feet…”).

It’s as bad as a Biology student translating “order” (in taxonomy) to “pesanan” (as in “pizza order”).

Substandard translation is also found in commercial translated English textbooks. Sometimes, it’s as if the translator didn’t check whether their sentence makes sense.

I’ve also found some funny translations in the Indonesian sub of movies containing techonological (e.g., “open source“), scientific (e.g., “string theory“), and mathematical (e.g., “group“) jargons. It caused some chuckles but sadly I don’t remember them. (IIRC it’s in, among others, “Antitrust”, “October Sky”, “A Beautiful Mind”, “Good Will Hunting”, and that movie about natural disaster where birds suddenly fall from the sky)

Paying tuition fees

2007 January 23

The procedure seems to change every semester. Once it was on Kinanti. Once it was in BNI. This time it’s in Bank Mandiri.

I arrived there around 10 AM and got number 56. I needed to wait until my number is called by the female worker. The currently called numbers were in the 200s, and simple inquiry revealed that the number will go to 500 before going back to 1. I was additionally told that mine should be called around an hour later (which means around 11 AM).

So I left to browse the web. I took note that I left at 10:18 and the last called number was 285. I arrived back at 10.56 and the last called number was 391.

I deliberately wrote all those info because I wanted to predict when I would be called. After finding a seat, I started to calculate.

The average rate of people processed during my absence was (391 – 285) people / (56 – 18) minutes, which is around 2.789 people/minute.

Let us assume that the processing rate is constant. My number is equivalent to 556, so I would be called around ((556 – 391) / 2.789) minutes later, which is around 59 minutes later. Because the time was 10:56, that means I will be called around 11:55.

I waited while practising writing some kanji. I got called at 11:57. Pretty close, eh? (compare with the 11 AM prediction which was pulled out of thin air)

Be careful in solving K-12 math problem: divison by zero

2006 December 23

When I passed “SD percobaan” this morning, I couldn’t resist the foods sold at the front gates (it was almost 12, and I was extremely hungry). So I went there and bought 2 lehers (lekers? forgot the name) which were put in a paper container.

The paper container was made from a used math book page, or a copy of it. I won’t talk about its hygiene but about a problem in it:

Simplify: a3 x a2 : a4

It was a “problems” page so there were no answers. However I could guess that the answer deemed “correct” by the book is a3+2-4 = a1 = a. That is also the answer that most, if not all teachers would give. However that’s wrong!

First, looking at other problems on the page, they use negatives (-13, -8, etc.) and fractionals (3/2, 4/9, etc.). So it is safe to say that the variable a is a rational number (as opposed to, i.e., a natural number). Now, remember that the set of rational numbers contains 0, and this is where we need to be careful.

If a is nonzero, the expression can be simplified to a3+2-4 = a. However if a is zero, division by a4 = 0 cannot be done so the result is undefined. Therefore the correct answer is:

a, if a is nonzero
undefined, if a is zero

Saying that the answer is unconditionally a is the same as saying that 03 x 02 : 04 = 0 which is of course wrong.

As a tangential side note, I once encountered a multiple-choice national (UAN) math test question about integrals. Because the problem question maker overlooked division by zero, none of the answers were correct. A math question then becomes a moral question: Is it a sin knowing that an answer is wrong, but answer it anyway because the question maker would probably regard it as correct? Indeed, ignorance is bliss.

I’m a computer scientist, not your tech support

2006 September 27

Problems worthy of attack
Prove their worth by fighting back
    – Piet Hein

A Princeton computer science major writes about his/her annoyance of being asked to fix computer problems.

What are problems in the field called “Computer Science” anyway? It’s NOT about how to make a web site. It’s NOT about fixing a machine that won’t boot up. It’s NOT about getting rid of worms investing your computer. It’s NOT about making the best hardware purchase.

Wikipedia has a list of unsolved computer science problems. It includes the famous P=NP. There is also a similar, significantly longer, list for mathematics and physics. This probably reflects the relatively young age of computer science. (those thinking mathematics is finished is dead wrong; the amount of mathematical research activity is in fact getting much bigger from time to time)

Anyway, does the reverse happen? (which means, people enrolling CS expecting to be taught about fixing computer problems)

Kanji as a form of data compression

2006 September 24

Using kanji, many ideas can be expressed using just a few characters. For example, here’s how we write the 12 months in various ways:

Kanji Hiragana Roomaji English Indonesian
一月 いちがつ ichigatsu January Januari
二月 にがつ nigatsu February Februari
三月 さんがつ sangatsu March Maret
四月 しがつ shigatsu April April
五月 ごがつ gogatsu May Mei
六月 ろくがつ rokugatsu June Juni
七月 しちがつ shichigatsu July Juli
八月 はちがつ hachigatsu August Agustus
九月 くがつ kugatsu September September
十月 じゅうがつ juugatsu October Oktober
十一月 じゅういちがつ juuichigatsu November November
十二月 じゅうにがつ juunigatsu December Desember
Average
character
2.17 4.17 8.83 6.17 6.25

Note that the average character count drops from roomaji to hiragana. That is expected, since each hiragana symbol expresses the idea of mora which for this discussion can be regarded as a syllable. If we use roomaji, most syllables must be written using two or more characters. Therefore hiragana can be thought to compress roomaji. As a character, hiragana is more high level than roomaji.

The average character count drops again when we go from hiragana to kanji. Kanji is even more high level than hiragana. Each kanji expresses a certain idea. Because most kanji expands to more than one character when written using hiragana, kanji can be thought to compress hiragana.

I’ve heard people say, “kanji is sooo ancient. They should abolish it and replace it with something simpler and modern like the latin alphabet.” It eventually boils down to the unwillingness to memorize lots of high level symbols.

However, kanji is a form of pictogram. What they don’t realize is they also use some pictograms. Ever saw 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? Great, let’s abolish them. Then we can all have fun writing “sixty five thousand five hundred thirty six” or “enam puluh lima ribu lima ratus tiga puluh enam”.

Anyway, it is natural to ask, “can we define even more higher level elements?”. I don’t see that happening in natural language, but there is one language in which simpler concepts (encoded in symbols) are used to consecutively build more complex ones: mathematics.

In modern mathematics, everything starts with the set theory. There we see symbols like “{“, “}”, “,”, and “⊆”. From sets, we can define things such as the natural number, and naturally (no pun intended) new symbols like “1” and “0” appear.

Going even higher level, there is calculus in which symbols like “∫” appears. Calculus is very high level so that using vector calculus, all electromagnetic phenomena can be written in only four equations (the so-called “Maxwell’s Equations“).

I think it is astonishing that using the more high-level symbols in Clifford Algebra, the Maxwell’s Equations can be written in only one equation.