Aftermathe

Christmas is over. Thanks to everyone who came. I hope you had a good time.

Finally completed the game the legend of Zelda : twilight princess after 60 hours of gameplay.
Fantastic game even though its a gamecube port. rating : 9.5/10

I realized it has been a very long time since i did some proper maths... Today lets talk about one of the most important aspects of modern mathematics, and indeed all of mathematics, isomorphism.

The concept of isomorphism is simple. When we add up 30 oranges with 20 oranges, we do not really add up all the oranges, rather we assign the number 20 to the 20 oranges and the number 30 to the 30 oranges and add up 20 and 30 to get 50 and thus our result would be 50 oranges.

It applies internally in mathematics as well.

consider the pair of numbers (x,y)

where x is the integers (mod 2), and
where y is the integers (mod 3).

if we list out all (x,y), they will look like this.

(0,0)

(1,1)

(0,2)

(1,0)

(0,1)

(1,2) ............. and so on.

notice that the elements repeat themselves after the sixth one.

If we assign each element to a number like this, where the right hand side is the integers (mod 6),

(0,0) -> 0

(1,1) -> 1

(0,2) -> 2

(1,0) -> 3

(0,1) -> 4

(1,2) -> 5

then something fun happens. Realize that if we add the sixth element (1,2) and the fourth element (1,0) we would get (2,2) but 2 (mod 2) is simply 0. s0 (1,2)+(1,0)=(0,2)

note that (1,2) is assigned to 5 and (1,0) is assigned to 3. If we know that the two columms are isomorphic, we can just add 3 to 5 to get 8, which is simply 2 under mod 6. and the element assigned to 2 is just the answer, (0,2).

Using the concept of isomorphism, areas of mathematics that are new can be "mapped" into areas of mathematics that are well known and mathematicians can use all the developed tools of this familiar area of mathematics to attack the new ones.

Think about geometry and algebra, in algebra it is a fairly tedious thing to solve a cubic equation, however if we port the problem over to geometry, we can find the roots simply by plotting the graph of the cubic and measuring the distance from the origin. Admittedly when we draw it the answer is never perfect, but in the ideal world of mathematics, that distance is exactly the root.

Retrospect

KK this christmas thursday at my house. 3pm onwards.

Present exchanging ceremony
1) 10 bucks and above
2) cannot be something that can be eaten
3) cannot be a book.. magazine.. etc..
4) cannot be a decoration item

haha i think it'll be fun

Damm i was watching this video of me playing pool bout two years ago... Din realise i was so cool back then. and shuai also k.

Mathematique

Running out of blog content. I assure you that I'll have something to write about by this weekend..

For the time being, I've decided to try my hand at poem writing.. Seems like a good idea cause Shermin always does it and I am a literature student after all :)

And i decided that the theme shall be mathematics since its challenging..

History of mathematics (part I)

Early thinkers seemed to know
that two from two would make zero
regardless of what they were

They used this knowledge to run their fields
To keep track of what was in the mill
They realized it was a useful skill
To know the numbers from nil

Then came the famous brotherhood
Who said that all numbers were good
But when they found irrationality
They threw a member into the sea

This anti-number number
Made all of them astounded
It destroyed the simple ideals
In which the sect was founded.

The world thus turned from number
Afraid of the irrational monster
And a thousand years thereafter
there were to be no maths on paper

Instead they focused all their minds
On points and squares, circles and lines
And it all seemed to work just fine
Till people started to repine

Then came descartes' x and y's
That linked numbers with points and lines
And showed all of the other guys
That Algebra is Geometry

Thus from then they knew the world
Was simply never rational
And any form of b/c
Is simply not to be

But that was only the beginning
Theorems and formulas were awaiting
For mathematicians to spend the evening
To discover, and to invent



Fine fine i know my poem lacks emotion and atmosphere. So what its factual (I hope)
and simple and easy to understand. I will continue this part I... better write it down..

To be continued.....

Xmas

Friday:

Teck Siang was supposed to come to HQ to play pool. In the end never come. So I stayed at home and watched tv and played pool the whole day.

Saturday:

Kenneth came over and played Zelda again. Very fun. Collected all the tears and gained a new suit. Also defeated a water bug with the help of a walkthrough. People who snigger at this have never played zelda before. After that about 8 went to pool fusion to play pool with daniel and kenneth. Played until like 5am..

Total bill : $73
Each pax pays : ~$24
Total games of nine ball played : 80+

Crazy pool night.




Sunday(which was yesterday)

Went to the esplanade with sm darren and daniel and yunhui to watch some chinese orchestra concert which sucked.
Lack of impact, lack of harmony and lack of memorable pieces. Anyway i took a cab there in order not to be late, but in the end they came like half and hour late and didn't even bother to call me when they reached, which pissed me off a lot.

Daniel kept refusing to pay me the six bucks he owed me and was very sore bout losing to me the night before, calling it beginners luck and that he could thrash me and stuff. I would love to see him beat me someday in pool, but the day is not today.

Afterwards they decided to go watch twilight which i refused. And i didn't want to waste my money and time watching a movie which i knew wouldn't interest me, pluss i was feeling pissed, so i deserted them and called my cousin out to watch the movie TDTESS, which was nice. Keanu Reaves is dam cool.

But it was quite an unhappy day for me.

Face it Daniel, you can't beat me yet.

Fun

Decided to do something on Shermin's blog cause must make up for the long absence.

Rules:

* Grab the book nearest you. Right now.

* Turn to page 57.

* Find the fifth sentence.

* Post that sentence along with these instructions in a note to your wall, and post your sentence in a comment here as well. Include the title of book and author.

"4.24. When a,b,c are elements of Z, prove a*(b - c) = a*b - a*c."

Abstract Algebra - Lloyd R. Jaisingh, Ph.D. & Frank Ayres, JR., Ph.D.

-.-'' Lemme try another book..

"Hey- he licked me!" Kermit exclaimed. "I think he licked me."

Goosebumps-Monster Blood IV
R.L. Stine

Lol...

closed

Kay lets see what happened...

Friday:
Went with soomin, darren, huimin and shixuan and yuan hong and suyi to east coast to cycle. Finally fulfilled my year-long wish haha. And no I'm not saying whats that. After cycling for bout two hours went to parkway parade to do nothing for a while then when to [the pool spot] to play pool. Quite an okay day. Oh yeah met jiayan at east coast too.

Sunday:
Went to Sam house for the 4E1,07 gathering to get my O level cert from Sin. Really a big failure. Thought there would be some activities but in the end we were just locked in a empty room with nothing except chairs and tables.. And nothing to eat but pizza and drumsticks.. And we had to pay ten bucks! Fuck it. A single pizza doesn't cost ten bucks. Good to know theres a worse organiser than me XD. Oh well but still, thanks for letting us go to your house anyway.

Imran came, expected a lot, disapointed in 10 minutes and had to go home in like half an hour. Feel damm sad for him. Fuck the fuckers who decided to desert him! And yes that includes me as well. I dam regret la, had the urge to alight the bus to go meet back and meet him.

Idris looks better than before. Wonder how her maths is?

Then after the gathering followed darren to tpy central to do nothing. Then dam bu gan yuan that the day will end so eventfuless, called kenneth to go Safra and play pool. Which he readily agreed haha.. So we played pool till like 2.30am I think i played quite well. developed a new stroke.

Monday:
Kenneth came to my house to play Zelda. Damm fun. Gordon mines. Leaping onto precarious sections of rock. Solving crazy puzzles. Collecting keys from wrestling elders. And battling FYRUS. After that it was jumping off burning bridges and flying dark dragons. Day well spent.

Tuesday:
Vincent and Soomin came to my house to play pool for a while. Then played a bit of big two and taught Vincent how to play mahjong. Then after that was GTA 4 and Super mario galaxy. Vincents dam good managed to defeat the boss in like a few tries. Not bad for someone new to the game. Then i follwed them to TPY to eat late dinner and asked my dad to fetch me home after. Quite a senseless detour....

And thats it. See so many things to do where got time to write.

And the set N of natural numbers and the set pZ of positive numbers are isomorphic. I dun get it.. always thought they were equal.. lol... apparently the natural numbers are defined as like {1,1*,(1*)*,....} where n* means the succesor of n defined by n+1... I guess the elements are different... hmm...

Pooling..

Tuesday:

Went to Kovan to meet Darren and Yong Siong. Yuan Hong was there too.. Played some pool, the pool parlour has changed hands, the tables are very good now. A far cry from when Supercue managed it. As usual I thrashed everyone. Boring.. But nice to just hang out la..

Here is Darren missing all his shots. And me clearing the rest of course.

But his stroke and stance looks good.. Remember- he's from TPA.

So we decided to play bowling after that since it was only downstairs. In the second game I decided to make a bet with Darren, the winner would get a free drink. So we bowled and I scored something like 130+ but Darren the god scored 170. But the third round came and i felt slightly more on form, So I decided to hustle Darren and up the bet. If he won, I would have to buy him a bowl of noodles and if I won there would be no free drink for him.



In the end i scored 140 and Darren losered at 123 hahahahaha.. No drink. Actually i knew he would lose cause i could see that he had no form after this chiobu came and played next to us. XD maybe i should have really gone for it and bet that if i won he would have to treat me instead... Ah well.. Hustle another day.

A shot of YH after a thousand dodges.

Yong Siong's hair looks like Chameleon. He looks even more yong-siongy than before lol... See that was the chiobu who came to play next to us. I swear I didn't purposely take.. She just happened to be in the pic :)

Wednesday:
Went out with tzeyon, my cousin to parklane to play pool... He was late for a fucking 3 hours! Dammit... I was praticing pool all alone and ate a plate of chicken rice. Luckily, he made up for it by bringing some not-too-bad looking girls along. He then decided to stay overnight at my house.. We played rollercoaster tycoon 3.. haha... fun game.

Him playing DJ max Classsic just now.

After the pool on wednesday, we went to mac... Yes I know how much i said i hated mac but... They all decided to go mac so i had no choice. Anyway I was curios and ordered the mega mcspicy cause tzeyong said he was going to treat me as an apology for being late. So i just chose the most ex one!

Just look at that man. Just carrying the burger up and the Chicken oil spills out onto the box. Eat this every other day and you'll get to die young. Monstrous piece of unhelathy junk. I swear I didn't finish it XD

Emo time:

Theres something I realized after playing pool for so many days. Back when i was in school it was always about being the best. I wanted to be richest, smartest, best in pool, bowling, drawing, even academics... But when playing pool outside these few days, seeing all the other tables and how other people all seemed to be having so much fun even though their pool skill sucked, and how they had pretty girls playing with them, even with my godly pool skills, I felt like the world's biggest loser. And maybe in a way, I am.

Life's not about being the richest or the strongest all the best cause when staring death in the face, all these things become trivial. Instead, when you feel you're at your last breath, all you have are memories. Are your memories happy? Or sad? Is it full of people having fun, or about you and your selfishness?

In the end, Life's about having a good time.

Mahjong and pool

Yesterday night Leeyen and Kenenth and Darren came to HQ to play mahjong... It was a really fun night. It would be nice to have more of these kinds of gatherings in the future.. Damm this sounds like something we would give our english teachers for the journal homework.. Just some mono comments... Ahh i've got nothing much to say... For a fuller account of the night go to www.leeyen.blogspot.com.. She's got some nice pictures as well.. Sm would say darren looks cute sucking on a chicken bone.

And by the way, i know how to clean and wash too okay leeyen. And theres no difference between sewing and knitting :P

I lost horribly in mahjong as usual.. Just dun understand why i can't play mahjong..

And Kenneth and I went for our investment after midnight... Go all the way to amk to play money pool. But in the end i lost to this guy in snooker, lost about 10 bucks total.. He was too good.. Four blacks in one visit.. forget it man...

There are many pretty girls at amk kpool...

busy

Totally surprising.. Why am i so busy that i don't seem to have time for myself? There are the library books to read.. Food to cook.. Balls to shoot.. Videogames to play... TV to watch... A week passes like a day..

Nothing much to blog about. Lets talk a bit about the universe.. Hahaha..

Did you know that the universe can actually be finite in volume?

If we accept the fact that the universe does not have to be restricted by three dimensions, then it is certainly probable that the universe can be finite in volume.. To show how this can happen, think of the earth as an analogy. From the ground, Earth looks like a plane (flat), that will stretch forever and ever. However, we know that the earth is finite in area, because it warps around itself in 3 dimensions. Similarly, if the universe exists in 4 dimensions or more, then mathematics permits the universe to wrap upon itself, in 4 or more dimensions. But because the universe is so huge, the fourth dimension is barely detectable by our equipment.. Of course, the human mind is only limited by 3 dimensions, no one can actually "see" a hypercube, but that doesn't mean it does not exist.

In maths, n-dimensional vectors are abstract arrays of numbers. for eg (2 , 3 , 4) is a 3 dimensional point which can be represented on a 3d graph (xyz). There is no way however to represent the point (2 , 5 , -5 , 10), which is a 4 dimensional vector, using an wxyz graph such that it is intuitively intepretable by the human mind.

Legend of Zelda

Worlds most pek chek game. Only played it for 8 hours, needed to refer to an online walkthrough 3 times. But although its angrifying, it is also surprisingly addictive. *swing swing stab swing*( with the wiimote.)

Got hustled on friday at Kpool. Actually i knew i would be hustled but i still go and let them hustle me.. For me, its about enjoying a good game of pool with an expert. Realized that my pool skills have deteriorated.. Must train from now on.. I shall go into seclusion for 2 months to practice the secret shooting arts of Tian.

THE SECRET SHOOTING ARTS OF TIAN:

- ART ONE
The Aimingless Technique; practice to a level that you can shoot intuitively without even aiming. Reverse cut shots, dun even have to look, your mind will tell you where the pocket is. A shot executed with this skill takes less than 2 seconds.

-ART TWO
Spirit Breaks; Meditate and release mental and physical energy when breaking. The Qi released will be converted into sound energy that can be detected by a human a hundred metres away. The total sum distance covered by all the balls will cover a length of a swimming pool. Beware of this dangerous technique, as the shockwaves can result in permanent tissue damage.

-ART THREE
Heaven's draw; Also known as Tian's screw, This involves keeping your cue at the perfect angle, then using a wavy technique to exert the most bottom spin on the cueball using the least momentum. In a dead on shot, the cueball should spin back with the speed of a motorcycle and traverse at least 27 feet, 3 times the distance of the table, before coming to a stop.

-ART FOUR
Railway Masse; Hone the skills of a curve so accurately that the cueball can travel down the length of the table, and just when it is outside the pocket, curve away towards the oposite pocket and pot any hanging balls. This skill can be trained to the degree that the player can discard the use of jump shots, kick shots and even bank shots. The curve is controlled so well by the player that it seems to be travelling on a virtual railway in the player's mind, hence the name.

-ART FIVE
Infinite Patterns; Train the player such that his knowledge and calculation of the angles are perfect. A branch of trickshot theory, this ultimate skill allows the player to spot and execute shots that would normally not be seen by another player. This can be advanced to the level where an impossible combination shot would be regarded as a straight pot for the user. A player with this skill can end the entire game of nine ball in less than 3 shots, thus eschewing the need for positioning and safety.

These are thus the five secret art of Tian that i have decided to practice. Wish me luck XD.



Random maths


" Now, i is a very proud number. It is aloof and doesn't care to mix much with the other numbers. If I add 3 to 4 I get 7; the 3-ness of the 3 and the 4-ness of the 4 disappear, absorbed into the 7-ness of the 7. If, by contrast, you add 3 to i, you get ... 3+i. It's the same with mutiplication. When you multiply 5 by 2 the 5-ness and 2-ness are swallowed up by the 10-ness of the result, and vanish without trace. Multiply 5 by i, and you get ... 5i. It's as if the i can't let go of its identity; or perhaps the real numbers know that i just isn't the same kind of thing they are."

-John Derbyshire; Prime Obsession-


Cutest way to put it.

Cooking master boy!

Been watching this anime lately. Very cool. "The goal of a chef is to spread happiness through food" Which i really agree with cause good food can really make people happy and satisfied XD.

Culinary is really an amazing art. When you eat something, you not only sense the flavour and the smell or the texture, but you can actually feel the emotions behind the dish. (If you put your heart into it, and not just eat to survive.) Which is why I think that fast food sucks cause all they do is dip everything into oil, and use crazy amounts of seasoning whithout a single good intention. Any Motherfucker of a cook can do something like that.

But when you eat for example home-cooked food or even food from a good hawker, you can feel the effort and heart poured into the food and it warms you up from within, leaving you with a sense of satisfaction and happiness.

Many people think that food is all about the taste and the smell, but the thought and spirit of the dish are equally important. Think about Ba Kut Teh, Pork bones boiled till the fats and juices in the bones degenerate into the soup and spiced up with pepper and garlic to whip up your appetite. Chicken rice, steamed rice first stir fried in chicken fats to soak in the aroma of the chicken, then boiled in chicken stock, coupled with fresh naturally-tenderized chicken with an amazingly tangy chilli sauce. How about prawn noodles, yellow noodles soaked in that otherworldy chicken-beef broth served with chilli oil.

Of course thats not to say that deep frying is not a respectable form of cooking. Deep frying in fact takes a lot of skill and experience. But when you look at that light and crisp breading of a good tempura prawn, or the flavourful, golden covering of a fresh red snapper in an expensive chinese restaurant, and then you look at the thick, starchy flour that mcspicy is covered with, and the fat filled oil is seeping out from the chicken itself, you will know that there was absolutely no skill involved. Food that is well deep fried is actually surprisingly light and NOT oily.

Thus I'm going to lead a mac revolution! Please vot on whether mac is good or not. I'll be the first to vote, mac suck my C#(K!

Wines.

Alright went to sakae sushi yesterday night at wheelock place. Wanted to go to the expensive jap restaurant upstairs, but then i suddenly saw the sushi banner and had a sushi crave :)

Seems like the prices of many hotel restaurants have gone up. We wanted to go eat buffet yesterday but everywhere was like $40++ which amounts to bout $180 after all the taxes..

The food at sakae sushi is well... Okay i guess.. Just remember that the sukiyaki is horrible.

Pics! Sorry i forgot to switch my phone camera back to normal mode after taking an mms-intended picture. Thus the quality is low.

Jap food is 40% presentation, 40% soya sauce, 20% about schoolgirls in kimonos fighting off pervertic aliens.

Damm the red plates which cost $5.59++ each!

This one is fucking worth. Chawanmushi with SHARKS FIN OMFG cost only $2.29++!!! Hiwamari sells it at $5.00++ without sharks fin. Steamed egg is the best. Soft and silky, comfortably wet and savourily sweet, its as good as a french kiss.

See the brown blob on the table right. Its damm good, rice cake. Its called Okonomiyaki and tastes exactly like the the small flour balls called takoyaki. Really good, one of the top national dishes of japan. Remember its called OKO-NO-MI-YA-KI. Remember to order it the next time you eat jap food.

And at the end right we ordered...
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Sake!

Hot somemore. Very shiok man, because its warm right it goes down your throat and sets it right on fire lol.. After drinking for a while you get a warm and fuzzy feeling. Goes well with the food but its a bit costly at $10.99++ for such a small bottle. Alcohol content is average at 12%. But i guess because its hot, it tastes much stronger than cognac. Shiok though, just swallow it!

I drank bout half of it with my mum, dad din wanna drink, just took one sip. And i became like erm.. Really high?? haha XD

So thats it. Weekend dinner at Sakae.

And erm, I'm not trying to be smart or anything, but can someone tell Cat she has her probabilities wrong...

The problem is actually non-elementary, The probability that in a group of N other people someone else will get the same result is actually 1-(15/16)^N . Thus if there were only another person doing the test, the odds would be 1/16. [Not 1/16 squared as claimed] But this rises quickly as shown below...

#of other people : % of match
1 : 6.25%

2 : 12.1%
3 : 17.6%
4 : 22.8%
5 : 27.6%
6 : 32.1%
7 : 36.3%
8 : 40.3%
9 : 44.1%
10 : 47.6%
.
20 : 72.5%
30 : 85.6%
40: 92.4%

This non-intuitive result is called the birthday paradox. Not that it is mathematically unsound, but that it is extremely non-intuitive.

Thus it was not really that "fateful" :X

kbox

Yeah and as you know if you've been reading, i went to kbox last friday with darren. As i don't have permission (yet) to upload the audio clips, I shall upload this solo by me.. Its Jacky Cheung's Qing Shu. And you can hear Darren swearing at the front XD.




K la i know its not my best voice-.-

maths...

Went to play pool with Darren and YH yesterday. After that went Karaoke with Darren alone. Darren has a better voice than me. Damn.

There was a Math problem last post.

a^x = -1
a^2x = 1
a^2x = a^0
2x = 0
x = 0

sub into first equation

a^0 = -1
1 = -1

Okay the error is really simple.

In secondary school you probably learnt that
if a^x = a^y, then x=y
this is, in general, not true.

For eq,
(-1)^6=1
(-1)^8=1

Thus (-1)^6 = (-1)^8
But that certainly does not mean that 6=8.

In general, if a^x=a^y
then y=x if you can take the logarithm of both sides and reduce the expression to x=y.

For example in the first example of this post,

a^x = -1

From this, it follows that a is negative, since a positive number to any real exponent would be positive.

then a^2x = 1
a^2x = a^0

But since a is negative, we certainly can't take the logarithm of it and thus the equation cannot reduce to 2x=0.

You might be wondering, how about the equation
1^6=1^8

In this case we can take the logarithm of both sides, since ln1 is defined, but doesn't that mean that 6=8?

Well no, lets do out the formal maths

1^6 = 1^8
ln(1^6) = ln(1^8)

bringing down the powers,

6*ln1 = 8*ln1

Realize that at this point, we cannot simply cancel out the ln1's because ln1 is zero. Doing so will mean doing a division by zero, which is simply undefined in mathematics.

In other words,

6*0=8*0

doesn't mean that 6=8.

MALA HOTPOT!!!

Everyone knows i like chilli. But i think this is too much, even for me.



Its the globally famous SICHUAN MALA HOTPOT !!! (麻辣火鍋)



I found it on thursday with my parents while searching for somewhere new to eat. Its actually just outside hexun's house, along the stretch of road where there are lots of eateries. Down the road from Thomson plaza, near macritchie park there.



Outside the shop theres an old signboard saying "天下第一刷" which translates loosely into "best steamboat under the heavens"


From the outside, it looks amazingly like some eatery from Beijing with the metal frame chairs and barely sufficient decorations.


Once inside, theres no doubt that this is in fact a china shop. That distintive smell of god-knows-what and flat faced, yellow skinned waiters with that recgonizable-at-once beijing curl accent reminded me instantly of the numerous restaurants I dined in at wang fujing, Beijing.



From my experience with cooking I have always heard of the mala hotpot and that it is a out-from-hell concoction of explosively hot chilli peppers and pepper seeds. One sip of the devilish soup is enough to numb your mouth instantaneously and break the ice between eaters, that i have been told is the main philosophy of the hotpot. It opens people up.





Enough of the talk, I'm sure you all want to know what the soup looks like.

On the left we have the traditional chicken stock that always tastes so good because of all the ingredients you cook inside it. But on the right is the real deal.

Looks like super oily curry right! I swear thats meant for you to drink.

That layer of oil is at least 3 cm thick.

My dad doesn't like spicy.

And actually i retract my statement from before, the soup isn't too much for me. In fact I LOVE it.. Oh yeah and this is supposed to be the not-so-oily version.. The waitress actually asked us whther we wanted less oil and less spice. My dad wanted both, but i heavily opposed to less spice so this was less oily only.. I bet the "normal" one would have like one litre of chilli oil on the top.

Oh yeah and if the place looks popular, thats cause it really is.. Got angmoh eating somemore, all laughing loudly with tomato red faces. Reminds me a bit of drunk people, guess the ice-breaking philosophy is really correct.

If you want to know what the soup tastes like, think bak kut teh with 3 times the pepper, add dozens of dried chilli and half a litre of chilli oil. I won't lie to ya, it is hot, but the hotness is somewhat different from the fresh chilli spice of tom yum soup, instead the hotness builds up gradually, along with the numbness.. its hard to describe la.

Haha for all of you who are looking for something special instead of your usual zichar or curry fish head, this is certainly a must try. Furthermore, its super duper extremely cheap. Only 18.80 for dinner buffet steamboat okay. The buffet menu includes super fresh beef, mutton and chicken slices, fresh prawns, cockles, soft shell crab, and your usual fishball, meatball, egg... etc etc.. got 70 different items in total.

Its really good, better than koka.

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And Hexun told me that he wanted to pick up pool again. That totally made my day. I always loved him and the way he played. It was a bit of a blow when I realized that he didn't want to play anymore. Although i did understand the logical reasons he gave, but i am, like him, an emotional person.

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Belldandy is the ideal beauty.

suprisingly busy

i know i know.. one week no update..

found out surprisingly that even i can be busy lol.. especially when you got 30+ games to play, mythbusters to watch everyday at 5pm on discovery, chase the anime "ah my goddess", learn fourier series, train your voice and on top of that, cook dinner every night. which is the worst i think.. (people who cook would understand..) But i managed to find some time to blog.. About maths again lol..

i came up with this in secondary three.. Mr Sin incorrectly dismissed it by stating that one equation cannot have two variables for it to be solved.. can you tell whats wrong with it?

we start with the equation,
a^x = -1

squaring both sides we would get...
a^(2x) = 1

since a is not zero and anything other than zero, to the power of zero is equal to one we have;

a^2x = a^ 0

and so,

2x = 0
x = 0

substituting this result into our original equation which was a^x = -1
we would get

a^0 = -1
1 = -1

which is of course impossible, but can you see where the error lies?

TPA Cup 2008

THE TPA

The TPA cup has ended. (For over two weeks now) A bit on my thoughts for the competition.

Before the cup i thought that this year everyone would have a more equal chance of winning instead of the usual "Kenneth, Daniel and Master sure win one.." And as it turns out, the later generation members did indeed come very close to almost clinching it. (yes teck siang I'm looking at you) But in the end our pure experience (and luck?) turned out to be too much for them.

Major surprise of the competition was yuan hong.. First non-TPA member to get into semi-finals?!? Teaches us something, play more pool simulations. Darren was a little disappointing.. Hexun was the worst.. Quit in the middle of the cup... But lets not talk too much about them :)

And here are the long awaited medias.Third prize
goes to Tan Yuan Hong, First non-TPA member and pool talent.
As such, he is pronounced the first honorary member of TPA.


Second prize
goes to Kenneth, Forefather and Coach of the TPA.


First prize goes to Daniel, who almost got banned for talking too much.


And heres the amazing match between teck and hex.

Teck Siang is godly.

TPA Cup

Okay the answer to the question in the previous post is its more likely to be a girl.. Cause theres two ways to have one girl and one boy (girl first or boy first) but only one way toget both boys.. Thus its a 66.6 % chance that the other's a girl. That is of course ignoring all other biological factors..


And YES the TPA cup is coming! Its actually on this Monday. Its quite sad though everyone seems to be worrying so much about school. Maybe it isn't a good time? I guess I'll have a "ask me anything about calculus corner" YES I AM THE ULTIMATE PRO.

TPA CUP PRIZES

1st : A box of double yolk mooncakes
2nd : A box of single yolk mooncakes
3rd : A box of yolkless mooncakes

Sponsored by Long Fung confectionery

And Hey people who know their mooncakes will know that mooncakes are super expensive okay. A box of double yolk mooncakes can easily cost up to 30 bucks. So I think they're the best prizes for the TPA cup thus far.

Here are some previous years pictures XDChampion Teck - Winner of the TPA cup 06 nine ball. My greatest pride.

Darren when he was young - taking some practice shots. ( I know totally akward stance but he can still thrash you.)



Kenneth posing with mr singh's bike. (who's actually his father)

Okay the last shot is not TPA cup related but I just decided to add it in anyway.

Haiz.. The birds have flew away.

Savant type questions

A guy has two children

One of them is a boy.

It is likely that the other one is a;
-girl
-boy
-both equally likely

(assume that the chances of having a boy or a girl is equal)

So what do you think is the answer? And no Vos Savant is an idiot.

Back

Damm everytime i do chemo i have to forfeit one weeks worth of blogging. But its okay, I'm back.

Olympics are here too ^-^ The other countries stand no chance against China this time. Yupp I'm proud to be chinese.

Happy Birthday Ray

Yes so Ray, Soomin.. Darren and two other Beatty people came to Hq today to celebrate Ray's birthday. With cake supplied by long fung. Here are the pics..



Yupp nice half-tucked-in shirt XD.. And he beat me in pool too!

WZ : Wah jc people all so busy, Sunday also need to go school.

WZ : Then if i Christian how ahh?

Soomin : Finish school already then go la! You might as well ask Muslim how..

WZ : Oh ya ah.. How?

Soomin : They get to end lessons earlier.

WZ: Ha next time i create one religion everyday have to pray one!

Ray : Wah then those people in your religion promos sure die one...

WZ : Won't cause God will bless you!

Yupp - Tianisticism

Hope.

Today was a fantastic day. Went to watch (in clear defiance of the doctor's advice) the Mummy: Tomb of the dragon emperor at Balestier. The show sucked big time la.. It seemed really childish compared to the previous mummy shows.. Then again, Mummy's are childish.. But the previous ones just seemed more believable ba.. Maybe its cause I'm chinese and i know that the Qin emperor definitely couldn't transform into a fire breathing dragon lol.. Then again.. I dun know for Sure...

After the show we went to Toa Payoh to eat the Taiwanese porridge at tpy park. Not bad laa but damm expensive.. 70 bucks for some vegetables and salted eggs.. lol.. Okay there was a fish too.. But all like poor man's stuff la.. Shouldn't cost so much.. Somemore also not say very nice.. I think goodwood park hotel's teochew porridge buffet is much more worth it..see this one is goodwood park hotel's one. DAMM fantastic spread right! Somemore Buffet leh!! Eat-all-you-can for 30 bucks. I think its damm worth it. And the bean sprouts rock.

Okay so once i was in tpy right.. Somemore so close to Safra.. I decided to go play pool.. So i called kenneth and of course he agreed.. But Ahhh!! The place was full no tables.. And i was like super disappointed.. What to do? Never play for eight months!

Then my Dad said got something to do at Long Fung so go lung fung dunno do what for some time.. Then after that i was like " I dun care i have to play today!" So they no choice call safra again.. Haha got one pool table!! So go play lor. Even though it was like 11.. Midnight pool ma..Home is still the same

I had super a lot of fun la.. felt so much like the Tpa era.. It was quite an emotional experience for me.. Eight months never go.. Somemore got to see miss tan and anthony.. And of course they got to know about my sickness.. They were very supportive and glad to see me la.. Miss tan looks older.. hahaha.. At first she like couldn't recgonise me cause i was wearing my mask and everything.. But eventually she was like weizhou ahh.. Haha of course la! Tian Weizhou is the greatest pool player Space Rhythm has ever seen!!!! How can dun recgonise me!!!?!?!?

Even though i lost alot of my skills.. And i mean like alot.. But still can beat Kenneth haha XD But i knew he was like giving me chance la.. Scratch so many times!! But okay still have bout 60% of my former skills. Still can beat any local player under eighteen.. (if i train for a week :D)

Random: Daniel super chicken la!! Dun dare to come HQ challenge me!! Scared lose only!

Okay so today i realized a lot of things.. And i know what i will be fighting for. Thanks everyone..

Really Tired

Soomin and Darren came to my house yesterday. They tried to help me solve some problems.
I think I realized why I have so little friends. If theres a problem I can't solve, nobody can help me because I'm the greatest problem solver within the radius. And friends are just eager helpers. Ahh how I wish I had a girlfriend who would support me blindy, like those cute cute innocent anime kinds XD..

Only when you understand why people commit suicide, then you will understand me. After all, is life worth hell?

On a brighter note, Kenneth and Ahma came to my house today. Nicole, my primary one cousin came too. Many people say she's stupid and all, like really slow in learning stuff. But i discovered that that was totally untrue la. Before Tian's tutorial she could only add two numbers together where the sum did not exceed twenty caused she used her fingers. But once i taught her the common addition algorithm she could add like up to 99 la. And i only spent like 20 minutes. I seriously must say she's quite a genius. And I'm not kidding la, even after one hour she still keep saying she wants to do more sums.

Then I learnt from Ahma that even at home, she often brings a book around but nobody bothers to teach her.. even her own parents. Fucking nabe chibye la! I mean if your child is like damm stupid damm hard to teach then maybe its understandable that you dun want to teach. But she's so smart lor. I bet anyone would find it a joy to teach her! Dammit man. If shes my daughter, can do calculus already!

Okay la must do some maths from time to time sorry.

I'll try not to write out the steps and just explain them verbally la..

Using Euler's formula we can obtain a expression that relates the tangent inverse function and the natural logarithm. Using the the infinite series from inverse tangent and then doing some algebra we can get an infinite series for lnx. After that it is not difficult to show that this series converges for all x. The series is,for all x > 0. I think its quite cool considering that the lnx has no taylor series about zero. (maclaurin series if you like). And like i discovered it on my own! Yay!

Okay la but I'm sure someone beat me to it. Damn i hate maths.

Lalalaaaaa...

I was on youtube yesterday night singing along to wang leehom's kiss goodbye. I turned the volume very low as it was late at night. My mum was facing her mirror behind me drying her hair.


Me : "Gei~wo~yi~fen~zhong~an~jing~hao~hao~xing~shang~ni~de
~mei.."

Mum : "Wah who is this singer ah the voice not bad.."

Me : *Stops singing* What singer?

Mum : No just now that singer. Why you turn down the volume?!

Me : Huh.. Just now that one is me leh.

Mum : What?!? Your head ahh, I don't believe.. Sing again la!

Me : Don't want I paiseh..

Mum : Its really you? Don't tell me you have mastered the secret..

Me : Haa haa haa haahaa!! (In that super high super falsetto japanesy arrogant woman laugh)

Mum : *Goes on to show off her amazing voice singing some operaish song*



Haha. I might just decide to go take singing lessons.

The blues

Haiz returned home from hosp again. Just had a 400ml serving of blood. Now my lips are looking redder. Mmmmm looks delicious.

Btw i've lost all my emo feel so doesn't look like i'll continue the "things i've wanted to say" post. For all my friends who were left out, next time maybe. Oh well most of them don't red anyway.

Here are some song lyrics I've penned. To be sung to the tune of "We are Singapore".
Okay la i know its lame and it sucks la. Nothing better to do mah!


There was a time when people thought that Singapore taught Maths well
They were wrong (they we wrong)

There was a time where people said that we could beat the Ang Mohs
Wait long long (wait long long)

We've built a system of exams
But after all the papers, we just don't give a damm

This one must memorize
That one got give

This one doesn't make sense
So just believe

You can forget all
After you leave

This is Singapore
i doesn't exist


Okay la. Reminds me of my primary school days where we would ridicule the national anthem...

Things I want to say (part one)

Back in my secondary school days, a considerable amount of my sleepless nights were spent thinking about death, and that if i were to die the next day, whether i had any regrets or things that i had not done. I had so many things to say about my life, and to my friends. I fantasized that if I knew my death was coming, I would leave a goodbye note of some sort, which would have contained my true feelings.

Since the probabilty of that happening has defintely increased now, i think it would be nice to go foolproof and materialize such a thing.

I think this is going to be somewhat grand mother^2(lame but nevertheless try to figure out what it meansXD) so people who can't stand tianemoism should probably close the page.

I think its best to start with my life now, and slowly go backwards.

Cancer. A frightening word for some people. Young people think it wouldn't happen to them, middle-aged people are too busy to think about such things, old people don't care cause they're going to die anyway.

I used to think (yes i do think about all kinds of stuff!) that if i got a serious illness like cancer or diabetes or something like that, I would probably be devastated and my world would fall apart. After battling lymphoma for eight months, I can assure you that Cancer is nothing like that. Maybe it is because i ain't afraid to die, having thought about it countless times (everyone has to die right?), or maybe I'm just stone-cold. Sure, the operations are always scary, and I've cried a good number of times. But it is definitely nothing like how it is in the movies.

But i guess its good to write a little about how chemotherapy is about. Chemotherapy is usually carried out in phases, with a standard treatment ranging from about 8 of these phases. In SGH, the treatment for my kind of lymphoma takes eight phases, each phase contains 2 weeks of chemo(4 days of actual chemo drugging, but you have to spend 2 weeks in the hospital). In KK however the treatment is quite different as they have a "day-therapy center" where you can go to take the chemo drugs and go home straight after. Eschewing the need for a hospital admission.

Before every chemotherapy session, a blood test is required. The numbers of each component in your blood has to satisfy a certain number to be able to administer chemo drugs. A syringe is used to suck blood out from your veins, the puncture site is usually on parts of your body where the veins can commonly be seen. Hands, wrist, inside of elbow.. etc. It is also not as simple as poking the skin and immediately getting blood. Amazingly, if the vein is not ruptured, no blood will be spilt. So the doctor, after punturing the skin, would have to shift the needle inside and push the needle further in to get the blood. If theres no blood, he'll shift the needle yet a bit more and push again. A common blood test would go like this. Poke the skin, no blood, shift the needle a bit, push further, no blood. Shift and push again, Which can take many cycles until a vein is punctured. I guess if a tiny ant bit through your skin and crawled inside and moved around and digs deeper and deeper into your flesh, that would probably be how it feels like.

Of course, that is only if you are lucky, if you are unlucky, the doctor, after digging in for blood for an extended period of time and unable to get it, would have to pull the needle out and puncture another area of the skin. In fact my highest record was four puncture sites in a single blood test! How cool. After chemotherapy, a blood test has to be done each day to check that the blood levels are returning to normal (Chemo sort of kills all the stuff in your blood). So a single chemotherapy session can have as many as six or more blood tests, depending on how fast the blood levels rise. In fact, I've had so many of these that the scars from each puncture line up nicely to form this nice dark black line thats along my major veins on my hand. Actually, a blood test is considered an ant bite compared to the actual chemo stuff.

Well apart from the lumbar puncture which i have descibed in detail already. There are the actual chemo drugs that go into your veins. These drugs are the stuff that makes all your hair drop off, lose 10 kg in a week and makes you nauseous like hell. An operation is first carried out to insert what is known as a port into your chest. It is actually a device that is placed just right under your skin that connects itself one of the major arteries near the heart. Think of it as a socket. So everytime they wanna pump chemo in they would just have to insert a big needle into this socket, which links to the artery and the chemo drugs would be flowing safetly through your arteries. Of course everytime this is done, your skin would have to be punctured as well.

The drugs themself well, fells like taking poison. makes you feel really sick and uncomfortable all over. Luckily it only lasts for a few days.

Mentally too, it does take its toll. One of the few scary moments was perhaps the time where i lied in bed unable to move because of an operation and i wanted badly to puke and pee and i was really hungry cause you're not allowed to eat six hours before the operation. And my parents were not around as they were at the foodcourt downstairs and i couldn't call the nurse to as i couldn't reach the callbell haha.. So in the end i just waited for my parents to come back and help me to pee. Or maybe it was that moment when the doctor came in with a serious look on his face, muttered something about a bone marrow and said "mummy and daddy, can we talk outside?". And my parent went out, and i thought at that moment that it was game over, i thought i was going to die. I though "wah so this is how it feels to know that you're going to die." It was nothing really, just a bit sad. In the end i found out they were only discussing about some bone marrow transplant stuff. Haha so it was not game over after all!

But i guess the most depressing moment is when you realize that all your frens are moving on with their lives, experiencing jc life and all, and that you are trapped in the memory of secondary school. That you are distancing ever so further away from them, and that even if you go to school next year, you would be all alone. But actually i wouldn't mind if my class was filled full of ij girls! I would devour them all! Muahahaha.

Okay enough of how chemo is and all. Time to list out my secondary school regrets. I'll bet everyone has a long list of them XD

#1 -not having a girlfriend (yes i am a romantic)
#2 -not having a CCA
#3 -having a bad reputation, in lower sec it was "haolian boy" which i think was coined by miss huang. In upper sec it was more or less "that idiot who always get full marks for math without studying..."
#4 -disappointing the TPA members. And losing the cohesiveness of the group thanks to my lousy attitude.
#5 -confessing to isabel
#6 -not making enough friends

Well six isn't a long list i think.. I bet you have more.

Okay now that thats settled its time to leave the individual messages. It would be good to read this only if something really happens to me since its incredibly corny. But i guess if you've read thus far through that crap, you probably couldn't resist.

TO;

Kenneth:
One of the forefathers of TPA! In fact most of the TPA students- Hexun, Daniel and even Darren were coached by him. So TPA should really be called KPA. "Best Friend" is probably a primary school phrase, but you're defintely in the set. What to do, playing pool everyweek for 6 hours does that to people.

Soomin:
One of the nicest person i know! Also one of the rare few to appreciate jacky cheung.. But why you don't like Leslie? So charismatic! I can become gay for him. Nah just kidding. I might consider though. Thanks for all your chocolates and visits. You are also definitely in the set.

Darren:
You are A~L~S~O in the set. But you exist too in the intersection of this set and another set called "most depressing friends". Well i guess the biggest issue with you is that you do not develop that sense of closeness to people. In most friendships both A(positive) and B(negative) move towards each other on this line of friendship eventually touching and reaching the state of "brotherhoodly closeness". For you, you're like this immovable point and its really frustrating for your more senstitive friends because no matter how close they try and move to you they cannot cross the zero point over to the next side because you won't budge- the state of "brotherhoodly closeness" can never be achieved. Or maybe you have changed. I don't really know because its been so long since our last good conversation. But i really like you cause you grasp mathematical concepts at an awesome rate. For most people its like:

Tian -explain explain*... do you understand?
student- er.. wait ah.. Whats this ah?
Tian -Thats the *blah blah* la!
student- But how do you know how to "blah blah"?!?
Tian - We don't, we derive it from here..
student- where?

By this time the student fails to take in anything i say because all his brain cells are dead and he's super turned off. For you however, its more of like.

Tian -explain explain*... do you understand?
You- okay.
Tian -so following from here, we get this and this..
You- okay.
Tian -so if we go back to the previous result and sub it in here and expand the thing..
You- Oh! then we get the identity!
Tian - Yah!
You - ok cool.

WALAN! The feeling is damm shiok man i never expand you already know expand already get what. Too good!!! haha..

See la you take up so much of the space.. Next!

Teck Siang:
The Champion; cool, calm, zai, chun. Your stance alone scares off any of your opponents. Your long pots are so accurate that it beats the computer aboard NASA's appolo spaceship. Also the only TPA member to master the Shen's snooker method of playing. Just awfully good. I will never forget the times playing snooker with you. In fact when i recover, thats the first thing I'll do. I'm really proud of the way you play.

Hexun
The Hustler; Unpredictable and dynamic. You're the only TPA member able to emulate the awfully difficult Tian's screw. Fast, furious and fearsome, your pure talent would put Ronnie O Sullivan to shame. Just a joy to watch. It must be one of the greatest loses in Singaporean pool that you quit.

OMFG its already FIVE... Guess I'll continue this tomorrow..
----To be Continued----

I'm so sorry but I'm really pissed off.

I don't see what the fuss is about. You catch them, they die. You cut off their fins and throw them back, they die. Both ways, we murder them for food. Too bad but well, we gotta eat right.

When we throw sharks with their fins cut off back into the water and leave them to die, we say its cruel and we spend millions of dollars protesting against it. But when we see a bull suffering nerve damage and internal bleeding from the toxin of a snake's bite, we say "Oh no we can't help the bull cause its all part of nature and we mustn't interfere, Oh see how the bull eventually suffocates and dies? Thats such a beautiful part of nature isn't it?"

Since when did we become God and not part of nature as well?

And those who say that shark's fin is worthless as an ingredient because it has little taste obviously know little about food and cooking. When working with foods like meat and seafood, the most important part of the ingredient is not the taste, but the texture. In fact, most of the taste of the dish comes from the sauce or seasoning or whatever you call it. Which is why it is so important that your fish is not too "stiff" and your meat can only be cooked medium-rare to have the best "bite".

As for shark's fin the amazing thing is how the glutens in the fin thicken the soup while cooking. Thus avoiding traditional soup thickeners like cornstarch which ultimately, produces a much more superior texture and consistency. Add that to the super rich broth of conpoy and other premium delights, plus the sweet taste and indescribable bite of the fins that just tickles your taste buds, and you have what i believe is the best soup in chinese cuisine.

Well,a good soup is good enough to kill for.

Too good!

Haha just heard that Darren got top in class for math. Congrajulations Darren, although its still a bit sad that there are still people from other classes toping him.. Always knew he had the stuff for mathematics. Too bad he's not passionate about it huh.. But I'm sure he'd make a fine mathematician:D

Did a bit of maths with Soomin yesterday. Realized that i'm falling behind them in terms of JC mathematics.. haha.. I guess i just do too little practice.. Oh well.. Its not fair anyway, they got teacher teach one :P

Guess what i dreamt of yesterday? I went to this strange school.. And i was attending math lesson where they had this weirdest math syllabus.. There was a chapter on symmetry, points and lines. And the teacher asked this really complex question, something like tell me what is the @#*$ property that both a group of #$*& points and a @(&# line shares? And i was like O.o And the teacher ordered me to go for extra lessons.. And i went. Then this damn chio maths teacher asked me.. So you dun like maths? And i said "yeah actually i dun really like maths" And then i woke up.. WALAU super sad laaa cannot stand it can!

So i decided its time to do some complex mathematics. To prove to myself that i still can do maths XD

Its common sense in jc mathematics that the expression 1/(1-x^2) can be split into partial fractions.
The denominator splits into (1+x) and (1-x) and thus we have
1/(1-x^2) = A/(1+x) + B/(1-x) and a quick mental sum will reveal A and B to both be half.

What is not so obvious is the expression 1/(1+x^2). While 1+x^2 is not factorable in the real domain, it is factorable in the complex domain. More precisely, 1+x^2 = (1+ix)(1-ix) and so we have 1/(1+x^2)=1/2(1+ix) + 1/2(1-ix).

Lets do some interesting things with this.. It is elementary calculus that the integral of 1/(1+x^2) is artanx

However, with our observation above the integral of 1/(1+x^2) = the integral of 1/2(1+ix) + 1/2(1-ix)= 0.5*(-i*ln(1+ix)+i*ln(1-ix))
=0.5i*[ln(1-ix)-ln(1+ix)]
=0.5i*ln[(1-ix)/(1+ix)]

This must be equal to arctanx since they are the same integral.
Thus we have arctanx = 0.5i*ln[(1-ix)/(1+ix)]

Or equally beautiful, e^arctanx = [sqrt[(1-ix)/(1+ix)]]^i

haha so i do like maths after all!

Another proof

d/dx e^ix = ie^ix
-----------------
d/dx cosx + isinx
= -sinx + icosx
= i (cosx + isinx)
-----------------
Both functions satisfy the differential equation dy/dx = iy

with initial value (x=0)equal to one

Thus they must be equal ( if two functions are the same at one point and change in the same way, their entire functions must be equal.)

Thus we have again, e^ix = cosx + isinx

fried rice!

my fried rice is good..
The otah bits taste great.

Even darren had 2 servings.

Tian can cook

Haha I've been real into cooking lately. Cancer patients are usually advised from eating hawker food because of cleanliness issues. Restaurant food is okay, but of course home cooked food is the best when you're as good a cook as me XD! You can add extra sumptous ingredients like prawns and scallops.

Anyway tonight i cooked chilli crabs. The recipe is really really simple. But you end up with a good sauce thats better than most of the coffee shops one. No need to buy those so-called pre-made instant chilli crab sauces they're such a waste of money (3 or 4 bucks a pack?!?!!) and way too salty. Homemade is the best. delcious too.

Here are some pictures. XD!

My Dad gave it a 72/100

My mum gave it a 75/100

I give myself 70/100 haha the crabs were not sweet enough. But that wasn't my fault sheng shiong just had lousy crabs.. Next time must buy good ones from wet market.. The sauce was awesome though- anyone who wants the recipe just leave a comment haha.. Yupp and i decided to leave a picture of myself.. Monk head! now can go and lian the iron skull marital art.. Very airy maybe i should get a temporary snakehead tattoo or something.. that'll be cool..

Yupp gotta go watch soccer now. Austria will win Crotia.

The Greatest Identity on Earth

Euler's Identity

The greatest identity on Earth, relating the base of the natural logarithm e, with the elementary trigo functions. substituting pi into the identity immediately yields;An equation relating the five most important numbers in mathematics. Nothing else needs to be said. I will attempt to give a single proof of this today. Which is not really totally formal, but works nonetheless.

Proof.

Consider the function

f(x)=

Differentiating we will have,

f '(x)=f '(x) = 0

Take a moment to think about what this means. This tells us that the gradient of f(x) at Every point of its graph is equal to zero. Thus is we draw the graph it must look like a straight horizontal line. Which means that it must be a constant function.

For every value of x, fx is a constant. Thus we can find out what this value is by substituting any value for x. Lets try zero.

f(x)=constant=f(0)= which reduces to one.

Thus we have shown, f(x)=1 for all x,

is equal to one, for all x

Yupp. Too chio.

Apologies

Ahhh... Finally got my PC restored back to normal.. Been downloading too many illegal psp games from utorrent.. Siao liao la. Half a month never update...-.-'''

Guess its time i talk about how I'm coping with chemotherapy.. Everything's going real fine now. I'm near the end.. Just discharged from hospital yesterday. Side effects this time were minimal too.. Doctor says after two more weeks or so i'll have a two months break from chemo to regain back my health. Then its off to another two more months of chemo and i'll be into a sort of remission state.

Its certainly been a long journey.. Many thanks to my parents and supportive friends.. I could not have gone this far without you all.

I'm also real into cooking now.

My chili crab rocks. When i'm well i'll have a party and we'll snack on chilli crabs and barcardi. XD

Life's good when theres food.

Incomplete TPA portrait

Haha wanted to post this for a long time. Too bad Kenneth isn't inside. I think he was sick or something. I should continue to post more TPA stuff.. xD

PS3

The power of the Playstation 3 is simply amazing. In my entire life of dealing with technology i've seen no hardware that can compare to the Playstation 3's simply striking processing power. It quite simply does everything that a normal household pc can normally do, and more. Let me come up with a short list of what i think are the most attractive features of the PS3.

- Games
With a lineup like Grand Theft Auto 4, Final Fantasy XIII and Gran Turismo 5, there is no other console that can match it when it comes to the greatest games on the planet. Even the best PC games lack the pure polish and feel of console games. Plus the fact that the PS3 now supports online multiplayer (I've just played GTA4 online and i can tell you that its more enjoyable than HALO), It seems like the PC will finally be obsolute as a gaming system. There's simply no reason to get a high end PC to play not-so-good games when you just need to spend 500+bucks for a PS3. And the 360 is simply a lousier PS3 with even suckier games. The only contender I can see now is the Wii. Although it certainly sounds a lot like a gimmick to cheat naive children,it actually is extremely enjoyable. Theres just so much fun in swinging the wiimote like its an actual sword instead of pressing a button, poking the wiimote forward like a cuestick when you're playing pool, or simply shaking it to trigger a special move. The PS3's sixaxis simply doesn't quite match up to it.

-Visuals
1080p, the highest output resolution now. What else needs to be said? After a mere three days of playing the PS3, i found my normal cable TV resolution to be a joke. And youtube was just unwatchable. Plus the PS3's processing power makes spheres, for once, look perfectly round. The ability to calculate thousands of trigonometric equations simultaneously means that in game water is now indistinguishable from real water. But the boobs still look a bit fake.

-Blueray Player
Now that the blueray has crushed the HD-DVD like it was made of paper, you will obviously Need to get a blueray player somewhere in the future. Why settle for only a blueray player when just by spending a bit more, or in some cases less, you can get a blueray player And a fantastic gaming console?.. I just downloaded a free trailer from the playstation store yesterday of kung fu hustle in blueray format and its simply breathtaking. You can even count the number of hairs on stephen chow's face.

-Its pure processing power.
Now, Homebrew developers are able to utilize the playstation to carry out tasks that it was never designed for. For an amatuer mathematician like me, it means that i can borrow the PS3's processing power to calculate pi or e to a million digits. Or do even more advanced things like simulate a chaotic function XD.

-Home
For most of you who do not know what Home is, Home is a free application that will be released by Sony at the end of this year. It is essentially a living virtual world where playstation users all over the world can live and play in. Each will have their own rooms which they can decorate etc.. There are also various minigames like bowling and pool. You can even pay to watch movies at the theatre which runs real life movies that are currently in the cinemas. Imagine if I'm bored at home, i just call up a few friends with a PS3 and its midnight pool till 5 man! We might even hustle some players into giving us their prized items. Like purchasable clothing.. etc.. Haha

Nows a great time to get a PS3. What are you waiting for!?

Modular arithmetic

Consider the following theorem from secondary two.

A number is divisible by 3 if and only if the sum of its digits are also divisible by 3.

for eg, 1001020344 is divisible by 3,
because the sum of its digits, 1+0+0+1+0+2+0+3+4+4=15
and 15 is certainly divisible by 3.

To prove this theorem, know first that any number can be written as (a+10b+100c+...), where a,b and c are positive integers (I've notice that a surprising number of people do not know what integers are.. integers are also known as whole numbers. The set of integers is {....-3,-2,-1,0,1,2,3...})

For example, the number 345678 can be represented as;
8+10(7)+100(6)+1000(5)+10'000(4)+100'000(3)

Now that this is clarified, lets continue

suppose we have a number that is in the above form (a+10b+100c+1000d+...) where a,b,c are positive integers. and suppose that the number is divisible by 3.

Since it is divisible by 3, its remainder,when divided by 3 must be zero. In modular arithmetic this is written as;

a+10b+100c+1000d+... = 0 (mod3)

but from the previous post we also have 10^k=1 (mod 3)
which means that any power of ten equals to 1 under mod3.

Thus, we can replace the 10,100,1000.... in our equation above

We will get a+b+c+d+.... = 0 (mod3)

And hey the proof is complete! This shows that if any number is divisible by 3, the sum of its digits must also be divisible by 3. And all the action happens so fast!XD


Now for a deeper use of modular arithmetic.

we will show that if
7a²+7b²+6 = d²

then a,b and d can never be integers.
Under mod7, 7=0

thus the left side of the equation can be rewritten as
0a²+0b²+6=6

and the right hand side stays unchanged. Thus we have
6 = d²(mod 7)

now we just have to test for the possible values of d and show that d² cannot be 6 under mod 7.
since we restrict d to be an integer, d can only be 0,1,2,3,4....

0²=0=0(mod7)

1²=1=1 (mod7)

2²=4=4 (mod7)

3²=9=2 (mod7)

4²=16=2 (mod7)

5²=25=4 (mod7)

6²=36=1 (mod7)

haha and you actually thought i was going all the way to infinity didn't you! In fact we can stop here because 7 which is the next number, is equal to 0 under mod7 which means the the list of numbers repeat themselves.

to show you what i mean, see that
7²=0²=0(mod7)

8²=1²=1 (mod7)

9²=2²=4 (mod7)

notice that the second number in the equality is the same as the first list. Thus we only need to test for d=0 to d=6. Since in all these 7 cases, the numbers are not congruent to 6 (mod7), thus there is no positive integer that satisfies the congruence d²=6 (mod)7

And thus, there is no positive integer that will satisfy the equation 7a²+7b²+6 = d²

Haha.. damn chio man..Imagine know, out of the whole infinity of positive integers(1,2,3,4,5,6,7,8,9....) there is no combination that will solve the equality 7a²+7b²+6 = d²

Proving such a thing can be of no pratical use, but it is certainly an elegant display of the power of mathematics.

Kay off to continue killing poeple in GtaIV!!!

Dreams

Phew... Back home after an excrutiating 5 days in the hospital. Got an infection from dunno-where, and they needed to pump in super solid antibiotics into my veins to keep the bacteria from killing me.

Anyway, i realized that i've been dreaming a lot recently.. And a lot means like on average once everyday. Which i would think is justifiable to be called a lot. I also discovered that dreams are easily forgotten. For eg i dun remember any dreams that i had last year.. Although the probability shows it highly likely that i must have had at least one last year.. Thus i've decided to record my dreams in this blog:D

I would think my dreams to be more interesting than my life. So it should be alright.

Okay i had two dreams this morning.

-I dreamt that i had a big house party. Everyone i knew was there.. Even Mr Boo lol.. And he was betting on some F1 stuff with my smallest uncle.. And there was this Huge fridge in the middle of my living room that had lots of vegetables.. (Yuck!) And then i woke up..

-I dreamt that i was on the bus with darren. Then my second aunt boarded the bus. I was lazy to acknowledge her and so pretended not to see her.. With some success.. After a short while, my first aunt boarded the bus.. I also tried to pretend i din see her. Unfortunately, my converstaion with darren got so loud that my second aunt noticed and turned back. "Aye Ah boy!" To which i pretended an enthusiastic reply. "Aye er gu!" Sadly, my first aunt heard this and called me.. And after that they were like telling me what to do.. Eg.. Eat more vegetables.. And i got so pissed off and stressed out that i fainted. When i woke up, i was in my room and my first aunt was talking to me. Though i cannot remember what she said. I woke up and looked around. There was a cabinet with a hell lot of pokemon cards.. Anyway i then asked my aunt,"wheres my friend?" and she told me that he had left.

Haha even in my dreams, Darren has no brotherhood.

Now that i think of it though, it is certainly strange that in our dreams, the places we know will always look a bit different from the actual places. But we will never fail to recognize it. Its like, in reality my room has no glass cabinet. But when i was in that room in my dream, i knew it was My room. Strange isn't it?

Anyway i woke up after that, went down, switched on the tv and began browse thru the channels. And i was reaching chnnel 5 when i saw kids central. And then i realized. SHIT! Just missed Pokemon!! ARGHHHH NOOOO...

I Totally forgot!!! Its a bit mysterious to find the relation between pokemon on kids central and pokemon in my dream, When my conscious mind has totally forgot about it... Somemore todays episode is the beginning of the fourth generation series leh. and for a few months now on kids central there has been NO pokemon. Its almost as though my subconscious mind told me, " Wake up you sleepyhead, theres pokemon now!!!"

Damn this is certainly scary. What if my subconscious mind had already formulated a Hell-Breaking maths theorem and my conscious mind just didn't knew about it!?!?!

Oh well.. I said in the last post that i'll begin modular arithmetic today, i'm too lazy to go thru all of it, so i'll start with something light.

In fact we use modular arithmetic in everyday lives.

For eg on the clock, 14 o'clock is almost always referred to as 2 o'clock. Extending this concept, 25 o'clock would be 1 o'clock and 120 o'clock would be 0 o'clock.

And thus we write 120 is congruent to 0 (mod 12), where 12 means the number in which the following numbers tend to "reset". And mod just means well, modular.

This is in fact equal to saying that when you divide 120 by 12 and 0 by 12, theire remainders are the same.

So using this we can can come up with whole systems of numbers. For eg.

13=3 (mod 5)
17=1 (mod 4)
100=1 (mod 9)

As my previous reader meticulously pointed out, the actual notation is not an equals sign. Rather it is something like that but with three strokes. To which mathematicians read "is congruent to" I can't find it in my computer. And even if i can i'm too lazy to use it. And besides if we consider (mod n) to be a function of x, then it would certainly be legitimate to write something like 5=14 (mod 9) , with he equal sign REALLY meaning equals..

I was really quite surprised that she knew because modular arithmetic is not taught in Singapore at either secondary or jc levels...At least not in the formal curriculum.. I was pretty sure few people would know it haha..

Anyway to show how this might be useful, know that substitution and adding and multiplying to both sides an equal integer is a legitimate operation. While dividing by a common integer isn't.

for eg 10=1(mod3)----------------------(1)
we want to find out that if k is an integer,
whats 10^k under mod 3.

The answer is just one cause we can substitute the 10 in 10^k with 1( as shown in the first equation.) and 1^k is certainly equals to one. so we have 10^k=1(mop 3)

This in fact proves that when you divide any power of 10 by 3, you will always have a remainder of one.

for eg 100=1(mod3)
1000=1(mod 3)
10'000=1(mod 3)
And so on ...

I'll show how this can be immensely powerful in mathematics.

For now see if you can solve the following congruences


y=30 (mod 8)
x=51 (mod 5)
z= 288(mod 4)

-A few extra facts for the inclined.

-notice that if p is a prime number, p will never be zero under (mod n), where n can be from 2 to (p-1)

-Also any number equals to 0 under mod 1.

- And mod 0 just doesn't exist..

Oh well i'll stop here for now.. Wow doing this certainly is not effortless..