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..

Revealing the solution

Okay if we take the distance between A and B to be d
and Bird's speed=V
Wind speed=W

Then the time taken is given by the table below.

V is always bigger than V-W²/V,

thus 2d/v is always smaller than 2d/(V-W²/V)

(if more people share the same pie, the individual pieces get smaller.)

And so we can see that the time taken with wind is always longer.

Next topic (damn interesting): modular arithmetic and how 9+5=2