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!
Sunday, July 6, 2008
Monday, June 30, 2008
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
-----------------
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
Sunday, June 15, 2008
Sunday, June 8, 2008
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!
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.
Saturday, June 7, 2008
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.
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.
Tuesday, May 20, 2008
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
Friday, May 16, 2008
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!?
- 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!?
Wednesday, May 14, 2008
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!!!
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!!!
Saturday, May 10, 2008
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..
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..
Saturday, May 3, 2008
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,
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
Monday, April 28, 2008
THE ULTIMATE HUSTLER QUESTION!!!
Okay seems like everyone i know got tricked by this question. (Even Idris refused to attempt it) And of course the beautiful thing is that the problem is so simple to state and solve. Primary school maths really.
A bird flies from A to B then back to A again.
It does this twice. The first time there was no wind.
The second time, theres a constant wind blowing from A to B.
The Ultimate hustler Question is...
Which takes the bird a longer time to complete? The trip with or the trip without wind ?!?!?!
-remember that in both cases the bird flies from A to B then back again to A.
100% of people said the two trips take the same time to complete.
Their standard explanation goes something like this. In the trip with wind, the increase in speed given by the wind when the bird flies from A to B is offset by the decrease in speed when the bird flies back from B to A so in the end, it would be as if there was no wind at all. Formally put, with or wihout wind, the birds average speed is the same, so the time taken for both trips must be equal.
Then they get the greaatest shock of their lives when they find out that this is in fact WRONG!!
In reality, the trip with wind always takes longer than the trip wihout wind.. XD
Since this is a mathematics site, i'm going to leave you guys some time to try and prove this and leave the solution for the next post.
A bird flies from A to B then back to A again.
It does this twice. The first time there was no wind.
The second time, theres a constant wind blowing from A to B.
The Ultimate hustler Question is...
Which takes the bird a longer time to complete? The trip with or the trip without wind ?!?!?!
-remember that in both cases the bird flies from A to B then back again to A.
100% of people said the two trips take the same time to complete.
Their standard explanation goes something like this. In the trip with wind, the increase in speed given by the wind when the bird flies from A to B is offset by the decrease in speed when the bird flies back from B to A so in the end, it would be as if there was no wind at all. Formally put, with or wihout wind, the birds average speed is the same, so the time taken for both trips must be equal.
Then they get the greaatest shock of their lives when they find out that this is in fact WRONG!!
In reality, the trip with wind always takes longer than the trip wihout wind.. XD
Since this is a mathematics site, i'm going to leave you guys some time to try and prove this and leave the solution for the next post.
Monday, April 21, 2008
Fantastic
Wikipedia says that Kang Kong is healthy and has as much iron as spinach.
It seems incredible to me that in this modern age of science and technology, there are still people who believe age old myths about food that are ridiculous and fantastic. How many of you believe that prawns and shrimps are "toxic"? How about the idea that Kang Kong makes your legs weak? Or even that potato chips are "heaty" and give you sorethroat?
After a long 5 months of staying in the hospital and thus having an increased knowledge of medicine, I can assure you that no vegetable sold in the supermarket has the amazing abilliy to make your legs weak.
Also, "prawns and shrimps are extremely good sources of protein and yet very low in fats and calories, making them very healthy to eat"(http://www.helpwithcooking.com/) . Prawns are also not known to have any toxins in them, and certainly nobody has ever gotten sick from eating a fresh, healthy and well-cooked prawn.
Many chinese, on the other hand, believe in the heatiness and cooling theory. Which is a load of bullshit. In fact, i discovered this once when i had a sorethroat. i thought about how people always said that salt cures ulcers and sorethroat, and thought of how to get large amounts of salt into the throat. Haha i came up with an ingenious plan. Eat extra salted potato chips! Incredibly, my sorethroat instantly felt better and disappeared the next day. In fact, this method works most of the time. Proving that the heatiness theory is incorrect. In fact, don't you find it ridiculous that food can be classified into two such broad categories(heaty and cold)? Common sense will tell you that they are insufficient to understand nutrition. On top of that, sorethroats, colds, flu, fever.. etc are all known to be caused by bacteria and viruses. Most people know and accept this, but why is it then that they still think being too heaty can cause you to be ill? Aren't they contradicting themselves? Its like believeing in both Christianity and Hinduism. Choose one religion man!
In fact, after confirmation with a doctor, i found out that the reason many people have sorethroats after eating nuts and potato chips is because the rough and hard texture of the food tends to scratch against the walls of the throat, possible damaging it. Also the oil from the chips, if not washed away by drinking lots of water, tends to repel saliva, which has natural anti-bacterial properties. Thus being a more potent ground for bacteria to grow. So there you have it. I beg people to trust empirical evidence and not to place too much faith in far-fetched claims. Question every information before accepting it, it will make you smarter.
And no Maggi Mee does not cause you to lose your hair. Neither does it have the fictitious "wax" thats get trapped in your intestines.
Chilli is an excellent source of vitamin C, much higher than orange in fact. And as a fruit, is very healthy. And no, there is no such thing as heatiness. Again, the misconception probably came from the oil in those oily chili sauces that as i explained before, tends to be a good place for bacteria to grow. In fact i've been chewing on fresh cut chilli padis for years and i can tell you that not once have i gotten sick from eating it.
Also ingesting chilli seeds certainly will not lead to "stones" in the stomach. In fact the fruits of seeds are routinely eaten by birds, who pass it out in their faeces. This helps to disperse them. Primary school science hello..
So there you have it. As long as you watch your weight and cook your food well and keep proper oral hygeine after you eat, everything is healthy to eat.
I admit that while most of the claims have been confirmed by doctors and achieve regular consistency on nutrition websites, there are one or two that are deduced by myself using the most elementary aspects of medicine and science and logic.
Life is already as restricted as it is. Why restrict yourself further with unbased claims?
It seems incredible to me that in this modern age of science and technology, there are still people who believe age old myths about food that are ridiculous and fantastic. How many of you believe that prawns and shrimps are "toxic"? How about the idea that Kang Kong makes your legs weak? Or even that potato chips are "heaty" and give you sorethroat?
After a long 5 months of staying in the hospital and thus having an increased knowledge of medicine, I can assure you that no vegetable sold in the supermarket has the amazing abilliy to make your legs weak.
Also, "prawns and shrimps are extremely good sources of protein and yet very low in fats and calories, making them very healthy to eat"(http://www.helpwithcooking.com/) . Prawns are also not known to have any toxins in them, and certainly nobody has ever gotten sick from eating a fresh, healthy and well-cooked prawn.
Many chinese, on the other hand, believe in the heatiness and cooling theory. Which is a load of bullshit. In fact, i discovered this once when i had a sorethroat. i thought about how people always said that salt cures ulcers and sorethroat, and thought of how to get large amounts of salt into the throat. Haha i came up with an ingenious plan. Eat extra salted potato chips! Incredibly, my sorethroat instantly felt better and disappeared the next day. In fact, this method works most of the time. Proving that the heatiness theory is incorrect. In fact, don't you find it ridiculous that food can be classified into two such broad categories(heaty and cold)? Common sense will tell you that they are insufficient to understand nutrition. On top of that, sorethroats, colds, flu, fever.. etc are all known to be caused by bacteria and viruses. Most people know and accept this, but why is it then that they still think being too heaty can cause you to be ill? Aren't they contradicting themselves? Its like believeing in both Christianity and Hinduism. Choose one religion man!
In fact, after confirmation with a doctor, i found out that the reason many people have sorethroats after eating nuts and potato chips is because the rough and hard texture of the food tends to scratch against the walls of the throat, possible damaging it. Also the oil from the chips, if not washed away by drinking lots of water, tends to repel saliva, which has natural anti-bacterial properties. Thus being a more potent ground for bacteria to grow. So there you have it. I beg people to trust empirical evidence and not to place too much faith in far-fetched claims. Question every information before accepting it, it will make you smarter.
And no Maggi Mee does not cause you to lose your hair. Neither does it have the fictitious "wax" thats get trapped in your intestines.
Chilli is an excellent source of vitamin C, much higher than orange in fact. And as a fruit, is very healthy. And no, there is no such thing as heatiness. Again, the misconception probably came from the oil in those oily chili sauces that as i explained before, tends to be a good place for bacteria to grow. In fact i've been chewing on fresh cut chilli padis for years and i can tell you that not once have i gotten sick from eating it.
Also ingesting chilli seeds certainly will not lead to "stones" in the stomach. In fact the fruits of seeds are routinely eaten by birds, who pass it out in their faeces. This helps to disperse them. Primary school science hello..
So there you have it. As long as you watch your weight and cook your food well and keep proper oral hygeine after you eat, everything is healthy to eat.
I admit that while most of the claims have been confirmed by doctors and achieve regular consistency on nutrition websites, there are one or two that are deduced by myself using the most elementary aspects of medicine and science and logic.
Life is already as restricted as it is. Why restrict yourself further with unbased claims?
Sunday, April 20, 2008
Calculus
Well its been a long time since i did some real mahs.. Besides i promised Teck Siang. So here it is.
y=f(x)
dy/dx=f '(x) (1)
On the other hand,
y=f(x)
F(y)=x , where F is the inverse function of f.
F '(y)=dx/dy
dy/dx=1/F '(y)
dy/dx=1/F '[f(x)] (2)
Combining equations (1) and (2), we have
f '(x)=1/F '[f(x)]
F '[f(x)]=1/f '(x)
Well most of you (including myself when i came up with this) must be thinking, wtf is that, and how can something like that actually be useful?
Well my original intention was to find an equation to relate the inverse function with its derivative. Which would have made me famous overnight..XD Sadly, this cannot be manipulated to give me such an equation, but it does have interesting uses.
for eg to find the integral of 1/x, we let f '(x)=1/x
then, f(x) must be the integral.
F '[f(x)]=1/f '(x)
F '[f(x)]=1/1/x=x
since the equation is an identity in x, we can replace every x in the equation with F(x)
F '[f(F(x))]=F(x)
since F is the inverse of f, f[F(x)] must be equal to x
F '(x)=F(x)
Now think about what this equation is trying to say. Its simply telling you that the derivative of F(x) is equal to itself. In fact we only know one such function , which is e^x. Haha..
So since F(x)=e^x
f(x) must be equal to the inverse of e^x, which is lnx.
f(x)=lnx
Thus we have shown, without doing ANY differentiation or inteegration of both 1/x and lnx, that the integral of 1/x is lnx. Quite cool. Don't you think? In fact the only two important properties we have used are,
1/dy/dx=dx/dy
and that
e^x is its own derivative.
Sadly, this "trick" that i've used doesn't work for other functions.. Maybe i can develop it further.. hmm...
In any case, I think either i'm going bonkers, otherwise I'm getting smarter.
A little sidenote: A lot of people coming to my site tell me they don't understand the mathematics on my site. In fact, that is perfectly normal. Even for professional mathematicians, they do not understand new mathematics the first time they read it. Most of the time, hey have to analyze and go through the text many times before they can fully grasp it. This is where the problem with many people lie. When they do not understand the text, they do not take the effort to properly anaylyze or think about the text. Just like music and art and literature, effort must be made to understand the subject before one can fully appreciate its beauty. And i promise you, the mathematical beauty that we speak of is certainly real. (with the exception of complex mathematics, where things are imaginary XD)
In fact i find it a waste that people get put off by mathematics because it seems so hard to grasp. Mathematics is a language more useful than any other.
y=f(x)
dy/dx=f '(x) (1)
On the other hand,
y=f(x)
F(y)=x , where F is the inverse function of f.
F '(y)=dx/dy
dy/dx=1/F '(y)
dy/dx=1/F '[f(x)] (2)
Combining equations (1) and (2), we have
f '(x)=1/F '[f(x)]
F '[f(x)]=1/f '(x)
Well most of you (including myself when i came up with this) must be thinking, wtf is that, and how can something like that actually be useful?
Well my original intention was to find an equation to relate the inverse function with its derivative. Which would have made me famous overnight..XD Sadly, this cannot be manipulated to give me such an equation, but it does have interesting uses.
for eg to find the integral of 1/x, we let f '(x)=1/x
then, f(x) must be the integral.
F '[f(x)]=1/f '(x)
F '[f(x)]=1/1/x=x
since the equation is an identity in x, we can replace every x in the equation with F(x)
F '[f(F(x))]=F(x)
since F is the inverse of f, f[F(x)] must be equal to x
F '(x)=F(x)
Now think about what this equation is trying to say. Its simply telling you that the derivative of F(x) is equal to itself. In fact we only know one such function , which is e^x. Haha..
So since F(x)=e^x
f(x) must be equal to the inverse of e^x, which is lnx.
f(x)=lnx
Thus we have shown, without doing ANY differentiation or inteegration of both 1/x and lnx, that the integral of 1/x is lnx. Quite cool. Don't you think? In fact the only two important properties we have used are,
1/dy/dx=dx/dy
and that
e^x is its own derivative.
Sadly, this "trick" that i've used doesn't work for other functions.. Maybe i can develop it further.. hmm...
In any case, I think either i'm going bonkers, otherwise I'm getting smarter.
A little sidenote: A lot of people coming to my site tell me they don't understand the mathematics on my site. In fact, that is perfectly normal. Even for professional mathematicians, they do not understand new mathematics the first time they read it. Most of the time, hey have to analyze and go through the text many times before they can fully grasp it. This is where the problem with many people lie. When they do not understand the text, they do not take the effort to properly anaylyze or think about the text. Just like music and art and literature, effort must be made to understand the subject before one can fully appreciate its beauty. And i promise you, the mathematical beauty that we speak of is certainly real. (with the exception of complex mathematics, where things are imaginary XD)
In fact i find it a waste that people get put off by mathematics because it seems so hard to grasp. Mathematics is a language more useful than any other.
Tuesday, April 15, 2008
Back home.
Phew. Finally back home after 1 week of hell.. Luckily the Indian Lumbar Puncturer was quite skillful. I like Indian doctors. They're friendly and really really professional. Unlike some chinese doctors wth that stuck up attitudes.. :X I guess its due partly to their family upbringings... How chinese mothers always tell their kids "you'd better become a doctor/lawyer!" And hows its considered a big deal if you become one.. I've always disliked doctors.. They place too much faith in their science.. Which like all sciences, cannot be proven to be true.. How can you place a human life on uncertainty?
Well. Thats what i like about maths. Things are nearly always certain.
Anyway, not much to write about.. Just returned home after yet another session of chemo.. Its good to know that i've only bout two months of chemo left if nothing goees wrong. After that i should be able to get back to my life. Although it is a bit sad to know that so many things have changed. And how some things can never be like the way it was. But well life goes on.. Hopefully, that is..XD
And this morning i dreamt that i had grown back my hair! I spent the whole of my dream styling and combing it.. Haha turns out i really miss it..
I also dreamt of Shi Xiong and Isabel and Catherine.. And we were all playing tag in an arcade.. -.-
Anyone who can crack dreams?
Well. Thats what i like about maths. Things are nearly always certain.
Anyway, not much to write about.. Just returned home after yet another session of chemo.. Its good to know that i've only bout two months of chemo left if nothing goees wrong. After that i should be able to get back to my life. Although it is a bit sad to know that so many things have changed. And how some things can never be like the way it was. But well life goes on.. Hopefully, that is..XD
And this morning i dreamt that i had grown back my hair! I spent the whole of my dream styling and combing it.. Haha turns out i really miss it..
I also dreamt of Shi Xiong and Isabel and Catherine.. And we were all playing tag in an arcade.. -.-
Anyone who can crack dreams?
Tuesday, April 8, 2008
Video Games
I love video games. Been playing them a lot recently, especially since God of War and FFVII:CC have both just been recently released.
I remember my first ever video game console to be a pirated version of NES my Dad got for me in 1998. It came at a price of about 70 bucks and had games like Duck Hunt and Mario. I only had about 2-3 cartridges and of course, the only real game was Mario. But i sucked at it though. and hence didn't like the console very much.
My first real step into the world of real gaming was probably simultaneous with thousands of boys islandwide, with one of the greatest games of all time,
Pokemon! My Dad first got me a game boy colour in late 1999, then when the game became famous in 2000 i got myself a yellow version and started my badge-collecting adventure in the lands of Kanto. The yellow version was really great since it allowed players to obtain all three staters (Bulbasaur, Squirtle and Charmander) without trading. The only downside was that Raichu was unobtainable in the game. i remember desperately trying to evolve my Pikachu with a thunderstone only to realize that it could not be done.
Pokemon yellow was really one of my favourite games. In fact i loved it so much that when i cracked my teeth in primary 3 and my mum asked me what i wanted after the tooth surgery i told her "pokemon yellow guidebook". And with that tool in hand i almost completed the pokedex, with the exception of Mew and one other pokemon i cannot remember. Of course i still continued to play the game and i was hustled once at Macdonalds when a group of secondary school students tricked me into trading my level 70 Mewtwo for a level 100 dragonite, telling me how it was very rare and unobtainable in Singapore cartridges. -.- However, that wasn't the only stupid thing i did. After obtaining the Master Ball in Vermillion, i set out like normal players to the seafoam islands. Upon reaching the twin islands, i saw a Seel and thinking that it was an extremely rare Pokemon(since it used ice beam and that just sounded so powerful since prior to that ice moves were unheard of), used the one and only most powerful pokeball in the game to capture it.. Idiot.
And with that Pokemon remains one of my favourite games till now. In fact even after most of my friends grew out of it, i continued collecting and battling with my neighbour-friends, Royce and Loic. (most of my classmates follwed at most only up till the crystal version)
Continuing with my video gaming adventure, the next huge step was in getting a Sony Playstation. My dad bought me the playstation in 2000 when he won money playing mahjong. But admittedly, the console failed for me because i did not know which games to buy. For eg, at that time i never even knew that final fantasy existed. And all my games were real lousy. My only real games were Metal Slug ,a Monster Rancher card game ,Digimon World(which sucked like fuck), and Gran Turismo which admitedly if you're not a fan, isn't real exciting.
And so for a while i switched my attention to Tamiya mini 4WD cars and Beyblades(Let it rip!!!). Until the arrival of Super yoyo in 2002. Before that, the only thing that i knew about yoyo's was that you bought them in cheap 1 dollar goodies machines. Those that you put a coin into the slot and twisted a knob that would release a single "goodie ball", kind of like a pokeball, and woud open to release a small yoyo inside. (Damm what are those things called?!) And that you could perform basic tricks like the sleeper, walk the dog and rock the baby with it. My dad happened to be quite good at it, showing and teaching me all the basic tricks. In fact my first brush with yoyo was in 1997 when proyo was released, but at the time i was simply too young and too small to know anything. Of course the main thing with Super yoyo was the anime on kids central. At first, i didn't actually think that the tricks shown on tv could actually be done. I thought that it was all a cartoon exageration, like how they could control beyblades with their mindXD, until i saw a trick being done at Ang Mo Kio games shop Rowell. I remember asking the boy how it was done. He was very cocky and showed off to me some other complex string tricks and told me "you cannot get it one la". Haha. That made me very angry and i spent that entire month training and practicing on my own, going to the internet for tips. In the end, i mastered all the complex tricks the cartoons did and was more pro than the boy. I went back to Rowell to show off, but i never did meet up with him again.
Oh my, i seemed to have strayed off the topic haven't I? Actually no, because in early 2003 while strolling along Orchard road with Royce we stumbled across a yoyo competition hosted by Toys 'R' Us for young kids till 13. Both Royce and me signed up. Royce wasn't good enough, but I got second:D.
Actually i should have gotten first, because the last match was between me and some primary 5 chiobu. And the one who did more loop the loops won. I was first. I did a fucking hundred loops, to the audiences wild applause. (They were like counting with me "95-96.. 100!!!!!!" clap clap clap*) then the judge came in and checked my string configuration and told me, "the 3-knot is not allowed." Walau, the three-knot is a professional string configuration used worldwide by pros to help looping. and you didn't even tell me beforehand that you didn't allow it. Whats worse they did not allowed me to have time to change my string(the yoyo string gets fucked up when you do that many loops) and straightaway asked me to redo the looping. Faced with a useless string and a string configuration i was not used to, i only managed to do 14 loops. While the girl, they never did check her string configration, but i realized only when my dad told me after the competition, did 28. But in fact the girls string was so short that she was more of like doing inside around the worlds. yoyo players would know what i mean. And she got first. Obviously, the judges were favouring the little girl.
Anyway with that i got second and a $250 voucher. I used the voucher wisely to get myself a Game Boy Advance SP and my mum bought me my first game, Pokemon Sapphire. Phew finally back to videogames.
With the game boy advance, i discovered a term that would be my most favourite genre of games till now. RPG's aka role playing games. Also i had my first brush with another of the greatest games of all time, FINAL FANTASY. This was in the form of FF:Tactics Advance for the GBA. i quickly became a pro at it.
When i transitioned into secondary school life, i became a true gamer, extensive use of the internet meant that i now knew which games were hot and which were not, and about the latest gaming developments. i got myself the psp(before its official release) and ps2 in 2005. The nintendo ds in late 2006 and of couse the Wii late last year. Playing games like Final Fantasy X(the only game to which i cried playing it.), Final Fantasy X-2 , Start Ocean 3, Tekken, Gran Turismo, Kingdom Hearts and of course, the ever popular Pokemon! I regret a bit about getting the Wii due to a decreased lack of support from Square enix.. But i would probably be getting a Playstation 3 late this year.
Of course i'm not shy to admit that while i am an avid gamer, i am clearly not good in it. Especially in FPS games like Halo which requires ultra fast reflexes and strategy and games like Age of empires which requires managing resources, i routinely get thrashed by others even non-gaming individuals. The place where i'm good at are rpg's but even then i loseout to people like Kenneth who can complete FFX in a few weeks. I'm sure he would lose to me in a batlle though! Too bad most rpg's are solo. As for online mmorpg's, i just don't have the time nor money to play them.
Haha with that its time to say goodbye. What a long story! Ahh tomorrow go back to hospital again... NOOOOO!!! :'(
I remember my first ever video game console to be a pirated version of NES my Dad got for me in 1998. It came at a price of about 70 bucks and had games like Duck Hunt and Mario. I only had about 2-3 cartridges and of course, the only real game was Mario. But i sucked at it though. and hence didn't like the console very much.
My first real step into the world of real gaming was probably simultaneous with thousands of boys islandwide, with one of the greatest games of all time,
Pokemon! My Dad first got me a game boy colour in late 1999, then when the game became famous in 2000 i got myself a yellow version and started my badge-collecting adventure in the lands of Kanto. The yellow version was really great since it allowed players to obtain all three staters (Bulbasaur, Squirtle and Charmander) without trading. The only downside was that Raichu was unobtainable in the game. i remember desperately trying to evolve my Pikachu with a thunderstone only to realize that it could not be done.
Pokemon yellow was really one of my favourite games. In fact i loved it so much that when i cracked my teeth in primary 3 and my mum asked me what i wanted after the tooth surgery i told her "pokemon yellow guidebook". And with that tool in hand i almost completed the pokedex, with the exception of Mew and one other pokemon i cannot remember. Of course i still continued to play the game and i was hustled once at Macdonalds when a group of secondary school students tricked me into trading my level 70 Mewtwo for a level 100 dragonite, telling me how it was very rare and unobtainable in Singapore cartridges. -.- However, that wasn't the only stupid thing i did. After obtaining the Master Ball in Vermillion, i set out like normal players to the seafoam islands. Upon reaching the twin islands, i saw a Seel and thinking that it was an extremely rare Pokemon(since it used ice beam and that just sounded so powerful since prior to that ice moves were unheard of), used the one and only most powerful pokeball in the game to capture it.. Idiot.
And with that Pokemon remains one of my favourite games till now. In fact even after most of my friends grew out of it, i continued collecting and battling with my neighbour-friends, Royce and Loic. (most of my classmates follwed at most only up till the crystal version)
Continuing with my video gaming adventure, the next huge step was in getting a Sony Playstation. My dad bought me the playstation in 2000 when he won money playing mahjong. But admittedly, the console failed for me because i did not know which games to buy. For eg, at that time i never even knew that final fantasy existed. And all my games were real lousy. My only real games were Metal Slug ,a Monster Rancher card game ,Digimon World(which sucked like fuck), and Gran Turismo which admitedly if you're not a fan, isn't real exciting.
And so for a while i switched my attention to Tamiya mini 4WD cars and Beyblades(Let it rip!!!). Until the arrival of Super yoyo in 2002. Before that, the only thing that i knew about yoyo's was that you bought them in cheap 1 dollar goodies machines. Those that you put a coin into the slot and twisted a knob that would release a single "goodie ball", kind of like a pokeball, and woud open to release a small yoyo inside. (Damm what are those things called?!) And that you could perform basic tricks like the sleeper, walk the dog and rock the baby with it. My dad happened to be quite good at it, showing and teaching me all the basic tricks. In fact my first brush with yoyo was in 1997 when proyo was released, but at the time i was simply too young and too small to know anything. Of course the main thing with Super yoyo was the anime on kids central. At first, i didn't actually think that the tricks shown on tv could actually be done. I thought that it was all a cartoon exageration, like how they could control beyblades with their mindXD, until i saw a trick being done at Ang Mo Kio games shop Rowell. I remember asking the boy how it was done. He was very cocky and showed off to me some other complex string tricks and told me "you cannot get it one la". Haha. That made me very angry and i spent that entire month training and practicing on my own, going to the internet for tips. In the end, i mastered all the complex tricks the cartoons did and was more pro than the boy. I went back to Rowell to show off, but i never did meet up with him again.
Oh my, i seemed to have strayed off the topic haven't I? Actually no, because in early 2003 while strolling along Orchard road with Royce we stumbled across a yoyo competition hosted by Toys 'R' Us for young kids till 13. Both Royce and me signed up. Royce wasn't good enough, but I got second:D.
Actually i should have gotten first, because the last match was between me and some primary 5 chiobu. And the one who did more loop the loops won. I was first. I did a fucking hundred loops, to the audiences wild applause. (They were like counting with me "95-96.. 100!!!!!!" clap clap clap*) then the judge came in and checked my string configuration and told me, "the 3-knot is not allowed." Walau, the three-knot is a professional string configuration used worldwide by pros to help looping. and you didn't even tell me beforehand that you didn't allow it. Whats worse they did not allowed me to have time to change my string(the yoyo string gets fucked up when you do that many loops) and straightaway asked me to redo the looping. Faced with a useless string and a string configuration i was not used to, i only managed to do 14 loops. While the girl, they never did check her string configration, but i realized only when my dad told me after the competition, did 28. But in fact the girls string was so short that she was more of like doing inside around the worlds. yoyo players would know what i mean. And she got first. Obviously, the judges were favouring the little girl.
Anyway with that i got second and a $250 voucher. I used the voucher wisely to get myself a Game Boy Advance SP and my mum bought me my first game, Pokemon Sapphire. Phew finally back to videogames.
With the game boy advance, i discovered a term that would be my most favourite genre of games till now. RPG's aka role playing games. Also i had my first brush with another of the greatest games of all time, FINAL FANTASY. This was in the form of FF:Tactics Advance for the GBA. i quickly became a pro at it.
When i transitioned into secondary school life, i became a true gamer, extensive use of the internet meant that i now knew which games were hot and which were not, and about the latest gaming developments. i got myself the psp(before its official release) and ps2 in 2005. The nintendo ds in late 2006 and of couse the Wii late last year. Playing games like Final Fantasy X(the only game to which i cried playing it.), Final Fantasy X-2 , Start Ocean 3, Tekken, Gran Turismo, Kingdom Hearts and of course, the ever popular Pokemon! I regret a bit about getting the Wii due to a decreased lack of support from Square enix.. But i would probably be getting a Playstation 3 late this year.
Of course i'm not shy to admit that while i am an avid gamer, i am clearly not good in it. Especially in FPS games like Halo which requires ultra fast reflexes and strategy and games like Age of empires which requires managing resources, i routinely get thrashed by others even non-gaming individuals. The place where i'm good at are rpg's but even then i loseout to people like Kenneth who can complete FFX in a few weeks. I'm sure he would lose to me in a batlle though! Too bad most rpg's are solo. As for online mmorpg's, i just don't have the time nor money to play them.
Haha with that its time to say goodbye. What a long story! Ahh tomorrow go back to hospital again... NOOOOO!!! :'(
Friday, April 4, 2008
Oh no...
Just returned home on Monday. This round of chemo had a bit of problems. The sedation caused me to feel sick. And i had a bad backache due to the bone marrow extraction. Siao liao la.. One week plus never post anything! (Nobody's going to visit my blog anymore!)
Daniel came over to HQ yesterday to visit me, though i think playing pool was his most imporant objective. Anyway, glad to know that i Can still beat him in pool, or maybe it was just him pitying a sick patient, but i doubt so XD. (Anyway i wasn't playing seriously daniel!) Goes to show that the cue really doesn't make much difference in gameplay. At least thats what i think. Besides, I feel that I am living proof that n the hands of a capable pool player, all shots that can be done with a thousand-over-dollar 314 predator cue can also be done with a cheap 10 dollar 2nd-hand cash converters cue, albeit with as much effort. Thus i despise people with expensive cues XD. In fact a normal new cue worth about 100-200 dollars is about as good enough a cue as you'll ever need unless you're a professional pool player. And no, i don't think anyone in Singapore qualifies to be proffesional. My idea of a professional pool player is a person who can survive solely on playing pool. And that excludes coaching, setting up pool parlours.. etc...
Okay, enough of all the pool and hot air. Lets move onto the real fun and the real world. Mathematics. Indeed, Mathematics is the code that the world is designed with. Equations and laws in physics, chemistry, engineering that so accurately describes the real world are written and explained not philosophically nor spiritually, but mathematically.
If there was indeed a God, he would have hidden mathematical secrets into the laws of the real world. By studying and revealing these laws, we are discovering the creator's little treats. And that makes Euclid's Elements more religious than the holy bible!
Jokes aside, lets get onto some serious fun.
Today, we will show that the harmonic series diverges. Known since about the 1300's. If you can't follow this proof, either i am a very bad explainer or that your level of intelligence is 700 years old!
The harmonic series is 1+1/2+1/3+1/4+1/5+1/6.... and so on. Upon looking at this series, most people's reaction would be that the series converges to a value, since the terms ultimately becomes zero. Indeed, the fact the the individual terms become zero is a neccesary but not sufficient condition of the convergence of a series. If the terms do not zoom off to zero fast enough, it is still possible that the entire value of the sum diveges(becomes infintiy).
Daniel came over to HQ yesterday to visit me, though i think playing pool was his most imporant objective. Anyway, glad to know that i Can still beat him in pool, or maybe it was just him pitying a sick patient, but i doubt so XD. (Anyway i wasn't playing seriously daniel!) Goes to show that the cue really doesn't make much difference in gameplay. At least thats what i think. Besides, I feel that I am living proof that n the hands of a capable pool player, all shots that can be done with a thousand-over-dollar 314 predator cue can also be done with a cheap 10 dollar 2nd-hand cash converters cue, albeit with as much effort. Thus i despise people with expensive cues XD. In fact a normal new cue worth about 100-200 dollars is about as good enough a cue as you'll ever need unless you're a professional pool player. And no, i don't think anyone in Singapore qualifies to be proffesional. My idea of a professional pool player is a person who can survive solely on playing pool. And that excludes coaching, setting up pool parlours.. etc...
Okay, enough of all the pool and hot air. Lets move onto the real fun and the real world. Mathematics. Indeed, Mathematics is the code that the world is designed with. Equations and laws in physics, chemistry, engineering that so accurately describes the real world are written and explained not philosophically nor spiritually, but mathematically.
If there was indeed a God, he would have hidden mathematical secrets into the laws of the real world. By studying and revealing these laws, we are discovering the creator's little treats. And that makes Euclid's Elements more religious than the holy bible!
Jokes aside, lets get onto some serious fun.
Today, we will show that the harmonic series diverges. Known since about the 1300's. If you can't follow this proof, either i am a very bad explainer or that your level of intelligence is 700 years old!
The harmonic series is 1+1/2+1/3+1/4+1/5+1/6.... and so on. Upon looking at this series, most people's reaction would be that the series converges to a value, since the terms ultimately becomes zero. Indeed, the fact the the individual terms become zero is a neccesary but not sufficient condition of the convergence of a series. If the terms do not zoom off to zero fast enough, it is still possible that the entire value of the sum diveges(becomes infintiy).
To the actual calculations.
S=1+1/2+1/3+1/4+1/5+1/6..
notice that we can group the terms like this.
S=[1]+[1/2]+[1/3+1/4]+[1/5+1/6+1/7+1/8]+.....
S=1+1/2+1/3+1/4+1/5+1/6..
notice that we can group the terms like this.
S=[1]+[1/2]+[1/3+1/4]+[1/5+1/6+1/7+1/8]+.....
Each of the grouped terms is more than 1/2.
Since there are an infinity of grouped terms, there an infinite number of terms more than half, and thus S is infinity as well..
But the divergence is extremely slow.. Indeed no computer would even hint at its divergence in a short period of time. For example. if we sum up the first 10^43 terms we would only get something less than 100.
But the divergence is extremely slow.. Indeed no computer would even hint at its divergence in a short period of time. For example. if we sum up the first 10^43 terms we would only get something less than 100.
Which is like. tiao-.-
Just goes to show that mathematicians are smarter than computers:D.
(more hot air)
In the next post i'll show why people who can go to jc but choose to go poly are (in my opinion) idiots-Mathematically..
Monday, March 24, 2008
17
Ahh.. Been busy over the weekend. Still studying transformation of double integrals to polar forrm.. Maths is a tough subject. Haha but i mastered partial fractions in a day yesterday. Really not much to it.
And i guess i should talk a lttle bit about it. I'm Seventeen!! But whats so special about being seventeen anyway? You still can't drink. Can't drive. Can't watch porno movies. So i guesss theress really nothing special about being 17. Except that its a prime numberXD Sorry i'm number sensitive.
So my cousins and a few friends came over to HQ on Friday to wish me happy birthday. I guess i should thank them for coming but Soomin kept snatching my blackboard and thus preveented me from doing maths with darren-the only person who has the intelligence and patience to do tians mathematics. And seriously whats up with the chocalate gift? I hate chocalate. Its totally insincere
and a waste of your money. (unless you're a pretty girl of course:D) I would have liked a maths book sooooo much better. But still, thanks everyone for visiting. I promise i'll treat you guys to kbox when i'm well.
And thanks to zuan cong for getting me the FIR album(see this is what you call a Real gift.) Its really nice.
Well whats a post in joeymath without math?
Today we will prove that there are an infinite number of primes.
But first you will have to know that every integer can be broken down into a product of primes, or it is itself prime. For eg. 15=3x5 This is called the fundamental theorem of arithmetic.
Okay we are ready for the proof.
imagine that there are a finite number of primes.
The product of all primes would be
(p1 x p2 x p3 x p4...........x pn) where p1 is the firsst prime and pn is the last.
we add one one this number and call the new number S
the new number S is an integer, and it is not a prime nnumber since the last prime was pn. Thus, by the fundamental theorem of arithmetic, it must be divisible by one prime.
S=(p1 x p2 x p3 x p4...........x pn)+1
But we realize that S is not divisible by any primes from p1 to pn since it will always give a remainder of 1. And we have a contradiction. Thus our original assuption that there are a finite number of primes must be false=There are an infite number of primes.
I love mathsXD
And i guess i should talk a lttle bit about it. I'm Seventeen!! But whats so special about being seventeen anyway? You still can't drink. Can't drive. Can't watch porno movies. So i guesss theress really nothing special about being 17. Except that its a prime numberXD Sorry i'm number sensitive.
So my cousins and a few friends came over to HQ on Friday to wish me happy birthday. I guess i should thank them for coming but Soomin kept snatching my blackboard and thus preveented me from doing maths with darren-the only person who has the intelligence and patience to do tians mathematics. And seriously whats up with the chocalate gift? I hate chocalate. Its totally insincere
and a waste of your money. (unless you're a pretty girl of course:D) I would have liked a maths book sooooo much better. But still, thanks everyone for visiting. I promise i'll treat you guys to kbox when i'm well.
And thanks to zuan cong for getting me the FIR album(see this is what you call a Real gift.) Its really nice.
Well whats a post in joeymath without math?
Today we will prove that there are an infinite number of primes.
But first you will have to know that every integer can be broken down into a product of primes, or it is itself prime. For eg. 15=3x5 This is called the fundamental theorem of arithmetic.
Okay we are ready for the proof.
imagine that there are a finite number of primes.
The product of all primes would be
(p1 x p2 x p3 x p4...........x pn) where p1 is the firsst prime and pn is the last.
we add one one this number and call the new number S
the new number S is an integer, and it is not a prime nnumber since the last prime was pn. Thus, by the fundamental theorem of arithmetic, it must be divisible by one prime.
S=(p1 x p2 x p3 x p4...........x pn)+1
But we realize that S is not divisible by any primes from p1 to pn since it will always give a remainder of 1. And we have a contradiction. Thus our original assuption that there are a finite number of primes must be false=There are an infite number of primes.
I love mathsXD
Wednesday, March 19, 2008
hyperbolic trigonometric functions!
Okay since i have nothing to write about these few days, (been enjoying my time at home watching Shen Diao Xia Lu and relaxing...) i shall do a bit of maths here..
Ever wondered what cosh and sinh and tanh mean? Find it , its on your calculator. Well, they can be equated using the expression below.
Where x is a real variable and i is the imaginary unit,
Cos(ix) = Cosh(x)
Sin(ix) = iSinh(x)
and of course,
tanh(x) = Sinh(x)/Cosh(x)
= -iSin(ix)/Cos(ix)
= -itan(ix)
They are interesting for they extend the range of the cosine and sine functions.
For eg,
Cos(10i)=Cosh(10)
=11013.23292..
(And yes the cosine of a purely imaginary number is real)
Sin(½π+6i)=Sin(½π)Cos(6i)+Cos(½π)Sin(6i)
Since the red part equals zero, we have
Sin(½π+6i)=Sin(½π)Cosh(6)
=201.7156...
Haha who said Sinx and Cosx must be in between 1 and -1?
(yupp and if anyone is interested in the proof, which i certainly hope so, i will probably be giving one soon)
And arghh... of all the side effects.. Why must it be mouth sores and ulcers?!?!? Now i can't eat all my favourite food. Like super hot chilli!! :'(
Okay challenge of the day. Solve Sin(z)=2
Ever wondered what cosh and sinh and tanh mean? Find it , its on your calculator. Well, they can be equated using the expression below.
Where x is a real variable and i is the imaginary unit,
Cos(ix) = Cosh(x)
Sin(ix) = iSinh(x)
and of course,
tanh(x) = Sinh(x)/Cosh(x)
= -iSin(ix)/Cos(ix)
= -itan(ix)
They are interesting for they extend the range of the cosine and sine functions.
For eg,
Cos(10i)=Cosh(10)
=11013.23292..
(And yes the cosine of a purely imaginary number is real)
Sin(½π+6i)=Sin(½π)Cos(6i)+Cos(½π)Sin(6i)
Since the red part equals zero, we have
Sin(½π+6i)=Sin(½π)Cosh(6)
=201.7156...
Haha who said Sinx and Cosx must be in between 1 and -1?
(yupp and if anyone is interested in the proof, which i certainly hope so, i will probably be giving one soon)
And arghh... of all the side effects.. Why must it be mouth sores and ulcers?!?!? Now i can't eat all my favourite food. Like super hot chilli!! :'(
Okay challenge of the day. Solve Sin(z)=2
Friday, March 14, 2008
The thing i fear most
Woohoo. Finally home after 4days in the hospital. I seem to be there more often than at home. Anyway, did a lumbar puncture this time. Did it a couple of times before, but none seemed to be as painful as this one. Luckily there were no side effects. Except a little bit of vomiting and headache, which were normal.
For all those who are wondering whats a lumbar puncture, it is a procedure where a needle is injected to the spine from the back so that the chemo medicine can reach the brain. (Yes, cancer cells can hide there too.) And normal chemo through the veins don't reach the brain well so.. Anyway i'm not a biologist.
Yupp thats the procedure right there. And if the picture doesn't show it, which i seriously think it does, it IS painful.
Doctor walks into room. "Okay weizhou are you ready?"
"No i'm not" Nah i didn't really said that i just kept quiet.
Doctor lifts up shirt and pulls down pants. Starts cleaning the area with iodine using circular motions. The area feels cold.
Doctor lifts up shirt and pulls down pants. Starts cleaning the area with iodine using circular motions. The area feels cold.
Dcotors prepares needles. (Damn long one later i show you.) "Okay weizhou going in ah" *PIERCE*
The Doctor continued pushing the needle deeper into the flesh when suddenly the syringe hits a sensitive spot, might be a nerve or bone or muscle mass. And i scream fuck repeatedly in my head. All the while in the bloody head-touching-knee crouching position.
The pain becomes too intense. I let out a groan. The chibye Doctor doesn't care, continues pushing the needle in. You were the one who said you didn't need sedation after all. (Sedation harms the body and has side effects like headache and nauseous. Usually used only on babies and young kids.) The doctor cannot finds the spot. Pulls the needle out and the horror begins again.
After two to three succesive tries, the doctor finally hits the spinal space. The medicine starts flowing in. That takes bout 1 minute. Finally the doctor pulls the needle out and applies a bandage on the now half swollen site. The procedure is complete. But the nightmare is not over.
After the physical torture, you are now asked to lie on the bed flat for 6 hours. And they mean totally flat. No pillow, no tilting the bed up even 10 degrees. No going to the toilet within the first 5 hours. Supposingly its to help restore the whatever pressure and help heal the wound. So being a good boy, i listen.
You try and get some sleep. Dreams of some chiobu with Edison chen. Wakes up. Shit. Only 1 hour has passed. But now you dun feel sleepy anymore. The excruciating backache is still there. You try to think of some maths formulas. Converting from cartesian to polar coordinates. Damn only 5 minutes has passed. You think about other things. All your other friends and what they are doing now. how you all used to play in the past. Okay 30 minute has passed.
And you wait and wait and wait for what seems like an eternity, but only 1 hour has passed. ITs freaking boring and freaking torturing but you can't move. You have to lie flat. You start singing some jay chou songs. Time still passes slowly.
And finally. 6 hours passed. You get up in excitement. Bad move. The chemo medicine makes you feel nauseous and the sudden movement makies it even worse. You vomit on the floor. The mee goreng from morning spills out from your gut in all its red glory. You start having a headache.
-End of story-
Okay luckily by the next day most of the side effects were gone except a little bit of nauseous and backache. And here are the needles used.
Advice: Eat more fruits and vegetables to prevent getting cancer.
Sunday, March 9, 2008
Sunday
Watched 倚天屠龍記 yesterday night till 4am.. Really nice story. Every chinese should know it. Anyway, today was a fun day. Soomin and Darren and Catherine came to hq today. (surprise, surprise) They watched a lot of jay chou and luo zhi qiang and S.H.E while darren and i played pool. Sad to say that Darren is still as thin.. "How to find girlfriend in JC"?? Thus as his good friend, i offered him a pack of ensure plus milk drink, "a nutritional supplement formulated for individuals with or at risk of developing malnutrition". Yupp, he definitely fits the bill. Glad to say that after taking the magic drink, he gained 1 kg. Nah, he didn't really. He needs to eat more. And my pool skills really suck now. Can't string 4-5 pots confidently now.
After about half and hour or so, Kenenth and Leeyen arrived as expected. Catherine left shortly after that, and i realized that the only words i said to her were "Hi" and "Bye". Sad. Isabel's right, useless male species. I Bet she was refering largely to me. Then we proceeded to play mahjong, while Soomin surfed the net on my beautiful bravia. Darren seemed to have lost some of his mahjong skills, while Kenneth and Leeyen were still as pro as ever. After that we had some of my Mum's beehoon. Then Soomin and Darren had to leave. After that was a bit of pool and mahjong and Kenneth and me relished the good old days by watching the TPA cup 06 videos. Its funny how time flies. It seemed like just yesterday Hexun and Teck Siang were asking me to teach them how to play. And the TPA was set up. And we were the best. Now Hexun is gone, I haven't seen Teck Siang for almost half a year, and Daniel's gone to join the pool fusion clique. Even Kenneth doesn't seem to play so much nowadays...
For dinner we had my Mum's first time chicken curry. It was really quite good, although i kept telling her its not hot enough. Curry and all chilli related dishes is supposed to be Burning hot, Spicy till the all the nerves in your tongue scream for help, and the heat from the chilli is so hot that it rushes up to your brain and numbs your tongue. Shiok man. Haha.. Anyway the curry was far from that, but since i loved curry, i decided to have 2 big servings of it.. Who said chemo patients had no appetite? After that its more pool and chitchat, and finally they had to leave. I was quite relived actually because i was very tired. Apparently I'm not a young kid anymore, can't play pool for 2-3 hours without feeling tired. After that i did a bit of Maths, still working on extending normal functions like sinx and e to the power of x to complex values of x. Its really fun. Finally know what the hyperbolic trigonometric functions are.
And here is the maths challenge for today. Who can tell me whats does this expression equal to?
√(x²) = ?
Please give your answers in the tagboard. :D
Clue: If its that easy it won't be a challenge.
P.S. to all those people who thought this blog was only bout math, i finally have non-mathematics dominated post.
After about half and hour or so, Kenenth and Leeyen arrived as expected. Catherine left shortly after that, and i realized that the only words i said to her were "Hi" and "Bye". Sad. Isabel's right, useless male species. I Bet she was refering largely to me. Then we proceeded to play mahjong, while Soomin surfed the net on my beautiful bravia. Darren seemed to have lost some of his mahjong skills, while Kenneth and Leeyen were still as pro as ever. After that we had some of my Mum's beehoon. Then Soomin and Darren had to leave. After that was a bit of pool and mahjong and Kenneth and me relished the good old days by watching the TPA cup 06 videos. Its funny how time flies. It seemed like just yesterday Hexun and Teck Siang were asking me to teach them how to play. And the TPA was set up. And we were the best. Now Hexun is gone, I haven't seen Teck Siang for almost half a year, and Daniel's gone to join the pool fusion clique. Even Kenneth doesn't seem to play so much nowadays...
For dinner we had my Mum's first time chicken curry. It was really quite good, although i kept telling her its not hot enough. Curry and all chilli related dishes is supposed to be Burning hot, Spicy till the all the nerves in your tongue scream for help, and the heat from the chilli is so hot that it rushes up to your brain and numbs your tongue. Shiok man. Haha.. Anyway the curry was far from that, but since i loved curry, i decided to have 2 big servings of it.. Who said chemo patients had no appetite? After that its more pool and chitchat, and finally they had to leave. I was quite relived actually because i was very tired. Apparently I'm not a young kid anymore, can't play pool for 2-3 hours without feeling tired. After that i did a bit of Maths, still working on extending normal functions like sinx and e to the power of x to complex values of x. Its really fun. Finally know what the hyperbolic trigonometric functions are.
And here is the maths challenge for today. Who can tell me whats does this expression equal to?
√(x²) = ?
Please give your answers in the tagboard. :D
Clue: If its that easy it won't be a challenge.
P.S. to all those people who thought this blog was only bout math, i finally have non-mathematics dominated post.
Friday, March 7, 2008
Math Magic demystified
Just returned home from the hospital. It has been a long stay(8 days), and its good to be home.
In the previous post, i showed a simple number trick. Let me now prove that it works for any three digit number.
Notice that all three digit numbers can be represented as 100a+10b+c where a,b and c are positive integers. This is simple stuff from primary school. So for eg, the number 684 would be 6(100)+8(10)+4. With that clarified, lets continue.
We start with the three digit number 100a+10b+c
Its reverse would be 100c+10b+a
their difference would be 100a+10b+c-100c-10b-a
which is equal to 100(a-c)+(c-a)=100(a-c)-(a-c)
=99(a-c)
since a-c can range only from 1 to 9 and is an integer, the difference of the two three digit numbers can only take 9 values . They are,
99 , 198 , 297 , 396 , 495 , 594 , 693 , 792 , 891
It is not difficult to see that the sum of the digits of all these numbers equal to 18, a pretty little coincidence, don't you think? But whats more beautiful is the technique used to prove the theorem. Originally, we had 990 numbers to test. But by using logic, we reduced that number to only 9 seperate cases. With todays technology, we could of course use a normal household computer to test all 990 numbers-it would probably finish the task before you can say "proven". However, there are certain theorems in mathematics that deal with an infinite number of numbers. And infinity is something not even the world's total processing power can attack. Besides, don't you think my short and simple proof is more elegant than the brute force of a computer?
In the previous post, i showed a simple number trick. Let me now prove that it works for any three digit number.
Notice that all three digit numbers can be represented as 100a+10b+c where a,b and c are positive integers. This is simple stuff from primary school. So for eg, the number 684 would be 6(100)+8(10)+4. With that clarified, lets continue.
We start with the three digit number 100a+10b+c
Its reverse would be 100c+10b+a
their difference would be 100a+10b+c-100c-10b-a
which is equal to 100(a-c)+(c-a)=100(a-c)-(a-c)
=99(a-c)
since a-c can range only from 1 to 9 and is an integer, the difference of the two three digit numbers can only take 9 values . They are,
99 , 198 , 297 , 396 , 495 , 594 , 693 , 792 , 891
It is not difficult to see that the sum of the digits of all these numbers equal to 18, a pretty little coincidence, don't you think? But whats more beautiful is the technique used to prove the theorem. Originally, we had 990 numbers to test. But by using logic, we reduced that number to only 9 seperate cases. With todays technology, we could of course use a normal household computer to test all 990 numbers-it would probably finish the task before you can say "proven". However, there are certain theorems in mathematics that deal with an infinite number of numbers. And infinity is something not even the world's total processing power can attack. Besides, don't you think my short and simple proof is more elegant than the brute force of a computer?
Tuesday, March 4, 2008
Mathematical magic.
Time to do some fun maths. Some really fun math.
This one's actually REALLY famous...
Think of a three digit number. for eg 348
Reverse the digits. 843
subtract the smaller number from the bigger number
843-348=495
add up all three digits and you will get 18.
Try it, it works for any three digit number. the only condition is that the two numbers must be different. for eg you can't take 555 because its reverse is also 555.
Haha try thinking bout it, i'll explain why this always work in the next post.
Fun right?
This one's actually REALLY famous...
Think of a three digit number. for eg 348
Reverse the digits. 843
subtract the smaller number from the bigger number
843-348=495
add up all three digits and you will get 18.
Try it, it works for any three digit number. the only condition is that the two numbers must be different. for eg you can't take 555 because its reverse is also 555.
Haha try thinking bout it, i'll explain why this always work in the next post.
Fun right?
Monday, March 3, 2008
Compressing the cubic.
Hi everyone. Still stuck in the hospital, can't got home. Apparently i'm still on antibiotics. (Not the kind that you swallow with water, the kind that goes directly into your veins.) Pretty much nothing to do here except maths and wahjong.. Thanks for the great site lee yen XD. I suck at wahjong tho, 4 wins, 17 losses?!?!?
Anyway i promised in the last blog that i would show how every cubic equation can be reduced to the form x³+px+r
we start with the general cubic x³+tx²+wx+z=0 (if the coefficient of x³ is not one we can just divide through to get one)
then we make the substitution x=y-t/3
(y-t/3)³+t(y-t/3)²+w(y-t/3)+z=0
y³-t³/27-y²t+t²y/3+ty²-2yt²/3+t³/9+wy-tw/3+z=0
see that the y²'s cancel out beautifully :D
y³+(t²/3-2t²/3+w)y+(-t³/27+t³/9-tw/3+z)
And there we have it! The cubic has been reduced to the form y³+py+r=0
with p=(t²/3-2t²/3+w) and r=(-t³/27+t³/9-tw/3+z)
I know it looks devilish, but basically all the technques used are from elementary algbra. In fact this particular solution was found in the early 17 century and obviously the techniques used were already known to the greeks over two thousand years ago. Actually it not all that intimidating if you do it with real numbers instead of t's,w's and z's. (really! i've done it countless of times!)
P.S. kk food sucks, my dad has been sneaking in laksa and mee rubus from downstairs much to the disapproval of the doctors. "Your immune is very low, outside food dunno whether clean or not, easy to get infected, then you'll become very sick"... heck care...
Anyway i promised in the last blog that i would show how every cubic equation can be reduced to the form x³+px+r
we start with the general cubic x³+tx²+wx+z=0 (if the coefficient of x³ is not one we can just divide through to get one)
then we make the substitution x=y-t/3
(y-t/3)³+t(y-t/3)²+w(y-t/3)+z=0
y³-t³/27-y²t+t²y/3+ty²-2yt²/3+t³/9+wy-tw/3+z=0
see that the y²'s cancel out beautifully :D
y³+(t²/3-2t²/3+w)y+(-t³/27+t³/9-tw/3+z)
And there we have it! The cubic has been reduced to the form y³+py+r=0
with p=(t²/3-2t²/3+w) and r=(-t³/27+t³/9-tw/3+z)
I know it looks devilish, but basically all the technques used are from elementary algbra. In fact this particular solution was found in the early 17 century and obviously the techniques used were already known to the greeks over two thousand years ago. Actually it not all that intimidating if you do it with real numbers instead of t's,w's and z's. (really! i've done it countless of times!)
P.S. kk food sucks, my dad has been sneaking in laksa and mee rubus from downstairs much to the disapproval of the doctors. "Your immune is very low, outside food dunno whether clean or not, easy to get infected, then you'll become very sick"... heck care...
Sunday, March 2, 2008
On how to solve a cubic equation.
In this post i will be explaining how to solve a cubic equation. While the quadratic formula was known to the babylonians since ancient times and should be easily derived by a good elementary algebra student (if you can't you can most probably stop reading..), the cubic equation has been considerably more difficult to crack.
At this point some of you might be wondering, "wait, haven't we already learnt how to solve a cubic equation in amaths? All you have to do is to guess one root and then divide the cubic by the guessed root and solve the resulting quadratic equation." Aha heres where the problem lies in this method.
1) You're guessing one root. which is cheating. You might as well guess all three roots and say "solved".
2) In effect you're actually only solving a quadratic.
3) Try guessing the roots to the cubic 61x³-5x²-3271x+305 . They are approximately equal to -7.328338841 , 7.317060569 , 0.0932454851!
And for those of you who rely on a calculator or computer to solve cubics without any idea of whats going on, congrats for having the understanding level of a primary school kid.
And thus i think it is worthwhile knowing the method to solve general cubics.
then we make the substitution x=u+v
(u+v)³+p(u+v)+r=0
u³+v³+3uv(u+v)+p(u+v)+r = 0
u³+v³+3uv(u+v) = -p(u+v)-r
it is easy to see that the equation would be solved if
u³+v³=-r and
3uv=-p
from the second equation, u=-p/3v
substituting this into the first equation,
-p³/(27v³) +v³ = -r
-p³/27 +v^6 = -rv³
this is in fact a quadratic equation in v³
(v³)² + rv³ - p³/27 = 0
which is solvable for v³ by using the quadratic formula
v³ = (-r+sqrt(r²+4p³/27))/2
v= cuberoot(-r+sqrt(r²+4p³/27))/2
doing the same for u,
u= cuberoot (-r-sqrt(r²+4p³/27))/2
(*note that when you solve for u and v you get the same expression,namely
cuberoot (-r+-sqrt(r²+4p³/27))/2
but it can be shown that one must be the positive root and the other must be the negative one.)
now the only thing left to do is to add u and v together to get x!
x= cuberoot (-r-sqrt(r²+4p³/27))/2 + cuberoot (-r-sqrt(r²+4p³/27))/2
note that the other two roots of the cubic are formally found by computing the imaginary cuberoots of (-r-sqrt(r²+4p³/27))/2 and (-r-sqrt(r²+4p³/27))/2 but we can easily navigate round that. Now that we know one root, we can just divide the original cubic by this root and solve the resulting quadratic(yes, the amaths method, though if you know how to find the imaginary roots its much easier...)
and thats how to solve a cubic equation XD
Don't you feel smarter already?
At this point some of you might be wondering, "wait, haven't we already learnt how to solve a cubic equation in amaths? All you have to do is to guess one root and then divide the cubic by the guessed root and solve the resulting quadratic equation." Aha heres where the problem lies in this method.
1) You're guessing one root. which is cheating. You might as well guess all three roots and say "solved".
2) In effect you're actually only solving a quadratic.
3) Try guessing the roots to the cubic 61x³-5x²-3271x+305 . They are approximately equal to -7.328338841 , 7.317060569 , 0.0932454851!
And for those of you who rely on a calculator or computer to solve cubics without any idea of whats going on, congrats for having the understanding level of a primary school kid.
And thus i think it is worthwhile knowing the method to solve general cubics.
We start with the equation x³+px+r=0
(later i will show how every cubic can be reduced to this form)then we make the substitution x=u+v
(u+v)³+p(u+v)+r=0
u³+v³+3uv(u+v)+p(u+v)+r = 0
u³+v³+3uv(u+v) = -p(u+v)-r
it is easy to see that the equation would be solved if
u³+v³=-r and
3uv=-p
from the second equation, u=-p/3v
substituting this into the first equation,
-p³/(27v³) +v³ = -r
-p³/27 +v^6 = -rv³
this is in fact a quadratic equation in v³
(v³)² + rv³ - p³/27 = 0
which is solvable for v³ by using the quadratic formula
v³ = (-r+sqrt(r²+4p³/27))/2
v= cuberoot(-r+sqrt(r²+4p³/27))/2
doing the same for u,
u= cuberoot (-r-sqrt(r²+4p³/27))/2
(*note that when you solve for u and v you get the same expression,namely
cuberoot (-r+-sqrt(r²+4p³/27))/2
but it can be shown that one must be the positive root and the other must be the negative one.)
now the only thing left to do is to add u and v together to get x!
x= cuberoot (-r-sqrt(r²+4p³/27))/2 + cuberoot (-r-sqrt(r²+4p³/27))/2
note that the other two roots of the cubic are formally found by computing the imaginary cuberoots of (-r-sqrt(r²+4p³/27))/2 and (-r-sqrt(r²+4p³/27))/2 but we can easily navigate round that. Now that we know one root, we can just divide the original cubic by this root and solve the resulting quadratic(yes, the amaths method, though if you know how to find the imaginary roots its much easier...)
and thats how to solve a cubic equation XD
Don't you feel smarter already?
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