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

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

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.

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

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.

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?

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?

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...

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.

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?