Pi (還是)不好用 (英)

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Pi (還是)不好用 (英) : 計算錯誤的地方抱歉。

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Let's say you're me and you're in math class
and you're supposed to be learning trigonometry
but you're having trouble paying attention because
is boring and stupid. This is not your fault.
It's not even your teacher's fault.
It's pi's fault, because pi is wrong.
I don't mean that pi is incorrect. The ratio of a circle's diameter
is still 3,14 and so on.
I mean that pi, as a concept, is a terrible mistake
that has gone uncorrected for thousends of years.
The problem with pi and pie days is the same as the problem with
Columbus and Columbus day. Sure Christopher Columbus was a real person who did some stuff,
but everything you learn about him in school is worked and overemphasised.
He didn't discover America, he didn't discover the world was round, and he was a bit of a jerk.
So why do we celebrate Columbus day?
Same with pi. You learn in school that pi is the all important
circle constant, and had to memorize a whole bunch of equations involving it.
Because that's the way it's been taught for a very long time.
If you found any of this equations confusing, it's not your fault,
it's just that pi is wrong.
Let me show you what I mean.
Radians, good system for meassuring angles when
it comes to mathematics. It should make sense, but it doesn't
because pi messes it up.
For example, how much pi is this?
You might think this should be one pi, but it's not.
The full 360° of pie is actually 2pi.
What?
Say I ask you how much pie you want, and you say pi/8.
You'd think that should be an eight of a pie, but it's not.
It's a sixteenth of a pie.
That's confusing.
You may be thinking, come on Vi, it's a simple conversion,
all you have to do is divide by 2. Or multiply by 2 if you're going the other way.
So you just have to make sure... you pay... attention... to which way you star...
No! You're making excuses for pi.
Mathemathics should be as elegant and beautiful as possible.
When you complicate something that should be as simple as
1 pi equals 1 pie
by adding all this conversions, something gets lost in translation.
But Vi, you ask, is there a better way?
Well for this particular example there's an easy answer
for what you'd have to do to make a pie be one pi instead of 2pi.
You can redefine pi to be 2pi. Or 6,28 and so on.
But I don't want to redefine pi because that would be confusing.
So let's use a different letter: tau.
Because tau looks kind of like pi.
A full circle would be one tau, a half circle would be a half tau. Or tau/2.
And if you want 1/16 of this pie, you want tau/16.
That would be simple.
But Vi, you say, that seems rather arbitrary. Sure tau makes radians easier
but it would be annoying to have to convert between tau and pi everytime you wanna work in radians.
True, but the way of mathematics is to make stuff up and see what happens.
So let's see what happens if we use tau in other equations.
Math classes make you memorize stuff like this, so you can draw graph like this.
I mean, sure you can derive this values everytime, but you don't,
because is easier to just memorize it, or use your calculator, because pi and radians are confusing.
This appaling notation makes us forget what the sin wave actually represents,
which is how high this point is when gone whoever far around this unit circle.
When your radians are notated horrificly, all of trigonometry
becomes ugly. But it doesn't have to be this way.
What if we use tau?
Let's make a sin wave starting with tau at zero.
The height of sin(tau) is also 0.
At tau/4 we've gone a quarter of the way around the circle.
The height, or y value at this point is so obviously one when you don't
have to do the extra step -- the "in your head" conversion of pi/2 is actually a quarter of a circle.
Tau/2, half a circle around, back at zero.
3/4 of tau, 3/4 of the way around, -1.
A full turn brings us all the way back to zero. And bam!
That just make sense. Why?
Because we don't make circles using a diameter,
we make circle using a radius.
The length of the radius is the fundamental thing that
determines the circunference of a circle, so why
would we define the circle constant as a ratio of the diameter to the circunference.
Defining it by ratio of the radius of the circunference makes much more sense.
And that's how you get our lovely tau.
There's a boat load of important equations and identities where
2pi shows up, which could and should be simplified to tau.
But Vi, you say, what about e to the i pi?
Are you really suggesting we ruin it by making it
e to the i tau/2 equals -1?
To which I respond, who do you think I am?!
I would never suggest doing something so ghastly as killing Euler's identity.
Which, by the way, comes from the Euler's formula, which
is e to the i theta equals cos theta plus i sin theta.
Let's replace theta with tau.
It's easy to remember that the sin or y value of a full tau turn
of a unit circle is zero. So this is all zero.
Cos of a full turn is the x value, which is 1.
So check this out, e to the i tau equals one.
What now!
If you're still not convinced, I recomend reading the Tau manifesto
by Michel Hartl, who does a pretty through job addressing
every posible complains at tauday.com.
If you still want to celebrate pie day, that's fine.
You can have your pie and eat it.
But I hope you all join me on June 28th, because I'll be making tau and eating too.
I've got pie here and pie there, I'm pie-winning.