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。
- 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.
- 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.