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# Pi is (still) wrong

## Video transcript

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 it's 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 to its circumference is still 3.14 and so on. I mean that pi, as a concept, is a terrible mistake that has gone uncorrected for thousands of years. The problem of pi and Pi Day 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 warped and overemphasized. 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 learned 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 these 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 measuring angles when it comes to mathematics. It should make sense. But it doesn't, because pi messes it up. For example, how much pie is this? You might think this should be one pie. But it's not. The full 360 degrees of pi is actually 2 pi. What? Say I ask you how much pie you want, and you say pi over 8. You'd think this should be an eighth of a pie. But it's not. It's a sixteenth of a pie. That's confusing. You may 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're-- No. You're making excuses for pi. Mathematics should be as elegant and beautiful as possible. When you complicate something that should be as simple as one pi equals one pi by adding all these 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 1 pi instead of 2 pi. You could redefine pi to be 2 pi 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 1 tau. A half circle would be half tau, or tau over 2. And if you want 1/16 of this pie, you want tau over 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 every time you want to 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 that you can draw graphs like this. I mean, sure you could derive these values every time. But you don't, because it's easier to just memorize it. Or use your calculator, because pi and radians are confusing. This appalling notation makes us forget what the sine wave actually represents, which is how high this point is when you've gone however far around this unit circle. When your radians are notated horrifically, all of trigonometry becomes ugly. But it doesn't have to be this way. What if we used tau? Let's make a sine wave starting with tau at 0. The height of sine tau is also 0. At tau over 4, we've gone a quarter of the way around the circle. The height, or y value of this point, is so obviously 1 when you don't have to do the extra step of the in-your-head conversion of pi over 2 is actually a quarter of a circle. Tau over 2, half a circle around, back at 0. 3/4 tau, 3/4 of the way around, negative 1. A full turn brings us all the way back to 0 and bam. That just makes sense. Why? Because we don't make circles using a diameter. We make circles using a radius. The length of the radius is the fundamental thing that determines the circumference of a circle. So why would we define this circle constant as a ratio of the diameter to the circumference? Defining it by the ratio of the radius to the circumference makes much more sense. And that's how you get our lovely tau. There's a boatload of important equations and identities where 2 pi 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 over 2 equals negative 1? To which I respond, who do you think I am? I would never suggest doing something so ghastly is killing Euler's identity. Which, by the way, comes from Euler's formula, which is e to the i theta equals cosine theta plus i sine theta. Let replace theta with tau. It's easy to remember that the sine, or y value, of a full tau turn of a unit circle is 0. So this is all 0. Cosine of a full turn is the x value, which is 1. So check this out. E to the i tau equals 1. What now? If you're still not convinced, I'd recommend reading The Tau Manifesto by Michael Hartl, who does a pretty thorough job addressing every possible complaint at tauday.com. If you still want to celebrate Pi Day, that's fine. You can have your pie and eat it. But I hope you'll all join me on June 28, because I'll be making tau and eating two. I've got pie here, and I've got pie there. I'm pie winning.