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## About pi and tau

# 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.