If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:16:22

What I want to do in this
video is revisit a little bit of what we know
about pi, and really how we measure
angles in radians. And then think about
whether pi is necessarily the best number to be
paying attention to. So let's think a little
bit about what I just said. So pi, we know, is
defined-- and I'll write defined as a
triple equal sign, I guess you could call it
that way-- pi is defined as the ratio of the
circumference of a circle to its diameter, which is
the same thing as the ratio of the circumference of the
circle to two times the radius. And from that, we get all
these interesting formulas that you get in geometry
class that, hey, if you have the radius and you want
to calculate the circumference, multiply both sides of this
definition, or this equation, by two times the radius. And you get two times
the radius times pi is equal to
the circumference, or more familiarly, it
would be circumference is equal to 2 pi r. This is one of those
fundamental things that you learn early
on in your career and you use it to find
circumferences, usually, or figure out radiuses if
you know circumference. And from that comes how we
measure our angles in radians once we get to
trigonometry class. And just as a review here,
let me draw myself a circle. Let me draw myself
a better circle. So there is my--
it'll do the job. And here is the
positive x-axis, and let me make some angle here. I'll make the angle
kind of obvious just so that it-- so
let me make this angle. And the way that
we measure angles, when we talk about
radians, we're really talking about
the angle subtended by something of a
certain arc length. And we measure the
arc length in-- well, the way I like to
think about it, is the angle is in radians
and the arc length itself is in radiuses, which
isn't really a word. But that's how I think about it. How many radiuses
is this arc length that subtends the
angle in radians? So let me show you
what I'm talking about. So this arc length right
here, if the radius is r, what is the length
of this arc length? Well, we know from
basic geometry the entire
circumference over here is going to be 2 pi r, right? This entire
circumference, that's really by definition,
this entire circumference is going to be 2 pi r. So what is just this
arc length here? And I'm assuming this is
a fourth of the circle. So it's going to
be 2 pi r over 4. So this arc length
over here is going to be 2 pi r over 4, which is
the same thing as pi over 2 r. Or you could say this
is the same thing as pi over 2 radiuses. One of those-- you
know, not a real word, but that's how I like
to think about it. Or you could say it subtends
an angle of pi over 2 radians. So over here, theta
is pi over 2 radians. And so really, when you're
measuring angles in radians, it's really you're saying,
OK, that angle subtended by an arc that has a length of
how many radiusi, or I don't even know what the
plural of radius is. Actually, I think it's
radii but it's fun to try to say radiuses. Radii, actually let me do
that just so no one says, Sal, you're teaching people the
wrong plural form of radius. Radii. So this arc length
is pi over 2 radii and it subtends an angle
of pi over 2 radians. We could do another
one just for the sake of making the point clear. If you went all the
way around the circle and you got back to the
positive x-axis here, what is the arc length? Well now, all of a
sudden the arc length is the entire circumference
of the circle. It would be 2 pi r, which is
the same thing as 2 pi radii. And we would say that the angle
subtended by this arc length, the angle that we
care about going all the way around the
circle, is 2 pi radians. And so, out of this
comes all of the things that we know about how to
graph trigonometric functions or at least how we measure
the graph on the x-axis. And I'll also touch on
Euler's formula, which is the most beautiful formula,
I think, in all of mathematics. And let's visit those right
now, just to remind ourselves of how pi fits in
to all of that. So if I think about our
trigonometric functions. Remember, if this was-- so
on trigonometric functions, we assume we have
a unit circle here. So in the trig functions, this
is the unit circle definition of the trig function. So this is a nice
review of all of that. You assume you have a unit
circle, a circle of radius 1. And then the trig
functions are defined as, for any angle you have
here, for any angle, theta, cosine of theta is how
far you have to move in-- or the x-coordinate of the
point along the arc that subtends this angle. So that's cosine of theta. And then sine of theta is
the y value of that point. Let me make that clear. Cosine of theta is the x value,
sine of theta is the y value. And so if you were to graph
one of these functions, and I'll just do sine of
theta for convenience. But you could try it
with cosine of theta. So let's graph sine of theta. Let's do one revolution
of sine of theta. And we tend to label
it-- so let's do sine. When the angle is 0,
sine of theta is 0. Let me draw the x- and
y-axis just so you remember. This is the y-axis and
this is the x-axis. So when the angle is 0, we're
right here on the unit circle. The y value there is 0. So sine of theta is going
to be right like that. Let me draw it like this. So this is our
theta, and this is-- I'm going to graph sine
of theta along the y-axis. So we'll say y is equal to
sine of theta in this graph that I'm drawing
right over here. And then we could
do-- well, I'll just do the simple points here. And then if we
make the angle go-- if we did it in
degrees, 90 degrees, or if we do it in radians,
pi over 2 radians. What is sine of theta? Well, now it is 1. This is the unit circle. It has a radius 1. So when theta is equal to pi
over 2, then sine of theta is equal to 1. So if this is 1 right here,
sine of theta is equal to 1. If theta-- and then if we
go 180 degrees or halfway around the circle--
theta is now equal to pi. Let me do this in
a color, orange. I have already used orange. Theta is now equal to pi. When theta is equal to pi, the
y value of this point right here is once again 0. So we go back to 0. Remember, we're talking
about sine of theta. And then we can go
all the way down here, where you can see
there's 270 degrees. Or you could view this
as 3 pi over 2 radians. So this is in
radians, this axis. So 3 pi over 2
radians, sine of theta is the y-coordinate on the
unit circle right over here. So it's going to be negative 1. So this is negative 1. And then finally, when you go
all the way around the circle, you've gone 2 pi radians, and
you're back where you began. And the sine of theta,
or the y-coordinate, is now 0 once again. And if you connect the dots, or
if you'd plotted more points, you would see a sine
curve over just the part that we've graphed
right over here. So that's another application. You say hey, Sal,
where is this going? Well I'm showing you--
I'm reminding you of all of these
things because we're going to revisit it with a
different number other than pi. And so I want to do
one last visit with pi. You say, look, pi is powerful
because-- or one of the reasons why pi seems to have some
type of mystical power, and we've shown this in
the calculus playlist-- is Euler's formula,
that e to the i theta is equal to cosine of
theta plus i sine of theta. This, by itself,
is just a crazy-- it's just one of those
mind boggling formulas. But it sometimes looks
even more mind boggling when you put pi in for theta
because then, from Euler's formula, you would
get e to the i pi is equal to-- well,
what's cosine of pi? Cosine of pi is negative 1. And then sine of pi is 0. So 0 times i. So you get this formula,
which is pretty profound, and then you say,
OK, if I want to put all of the fundamental numbers
together in one formula, I can add 1 to
both sides of this. And you get e to the i
pi plus 1 is equal to 0. Sometimes this is called Euler's
Identity, the most beautiful formula or equation
in all of mathematics. And it is pretty profound. You have all of the fundamental
numbers in one equation. e, i, pi, 1, 0, although
for my aesthetic taste it would have been even
more powerful if this was a 1 right over here. Because then this
would have said, look, e to the i pi,
this bizarre thing, would have equaled unity. That would have been super
duper profound to me. It seems a little bit of a
hack to add 1 to both sides and say, oh, look,
now I have 0 here. But this is pretty darn good. But with that, I'm
going to make-- well, I'm not going to argue for it. I'm going to show an
argument for another number, a number different than pi. And I want to make it clear
that these ideas are not my own. It comes from--
well, it's inspired by-- many people are on this
movement now, the Tau Movement, but these are kind
of the people that gave me the thinking on this. And the first is Robert
Palais on "Pi is Wrong." And he doesn't argue that
pi is calculated wrong. He still agrees that it is
the ratio of the circumference to the diameter of a
circle that is 3.14159. But what he's saying is
that we're paying attention to the wrong number. And also, you have Michael
Hartl, "The Tau Manifesto." All of this is available online. And what they argue for
is a number called tau, or what they call tau. And they define tau, and it's
a very simple change from pi. They define tau not as the
ratio of the circumference the diameter, or the ratio of
the circumference to 2 times the radius. They say, hey, wouldn't
it be natural to define some number, the ratio of the
circumference to the radius? And as you see here, this
pi is just one half times this over here, right? Circumference over 2 r, this
the same thing as one half times circumference over r. So pi is just half of tau. Or another way to think about it
is that tau is just 2 times pi. Or, and I'm sure
you probably don't have this memorized,
because you're like, wait, I spent all my
life memorizing pi, but it's 6.283185 and keeps
going on and on and on, never repeating just like pi. It's 2 times pi. And so you're
saying, hey Sal, pi has been around for
millennia, really. Why mess with such a
fundamental number, especially when you
just spent all this time showing how profound it is? And the argument
that they'd make, and it seems like a
pretty good argument, is that actually things seem
a little bit more elegant when you pay attention to this number
instead of half of this number, when you pay attention to tau. And to see that, let's revisit
everything that we did here. Now, all of a sudden, if
you pay attention to 2 pi, as opposed to pi. Or we should call it--
if you pay attention to tau instead of tau over 2. What is this angle
that we did in magenta? Well first of all, let's
think about this formula right over here. What is the circumference
in terms of the radius? Well, now we could
say the circumference is equal to tau times
the radius because tau is the same thing as 2 pi. So it makes that formula a
little bit neater, although it does make the pi r squared
a little bit messier. So you could argue
both sides of that. But it makes the measure of
radians much more intuitive because you could say that
this is pi over 2 radians, or you could say that this
is pi over 2 radians is the same thing as to
tau over 4 radians. And where did I get that from? Remember, if you go all the
way around the circle, that is the circumference. The arc length would
be the circumference. It would be tau
radii, or it would be to tau radians would be the
angle subtended by that arc length. It would be tau radians. All the way around
is tau radians. So that by itself is intuitive. One revolution is
one tau radians. If you go only one
fourth of that, it's going to be tau
over four radians. So the reason why
tau is more intuitive here is because it
immediately-- you don't have to do
this weird conversion where you saying, oh, divide
by 2, multiply by 2, all that. You're just like, look, however
many radians in terms of tau, that's really how many
revolutions you've gone around the circle. And so, if you've gone
one fourth around, that's tau over four radians. If you've gone halfway around,
that'd be tau over two radians. If you go 3/4 around that'd be
three tau over four radians. If you go all the way around
that would be tau radians. If someone tells you that
they have an angle of 10 tau radians, you'd go
around exactly 10 times. It would be much more intuitive. You wouldn't have to do this
little mental math, converting, saying, do I multiply
or divide by 2 when I convert to
radians in terms of pi? No, when you do it in terms of
tau radians, it's just natural. One revolution is
one tau radians. So that makes-- and it makes
a sine function over here. Instead of writing
pi over 2-- well, when you look at
a graph like this you're like, where was
this on the unit circle? Was this one fourth
around the circle? Was this one half? And this is actually one
fourth around the circle, right over here? But now it becomes obvious
if you write it in tau. Pi over 2 is the same
thing as to tau over 4. Pi is the same thing
as to tau over 2. 3 pi over 2 is 3 pi-- oh,
sorry, 3 tau over 4, 3/4 tau. And then one revolution is tau. And then immediately, now
when you look at it this way, you know exactly where you
are in the unit circle. You're one fourth
around the unit circle, you're halfway around
the unit circle, you're 3/4 of the way
around the unit circle. And then you're all the
way around the unit circle. And so the last thing
that I think the strong pi defenders would say
is well, look Sal, you just pointed out one of
the most beautiful identities or formulas in mathematics. How does tau hold up to this? Well, let's just try it
out and see what happens. So if we take e to the i tau,
that will give us cosine of tau plus i sine of tau. And once again, let's just
think about what this is. Tau radians means we've gone all
the way around the unit circle. So cosine of tau--
remember, we're back at the beginning
of the unit circle right over here-- so cosine of
tau is going to be equal to 1. And then sine of
tau is equal to 0. So e to the i tau is equal to 1. And I'll leave it
up to you to decide which one seems to be more
aesthetically profound.