Tau versus Pi Why Tau might be a better number to look at than Pi
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- 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 and 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 -- 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.
- If you've got the radius and you want to calculate the circumference
- multiply both sides of this definition 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 two 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 the circumference.
- And from that comes how we measure our angles in radians
- once we got to trigonometry class. And just as a
- review here, so 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 angles 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.
- 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? I'm assuming this is the fourth of
- the circle, so it's going to be 2 pi r over four. So this arc
- length over here, this arc length is going to be 2 pi r over four,
- which is the same thing as pi over 2r or you could say this is
- the same thing as pi over 2 radiuseseses.
- One of those -- y'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 "okay, that angle is subtanded by
- an arc of --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 I should 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-- so if you
- went all the way around the circle and you got back
- to the positive x axis here, what is the arc length?
- 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 do we graph trigonometric
- functions or at least how we measure the graph
- on the x axis and I 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 into all of that. So if I think about our
- trigonometric functions, remember, if this was a lesson
- on trigonometric functions, we assume we have a
- unit circle here. So on the trig. functions, this is the
- unit circle definition of the trig functions, 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 subtans this
- angle, so that's cosine of theta and then sine of theta is
- the y value of that point. 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 can 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, so when then angle is 0,
- sine of theta is zero -- let me draw the x and y axis
- just so you remember this, this is the y axis and this is the
- x axis, right here is 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, so let
- me draw it like this...
- so this is out theta and this is -- i'm going to graph
- sine of theta along the y axis, 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, i'll just do the simple points
- here, then if we make the angle go-- if we did in degrees
- 90 degrees, or if we do it in radians, pi over 2 radians
- what is sine of theta?
- Well, now it is one, this is a unit circle, it has radius one
- so when we get to pi over 2, when theta is equal to pi over 2
- then sine of theta is equal to 1. This is 1 right here,
- sine of theta is equal to one, then if we go 180 degrees
- or halfway around the circle, theta is now equal to pi.
- When theta is equal to pi, the y value of
- this point right here is once again zero, so we go back
- to zero, remember we're talking about sine of theta,
- and then we can go all the way down here, where you
- can either view it 270 degrees or you can 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 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's 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 pi
- seems to have some type of mystical power, and we've
- shown this in the calculus playlist,
- is there's Euler's formula. e to the i theta is equal
- to cosine of theta plus i sine of theta. This by
- itself is 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
- --what's cosine of pi? cosine of pi is negative 1,
- and then sine of pi is zero, so zero times i, so you get
- this formula, which is pretty profound and then you say
- "okay, if i want to put all of the fundamental numbers
- together in one formula, i can add one 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've been even
- more powerful if this was a one right over here.
- Because this e to the i pi, this bizarre thing, would've
- equaled unity. That would've been super-duper profound
- to me. It seems a little bit of a hack to add one to both
- sides, "oh look, now i have zero 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'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 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 the circle, that it's 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 to the diameter, the ratio of circumference
- to two times the radius, they say "Hey, wouldn't it
- be natural to define some number that's the
- ratio of circumference to the radius?" And as you
- see here, this pi is just one half times this over here, right?
- Circumference over 2 r, it's 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 two times pi or, if you -- and i'm sure
- you probably don't have this memorized, because, like
- "wait, i've spent all my life memorizing pi", but it's
- 6.283185 and it keeps going on and on and on never
- repeating just like pi, it's two times pi. And so you're
- saying "hey, Sal, pi has been around for millenia, really,
- why mess with such a fundamental number, especially
- when you've just spent all of 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, 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 can say that 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
- altough 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 this is pi over 2 radians is the same thing as
- 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
- 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 4 radians. So the reason why
- tau is more intuitive here is because you don't have to do
- this weird conversion where you say "divide by two,
- multiply by two" all that, 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 1/4 around,
- that's tau/4 radians, if you've gone halfway around,
- that'd be tau/2 radians, if you go 3/4 around, that's be
- 3tau/4 radians. If you go all the way around, that would
- be tau radians. If someone told you that they have
- an angle of 10tau radians, you'd go around exactly 10 times.
- It would be much more intuitive, you wouldn't have to do
- this mental math, converting the multiplier, divide by
- two 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. And it makes the
- sine function over here, instead of writing pi over 2
- when you look on the graph, where was this on the
- unit circle? It's 1/4 around the circle, was this one half?
- And this is actually one fourth of the circle, you're
- right over here, but now it becomes obvious if you
- write it in tau. Pau pi -- not pau-- pi over 2 is the same
- thing as tau over 4, pi is the same thing as tau over 2
- 3 pi over 2 is 3pi--sorry, 3tau 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 1/4 around the unit circle
- , you're halfway around the unic circle, you're
- 3/4 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.
- 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
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.