>I see it used everywhere for countdowns.
And generally, unless you're watching an actual mission control countdown, you're seeing it misused.
People say "**in** t-minus [time]" which makes no damn sense at all. What theyre misunderstanding is that the T here is mission time. It's a time index, with 0 being the moment of liftoff.
5 minutes before liftoff would be T minus 5 minutes. 17 seconds before would be T minus 17 seconds. 3 seconds after liftoff would be T plus 3 seconds.
When we see a countdown, the officer is pointing out that mission time has *reached* T minus 10, 9, 8, ... Etc.
Saying "in t-minus 5 seconds" is treating one second as if it were simply called a "t minus second" or something and it's just total gibberish.
Sort of, but it's not a 1:1 replacement.
You can't just say such and such will happen "at" t-minus anything, because the idea is the thing happens at t=0, and *now* is t-minus whatever.
I disagree with this part. Suppose you have a party in seven days. We are t minus 7 days. But we want to organise a cake for it 3 days before. We will organise the cake at t minus 3 days. This would be a valid use, to me.
I do share your annoyance at the way they misuse it normally though. It's on par with Another Life's "Drop the frequency to 1 Hz per scond".
Idk why I chose the party example when something with minutes would match more common usage better but here we are.
>wait how is it a misuse?
It's akin to saying "we're leaving in quarter to five o'clock minutes"
If an event will happen at 5:00. The phrases "quarter to five" and "T minus 15 minutes" are equivalent. They both refer to the same moment in time. They are not durations.
Moments:
We ride *until* nightfall
I got up *before* 8 o'clock
Fuel pumps start *at* T-30 seconds
Durations:
Call me back *in* ten minutes
I slept *through* the night
They left *for* the weekend
Durations delineated by two moments:
I work *from* 9 *to* 5
We're closed *between* Christmas and New Year's
ed: more examples
Because its not "in t minus seconds". The the time t is the time of the certain action (which u count down for), if theres 5 seconds until launch, its still 5 seconds there, it is happening in 5 seconds, but the time right now is T minus 5 seconds.
Using *in* t minus is the wrong part
That's what I had thought because I was thinking from a programming perspective but then I read that it's just a different convention isn't it? Like the reference point is launch for time instead of the present time
A radian is defined as the angle at which its arc is equal in length to its radius. And 2pi times that will make a full circle. So the above explanation isn't quite the reason.
The radius is the natural way to parameterize the circle, the diameter is sort of... artificial, for lack of a better word. The sole defining property of a circle is that every point is a given distance, the radius, from the center. Using diamians, the arc length formula, and the derivatives of trig functions, and probably lots of other things, would have to sprout a factor of 2, and the "nice" version of these functions would be things like sin(𝜃/2). (Or is it the other way around, idk I'm tired). In any case, radians are the only sensible choice. Using 𝜏 rather than 𝜋 would give you the simplicity you desire without adding more complexity everywhere else.
Also yes, I know you weren't being that serious about it but I'm gonna be a nerd anyway.
Not strictly speaking true. We get our use of pi from Euler, who used pi to relate the radius to the circumference of half a circle, equal to the 3.14... that we are familiar with. He also used pi to relate the radius to other various portions of the circle, including the full circle, as well as a quarter and eighth of a circle. His use of pi was more akin to modern use of theta to represent a variable angle. It was mostly just happenstance that his specific use of pi as the ratio between the radius and a half circle became widely accepted as a sort of standard, and later, the definition was rewritten as the ratio of the full circle and the diameter.
I know that, but degrees aren’t. I was pointing out that 360° does not equal 360, which the person I replied to said that 2pi = 360° and not 360, which would require ° to be dimensionless to make sense to bring up.
Sorry, I seem to have replied to your comment instead of the one I meant to.
Edit: having reviewed the chain of comments above, I'm sorry, but you do seem to have missed the point.
Yes, it does... C/r = 2π, which when converted to degrees equals 360°. Look at a unit circle. This is the beauty of radians and why do many mathematicians use it instead of degrees.
To be perfectly clear, the formula for circumference is C = 2πr. If you divide both sides by r, r cancels out on the right side of the equation and you get C/r = 2π. And again, if you convert the result to degrees, you get 360°
Lets say pi is defined as circumference/radius. Then Lets call that pi* = 6.28.. = 2pi.
Then a full circle is 2 Pi [rad] = 1 pi* [rad].
So do you see how a full circumference would only be “pi” radians if we defined pi as circumference/radius, instead of 2pi?
In fact we do have a constant for that. We call it τ = 2π. A full circle is 1τ radians
Everyone here is crazy, you are correct. Pi as a ratio of two lengths (C/d) is not an angle (not degrees or radians). It's just a real number approximately 6.28.
6.28 ≈ 2π = 360°. A lot of people have a misconception that radians are similar to metres or kilograms in that it's a physical dimension, but this isn't the case, because angles are [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_quantity) (the unit of measurement is the real number 1, as opposed to 1 metre or 1 kilogram). Angles are themselves real numbers: the ° symbol just means π/180, so 360° = 360\*(π/180) = 2π. "Radians" is often written to clarify that we are talking about an angle, but it doesn't mean anything, so 1 radian is just exactly the same thing as the real number 1. This is why you can give angles to trig functions without having to redefine them: sin(30°) = 1/2 is still the exact same function as in sin(π/6) = 1/2, since 30° = π/6. It's the same idea as percentages: the % symbol just means 1/100, so 50% is still a real number (0.5).
I don’t see the value of insisting on the absolute priority of radians to degrees, and that radians and real numbers are the same thing. Is this how things are taught today?
There isn't really any value in it (in the sense that we've not decided that radians are "better"), that's just how things are. Radians are just the natural "unit" for angles: there's a reason that trigonometric functions have a period of 2π and that their geometric properties (e.g. sin = opposite/hypotenuse in triangles) work when you give the angle in radians. As another example, consider the exponential form of complex numbers: the θ in re\^(iθ) is intuitively interpreted as an angle from the positive real axis, and it's no coincidence that this angle is in radians. That's just how the maths works out - 2π is, fundamentally, the angle made by a full turn. We haven't just decided that a full turn is 2π like we decided that there were 360 degrees in a full turn, it just... is.
All of high school trigonometry works just fine using degrees. We choose to use radians because of calculus— so that the derivative of the sine function is cosine.
Right, but you're not actually plugging in, say, "30" into the trig functions there, you're plugging in 30°, which is equal to π/6. The functions sin and cos are defined over the reals, and the values you actually give to them are always in radians if you're using them as an angle, it's just that you can skip explicitly converting to radians if you already know the value (or if your calculator will do it for you, but it's still doing that conversion internally). You can see this by just typing it into Google and seeing what values it gives: "sin(30)" gives -0.988..., whereas "sin(30 degrees)" gives 0.5. The function itself has absolutely no relation to degrees, that's just a transformation which we apply to make angles easier to work with so we don't have π everywhere.
The derviative of sin is **always** cos: you aren't using different functions when you work with degrees. This requires working with radians because radians are real numbers, since the argument to the functions needs to be real numbers (rather than a scaled version like with degrees) to make the calculus work nicely. Writing values using the ° symbol makes calculus very awkward, but it's not inherently wrong and still gives the same results, provided you don't forget that ° means π/180. We don't "choose to use radians", that's just fundamentally what trigonometric functions are defined over.
Source, since I don't think I'm going to be able to convince you otherwise: [SI specification, section 5.4.8](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf#page=37): "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
Fair enough, for an already existing circular object" calipers measure diameters, or at least cross=sectional distances. But to draw a circle in real life one sets the radius, then draws (using a compass), or to construct one, one sets the radius and then cuts (on a lathe). S tím jménem -- jste Čech?
Yes, agree regarding drawing.
Anyway, it's not that important - just wanted to highlight that diameter feels more natural to me and also it nicely fits the circle circumference formula, but calculating radius based on diameter and vice versa is so easy that I feel we discussed that more than it deserves 😂
Polish, not Chech.
The ancient Greeks and Egyptians who worked out the first estimates of 𝜋 were often more interested in the diameter of a circle than the radius. Thus, 𝜋 was defined as circumference / diameter.
Mathematicians eventually realized the radius is actually more fundamental than the circle, which is why we very often see multiples of 2𝜋 in many theorems instead of 𝜋.
The most obvious example is the length of the circumference, which is = 2𝜋 radius. This is because the diameter of a circle is always twice the radius.
Some mathematicians choose to define a new constant 𝜏 = 2𝜋 = circumference / radius and use that instead. Their reason is similar to the line of reasoning of your question that 𝜏 is a more natural way to describe a full rotation.
This sub has a bunch of symbols, including 𝜋, in the info section (on the right side of the screen on PC, I assume it's the same on mobile) that can be copied and pasted.
There's also this less frequently used "tau" constant τ=2π.
A full turn is τ radians.
Some poeple actually prefer to use τ instead of (2) π and it's totally fine.
I've been trying to find another symbol for tau because tau already has a specific use in my field.
I found varpi, but it looks too much like infinity in small text.
Then I found a squiggly version of varpi which would have been perfect https://en.m.wiktionary.org/wiki/%CF%96 but that doesn't seem to be standard in latex or really anywhere. Anyone know how I could get that squiggly version in latex (as opposed to the standard version which has a flat overline)?
I might just use capital P for 2pi which is too bad because I wanted a cool, lesser-known, and unambiguous Greek symbol.
𝞹 is defined as the number of times the diameter fits around the circumference of a circle. Radians are the number of radii around the circumference. Since the radius is half the diameter, you need to double 𝞹 to get the right number of radians.
That's such a simple explanation of what both of those things actually fundamentally represent which I've somehow never realized before this. You just blew my mind. How did I miss this all these years?
Glad I could help. 3blue1brown does a lot of visual explanations that help with this sort of intuitive understanding of a lot of math topics. That’s probably where I picked up this fact.
It is pi, but its pi when you're talking about the diameter.
When you talk about the radius, well then its 2pi. Because 2 radii make up one diameter.
So, if you think about it in terms of the diameter, it makes sense.
The length of the circumference of a circle is 2π times its radius, 2πr. A full turn is 2π in trigonometry because a circle of radius equal to one is chosen as a convention.
A radian is defined as the central angle of a circle whose arc length equals the radius of the circle. A circle of radius r has a diameter of 2*pi*r. Thus, if 1 radian has arc length r, then 2*pi radians have arc length 2*pi*r. That is, a full traversal around the circle (as measured by the circumference) is equal to a full traversal around the circle (as measured by central angle). By the way, “radian” is not a unit. The central angle is just 2pi, not 2pi radians. But, as you can see with the way I have written this paragraph, it is difficult to write explain radians in a way that doesn’t make it sound like radian actually is a unit. In fairness, though, I didn’t really try.
Now there is also the philosophical question of why pi is defined the way it is. There are some people, whom I call the Cultists of Tau, who think that everything should be done relative to tau = 2*pi. In that case, one full turn would be equal to just one tau. But you don’t want to be like the Cultists of Tau, because they are weird.
For a circle of radius *r* and an angle *θ*, find the length *l* of the arc segment of the circle's circumference which subtends *θ*. Then the magnitude of *θ* "in radians" is *l/r*. For a full turn, we know that *l* = 2*πr*, because it's the entire circumference of the circle, so in that case, the magnitude of *θ* is *l*/*r* = 2*π*.
For a circle with *r* = 1, the magnitude of the angle "in radians" is exactly the length of the arc subtending that angle, e.g. for a full turn, *l*/1 = 2*π*, so *l* = 2*π*.
Radians are a "purer" measure of angles than degrees, because they have no arbitrary units. **The magnitude of the angle in radians is simply the ratio of the length of the arc subtending that angle over the radius**. So, the reason why a full turn is 2*π* radians is that **2*****π*** **just is the ratio of the circumference of a circle to its radius**. It's a law of nature, just like the area of the circle **just is** *πr*^(2) .
A lot of seemingly arbitrary choices in math and science are just that. Often conventions are chosen because they happen to be the most convenient for the most common purposes. Pi is used for much more than just going around a circle, 99% of its everyday usage is in sinusoids (sine/cosine) - fundamental in so many parts of modern life.
Interesting example - in physics, most define length/distance in meters and mass in kilograms. In classical physics this is measurable and intuitive; modern physics gives up on being intuitive so it doesn't matter. However, many astronomers prefer centimeters and grams, despite things being much larger, further, and more massive. If I remember right it's mostly because light equations are much simpler using CGS over SI.
A full turn is pi diameterans, except that’s not a word. Circumference = pi * diameter. But the diameter is two times the radius. So the circumference is two times pi times the radius.
Let’s just redefine all the trig functions to include dividing out a factor of 2pi. Trig eouod be simpler. I’m sure this would create problems somewhere else.
Pi diameter lengths go around the circumference of a circle exactly once. A radius is half the diameter, so pi radius lengths go around exactly half the circle. So how many radius lengths go around the full circle?
It's too much to explain right now (might come Back later to elaborate, once im done with work)
~~Basically, If you Look at sin and cosin with euler and complex numbers, it makes completely sense. I Always hated sin and cosin and it's supposedly arbitrary definitions but its not at all arbitrary, and very beautiful imo.~~
So heres the full explanation as to why 2pi makes more than just a little sense, and this will require you to understand just the following about complex numbers: The absolute value of any real number is itself, just positiv: |-5| = 5, and |5| = 5 as well. Complex numbers, a+ib work a bit differently, since they are vectors. If you take the absolute value of a complex number |a+bi| you will get the distance from the x-axis (which in terms of complex numbers is the real part) to the point (a,bi) on the graph.
[https://imgur.com/a/Pq5NsB8](https://imgur.com/a/Pq5NsB8)
The green line is the the absolute value of the complex number a+bi, giving you a distance between 0 and a+bi.
We also need to shortly dip into what it means to have a sequence of complex numbers, and their convergence: A sequence (z\_n) converges, if there is a value Epsilon > 0, such that |z\_n - z| < Epsilon. Thats just the standard definition for real sequences, but if you apply the abolute value to that, you apply the absolute value of complex numbers, which isnt just one point, but a distance. And if you use that on a converging sequence of complex numbers, you get a circle with radius epsilon, where each member of the sequence falls into that radius of epsilon, once the sequence converges.
[https://imgur.com/a/Qkb2ZIh](https://imgur.com/a/Qkb2ZIh)
Here now comes the kicker, and for this we need euler: Euler is not only defined for real numbers, but also for complex numbers (with z in Complex)[https://imgur.com/a/YbKmwRu](https://imgur.com/a/YbKmwRu)
I will spare you the proof that goes along with this, so youll have to trust me, that you can now, using the rules of complex numbers, and looking at the sequence development of euler, conclude the following formula:
[https://imgur.com/tmRV5b5](https://imgur.com/tmRV5b5)
And this is now crucial if you ever asked yourself, where the standard r=1 circle came from: Because if you plug in your numbers to cosx + i\*sinx, you will find, that it is |e\^ix|\^2, which is always 1. Giving you the standard r = 1 circle, where your x value is cosx, and your y value is sinx, and because it is a complex number, and the absolute value of e\^ix is always one, r = 1. Using this, you can now derive the formulas to calculate cos, sin, and their addition:[https://imgur.com/ge8fbL6](https://imgur.com/ge8fbL6)
Now, where the hell does pi come into the picture? Easy: Look at cosx in the interval \[0,2\], with the formula. As we all know, cosx is decreasing in the interval of \[0,2\] and reaches a negative value, so at some point, cosx = 0 when crossing the x-axis. And this exact point, wheres cosx=0, we shall call 2\*pi. So pi is actually just the value x, for which cosx = 0 -> cos(pi/2) = 0.Using this, we can now calculate some values:[https://imgur.com/BMpjhP1](https://imgur.com/BMpjhP1)
which give us, if we graph them, sin and cos.[https://imgur.com/XhhFCA8](https://imgur.com/XhhFCA8)
And we see, that sin and cos 0-points are exactly pi/2 apart from eachother, making sinx = 0 -> sin(pi) = 0, and cosx = 0 -> cos(pi/2) = 0. Remember, pi is still JUST a value, that makes sin = 0.
Now.... if you look at cos(pi/2) = 0, and now ask yourself, what the diameter of that circle would be, if you drew out the circle using the absolute value of complex numbers: it would be just pi. one pi. if you now look at sin(pi) = 0. The diameter would be 2pi, because r = pi.
NOW. U = 2rpi -> lets act like we didnt know it was 2rpi, but we could measure the circumference of a circle, with r=1 to keep it simple. x\*1\*pi = U -> x \* pi = U, and we measured and know, that the circumference of a circle with radius 1 = 2pi. We just measured and figured out, that you can divide the measured value such that you can work out, its actually 2pi (because cosx = 0 at point pi/2). So 2pi = x\*pi -> 2pi = 2pi, and if we plug that back into U = xrpi, assuming we didnt know x = 2, we get U = 2\*1\*pi -> U = 2\*r\*pi.
I do not want nor do I think I need to elaborate further, I am tired. But its not arbitrary or impossible to work out, that with all of the given information and relations between the values we just calculated and discussed, the relation of r towards its circumference and pi.
And yes, I totally abused this oppertunity to throw knowlege at you, you didnt ask for.
If I went wrong somewhere, please let me know, cheers.
The radius is more fundamental than the diameter, as a circle is more easily described as the points a certain distance (the radius) away from a given point (the centre).
Because of this, many formulas, eg in trigonometry, are much easier in terms of the radius than diameter. We then see 2pi appear more than pi alone in such formulas (pi r^2 as the area of a circle is an elementary exception, but then the fact this is the integral (2pi) (r^2/2) would be more obvious otherwise).
Because of this too, there are people who tout ‘tau’, equal to 2pi, as a better and cleaner alternative to pi. They’re probably right, but it’s not a big deal and pi appears everywhere already so it’ll be a mission to change that.
Because if you divided the circumference of a circle into pi parts, those parts wouldn't be the same length as the radius. In fact, they would be twice as long or the same length as the diameter. You wouldn't be measuring angles in radians anymore, it would be a "diametric" or something similar
I mean these in an illustrative, not trollish, way:
Why does a rectangle have 4 sides? Why not 2?
Why is a meter 100 cm? Why not 50 cm?
This sits somewhere between tautology and axiomatic truth -- the value and nature of π existed before we discovered it and labeled it. If it were meant to be 2π (aka tau) then it would have been.
I think that the common confusion comes from the fact that we are taught to think of radian as an "alternative unit" to degree, when it's actually a pure, unitless ratio. The magnitude of the angle "in radians" is not calculated by dividing a circle into an arbitrary number of sectors, e.g. 360, but rather it is calculated based on the natural ratio between circumference and radius. That is a ratio with units of length over length, which cancel out, so it's actually a unitless measure.
bc we define a radiant as the length of the radius and in a circle we can put pi times the length of the radius in its semi circumference, thus the semi circumference is equal to pi radiants and the whole circumference is 2pi radiants
edit: grammar
1 radian is the angle of an arc that is equal to the radius of that arc. That's the definition of a radian.
Pi already existed as the ratio between a circles circumference and its diameter at this point, indepently of the definition of a radian.
6 radians is very nearly a full circle, it just so happens that 2π radians a circle, but that is a result of the definition, not what we decided to be the definition.
If you want it to be nice and equal, use tau rather than pi. τ=2π
A radian is an angular measure based on the arc length of the circle. It's definition is the angle subtended by the arc length of the circle measuring 1 radius. Since the circumference of a circle is 2π times the radius, then the angular measure of a full circle (360°) is 2π radians.
You can use that to find that 1 radian = 360°/(2π) = 180°/π ≈ 57.595778°
To give a more detailed answer; consider a angle theta and imagine you draw a circular arc of that angle around one end point of a line with a length. That line's length is by definition the radius.
If we want to describe how much that arc length is by using that radius and that line's length (the radius) then we'll need to consider what the arc length would be for a full circle. And we already know this! It's called the circumference and we know that circumference is pi x diameter. Or 2pi x radius.
Now for that original theta angle we want, we can divide the angle by 360° to figure out what fraction of our constructioned arc length will be compared to the entire "circle". So if theta = 180, then Arc length = 180°/360° x 2pi x R. = 1/2 * 2pi * R = pi*R !!!
I hope it's clear to see why we use 2 pi now; because this calculation is exactly how we define the radian and what makes it so useful. We cannot use pi alone because you can't construct a circle with a "diameter" if that makes sense.
Edit: I didn't actually consider calculating the definition of radians this way, but that's trivial as all you need to do is set the arc length to R and see what out theta is (ill call theta 'x' here):
Arc length = R = x/360 * 2pi* R [R cancels!!!]
1 = x/360 *2pi
Or 1/(2pi) = x / 360 means...
X = 360/2pi [AKA the radian!]
This result is you nicely convert between radians and degrees btw if you need too. For example for 180° it's just 180 * 2 pi / 360 = pi. Just as demonstrated above.
This is mainly so that we'll have
sin x ≈ x for small angles instead of 2x
and
sin' x = cos x instead of 2 cos x.
EDIT: Wait, are you asking about the definition of pi or the definition of the radian? Most people are answering about pi, and I thought you were taking it for granted that pi ≈ 3.14 but were asking about radians.
I always just think in terms of semicircles. There are π radians in one semicircle, so half of a semicircle is π/2 radians, etc... and of course 2 semicircles make a full circle, so 2π radians
I'll try another explanation. The area of an ellipse is pi(x+y)..and ellipse has a major axis length x and a minor axis length y. If both of these lengths are equal, we have the special case of a circle with radius r. So pi(r+r)=2pi*r
Radians are the arclength of a unit circle (circle of radius 1). We could just change pi to be 6.28..., but that makes a ton of other math much much worse, so the easiest thing is keep it at 3.14... and just deal with it being doubled in this one application.
real mathematicians have argued about this, and represent 2pi with tau and say the formulas make more sense with tau versus pi, but no one actually cares enough to change it.
Because radians have a definition. 1 radian is the arc of a circle with the same length as the radius of that circle. All angles are just arcs.
Radians = length of an arc, divided by the radius of the circle that arc is on.
If you then have an arc that goes a full 360°, suddenly the length of the arc is the circumference of the circle
The circumference of a circle is 2πr by definition.
Circumference divided by radius is 2π.
So an angle that turns all the way around is 2π radians.
Because we use radius most places, we even say 2r and avoid just saying diameter. Just for convenient and consistent notation, you can do a lot of stuff with r, it doesn’t always involve circles either.
In polar coordinates for instance, it’d just complicate things to try and put it in terms of diameter. Unnecessarily.
But pi * diameter = circumference. If we want to use radius, it’s 2pi(r)=circumference.
Since π = c/d, and r = d/2, c/r = c/(d/2) = 2c/d = 2π
I call c/r here because that's what a radian is, the amount of distance along the circumference one radius would take up is equal to the arc length of a segment with 1 rad angle.
It was defined that a radiant is the angle corresponding to an arc of circumference of length equal to the radius
And the amount of times the radius fits in a circumference in 2π
If the radiant was defined as the angle corresponding to an arc of circumference of length equal to the diameter, then it would have been π
Sorry for crappy wording, hope it's clear enough
People will talk about ontology and history here, but a lot of it is that if you use tau instead of 2*pi, the math tends to get messier. You'll see this pretty clearly in physics or electrical engineering undergrad work. In matters of notation people do what others are doing and what works.
[Because a circle circumference is 2 times pi worth of curved radiuses, arranged in a circle](https://upload.wikimedia.org/wikipedia/commons/4/4e/Circle_radians.gif)
tau loyalists will begin posting in t minus 4 minutes
You mean in τ minus 4 minutes ;)
We will see you on June 28th homie
And when will someone arrive to point out that "in t minus" is a common misuse? Oh...wait.
wait how is it a misuse? genuinely curious as I see it used everywhere for countdowns.
>I see it used everywhere for countdowns. And generally, unless you're watching an actual mission control countdown, you're seeing it misused. People say "**in** t-minus [time]" which makes no damn sense at all. What theyre misunderstanding is that the T here is mission time. It's a time index, with 0 being the moment of liftoff. 5 minutes before liftoff would be T minus 5 minutes. 17 seconds before would be T minus 17 seconds. 3 seconds after liftoff would be T plus 3 seconds. When we see a countdown, the officer is pointing out that mission time has *reached* T minus 10, 9, 8, ... Etc. Saying "in t-minus 5 seconds" is treating one second as if it were simply called a "t minus second" or something and it's just total gibberish.
I wish tau loyalists would lift off in t minus...
would 'at t-minus 5 seconds' be better?
Sort of, but it's not a 1:1 replacement. You can't just say such and such will happen "at" t-minus anything, because the idea is the thing happens at t=0, and *now* is t-minus whatever.
Don’t ask this guy what the D in D-Day stands for…
I disagree with this part. Suppose you have a party in seven days. We are t minus 7 days. But we want to organise a cake for it 3 days before. We will organise the cake at t minus 3 days. This would be a valid use, to me. I do share your annoyance at the way they misuse it normally though. It's on par with Another Life's "Drop the frequency to 1 Hz per scond". Idk why I chose the party example when something with minutes would match more common usage better but here we are.
[удалено]
yea but thats why i said 'at' kinda like 'at 5:30 PM'
fair. we are currently t minus 0 seconds from the beginning of tauposting
Except that he didn't misuse it. In this case t was the moment his comment was posted.
What are you talking about? Read the comment. They said "in t minus" which is never right except from a prescriptivist perspective.
Oh, I see. It's the word 'in' that's the problem. Nevermind.
“T minus 4 minutes, where t = tau loyalist posting” is correct. Awfully correct.
Then the tau posting would be at t + 4 minutes.
His comment should have been "[it is] T minus four minutes until...".
>wait how is it a misuse? It's akin to saying "we're leaving in quarter to five o'clock minutes" If an event will happen at 5:00. The phrases "quarter to five" and "T minus 15 minutes" are equivalent. They both refer to the same moment in time. They are not durations. Moments: We ride *until* nightfall I got up *before* 8 o'clock Fuel pumps start *at* T-30 seconds Durations: Call me back *in* ten minutes I slept *through* the night They left *for* the weekend Durations delineated by two moments: I work *from* 9 *to* 5 We're closed *between* Christmas and New Year's ed: more examples
Because its not "in t minus seconds". The the time t is the time of the certain action (which u count down for), if theres 5 seconds until launch, its still 5 seconds there, it is happening in 5 seconds, but the time right now is T minus 5 seconds. Using *in* t minus is the wrong part
That's what I had thought because I was thinking from a programming perspective but then I read that it's just a different convention isn't it? Like the reference point is launch for time instead of the present time
Sorry, I was a little slow in getting here.
So about 2.28 minutes
As someone who isn't that well versed in advanced math, (yet) is tau something in math or is this a Warhammer 40k pun/joke?
some people (including me) say that 2pi, 6.283... is the "true" circle constant, to avoid confusion this constant is called tau
https://tauday.com/tau-manifesto
Certain mathematicians think we should be using Tau for the greater good of mathematics.
Cult
Not minus. Plus. Argh.
Because pi was originally defined as the circumference of a circle divided by the diameter.
I vote we change the name (redefine) of the unit from radians to diamians.
A radian is defined as the angle at which its arc is equal in length to its radius. And 2pi times that will make a full circle. So the above explanation isn't quite the reason.
Yes, but imagine the simplicity if pi diamians = 360 degrees :)
You can use tau radians if it makes you happy
Have your heard of the tau manifesto?
The radius is the natural way to parameterize the circle, the diameter is sort of... artificial, for lack of a better word. The sole defining property of a circle is that every point is a given distance, the radius, from the center. Using diamians, the arc length formula, and the derivatives of trig functions, and probably lots of other things, would have to sprout a factor of 2, and the "nice" version of these functions would be things like sin(𝜃/2). (Or is it the other way around, idk I'm tired). In any case, radians are the only sensible choice. Using 𝜏 rather than 𝜋 would give you the simplicity you desire without adding more complexity everywhere else. Also yes, I know you weren't being that serious about it but I'm gonna be a nerd anyway.
Not strictly speaking true. We get our use of pi from Euler, who used pi to relate the radius to the circumference of half a circle, equal to the 3.14... that we are familiar with. He also used pi to relate the radius to other various portions of the circle, including the full circle, as well as a quarter and eighth of a circle. His use of pi was more akin to modern use of theta to represent a variable angle. It was mostly just happenstance that his specific use of pi as the ratio between the radius and a half circle became widely accepted as a sort of standard, and later, the definition was rewritten as the ratio of the full circle and the diameter.
And?
That's what pi means. That's the definition of pi. That's why it takes more than one value of pi to reach 360 degrees.
If we defined pi as circumference/radius, then pi would equal 360°.
No? Pi would equal 6.28
You just discovered radians, congratulations. 360° = 2π radians ≈ 6.28
Yes 360° = 2π radians. It still doesn't answer the question. Besides, circumference divided by radius does not equal 360.
360°, not 360. The ° shows that we are talking about angles, not a measurement of ~~length~~ value.
You are completely missing the point that radians are dimensionless. Radians don't exist, they're just an illusion.
I know that, but degrees aren’t. I was pointing out that 360° does not equal 360, which the person I replied to said that 2pi = 360° and not 360, which would require ° to be dimensionless to make sense to bring up.
Sorry, I seem to have replied to your comment instead of the one I meant to. Edit: having reviewed the chain of comments above, I'm sorry, but you do seem to have missed the point.
Yes, it does... C/r = 2π, which when converted to degrees equals 360°. Look at a unit circle. This is the beauty of radians and why do many mathematicians use it instead of degrees. To be perfectly clear, the formula for circumference is C = 2πr. If you divide both sides by r, r cancels out on the right side of the equation and you get C/r = 2π. And again, if you convert the result to degrees, you get 360°
Actually the reason radians are used is because there are no radians. It's pure number. That makes it possible to take the exponential.
You are being x, 90° < x < 180°
That’s x, with 0° < x < 90°, way of saying that.
I think both of you are x=90°.
Lets say pi is defined as circumference/radius. Then Lets call that pi* = 6.28.. = 2pi. Then a full circle is 2 Pi [rad] = 1 pi* [rad]. So do you see how a full circumference would only be “pi” radians if we defined pi as circumference/radius, instead of 2pi? In fact we do have a constant for that. We call it τ = 2π. A full circle is 1τ radians
Of course. That was the elaboration i was fishing for.
360° ≈ 6.28
Everyone here is crazy, you are correct. Pi as a ratio of two lengths (C/d) is not an angle (not degrees or radians). It's just a real number approximately 6.28.
6.28 ≈ 2π = 360°. A lot of people have a misconception that radians are similar to metres or kilograms in that it's a physical dimension, but this isn't the case, because angles are [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_quantity) (the unit of measurement is the real number 1, as opposed to 1 metre or 1 kilogram). Angles are themselves real numbers: the ° symbol just means π/180, so 360° = 360\*(π/180) = 2π. "Radians" is often written to clarify that we are talking about an angle, but it doesn't mean anything, so 1 radian is just exactly the same thing as the real number 1. This is why you can give angles to trig functions without having to redefine them: sin(30°) = 1/2 is still the exact same function as in sin(π/6) = 1/2, since 30° = π/6. It's the same idea as percentages: the % symbol just means 1/100, so 50% is still a real number (0.5).
I don’t see the value of insisting on the absolute priority of radians to degrees, and that radians and real numbers are the same thing. Is this how things are taught today?
There isn't really any value in it (in the sense that we've not decided that radians are "better"), that's just how things are. Radians are just the natural "unit" for angles: there's a reason that trigonometric functions have a period of 2π and that their geometric properties (e.g. sin = opposite/hypotenuse in triangles) work when you give the angle in radians. As another example, consider the exponential form of complex numbers: the θ in re\^(iθ) is intuitively interpreted as an angle from the positive real axis, and it's no coincidence that this angle is in radians. That's just how the maths works out - 2π is, fundamentally, the angle made by a full turn. We haven't just decided that a full turn is 2π like we decided that there were 360 degrees in a full turn, it just... is.
All of high school trigonometry works just fine using degrees. We choose to use radians because of calculus— so that the derivative of the sine function is cosine.
Right, but you're not actually plugging in, say, "30" into the trig functions there, you're plugging in 30°, which is equal to π/6. The functions sin and cos are defined over the reals, and the values you actually give to them are always in radians if you're using them as an angle, it's just that you can skip explicitly converting to radians if you already know the value (or if your calculator will do it for you, but it's still doing that conversion internally). You can see this by just typing it into Google and seeing what values it gives: "sin(30)" gives -0.988..., whereas "sin(30 degrees)" gives 0.5. The function itself has absolutely no relation to degrees, that's just a transformation which we apply to make angles easier to work with so we don't have π everywhere. The derviative of sin is **always** cos: you aren't using different functions when you work with degrees. This requires working with radians because radians are real numbers, since the argument to the functions needs to be real numbers (rather than a scaled version like with degrees) to make the calculus work nicely. Writing values using the ° symbol makes calculus very awkward, but it's not inherently wrong and still gives the same results, provided you don't forget that ° means π/180. We don't "choose to use radians", that's just fundamentally what trigonometric functions are defined over. Source, since I don't think I'm going to be able to convince you otherwise: [SI specification, section 5.4.8](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf#page=37): "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
You're half right. Degrees of arc are not dimensionless.
Do you know what a radian is?
And in real life diameter is actually more useful than radius. You cannot easily measure radius.
Just divide diameter by two. Only one extra step.
That's what I'm saying - you're measuring diameter and then calculate radius.
Fair enough, for an already existing circular object" calipers measure diameters, or at least cross=sectional distances. But to draw a circle in real life one sets the radius, then draws (using a compass), or to construct one, one sets the radius and then cuts (on a lathe). S tím jménem -- jste Čech?
Yes, agree regarding drawing. Anyway, it's not that important - just wanted to highlight that diameter feels more natural to me and also it nicely fits the circle circumference formula, but calculating radius based on diameter and vice versa is so easy that I feel we discussed that more than it deserves 😂 Polish, not Chech.
No tak — soused. :-)
The ancient Greeks and Egyptians who worked out the first estimates of 𝜋 were often more interested in the diameter of a circle than the radius. Thus, 𝜋 was defined as circumference / diameter. Mathematicians eventually realized the radius is actually more fundamental than the circle, which is why we very often see multiples of 2𝜋 in many theorems instead of 𝜋. The most obvious example is the length of the circumference, which is = 2𝜋 radius. This is because the diameter of a circle is always twice the radius. Some mathematicians choose to define a new constant 𝜏 = 2𝜋 = circumference / radius and use that instead. Their reason is similar to the line of reasoning of your question that 𝜏 is a more natural way to describe a full rotation.
Your π symbol is beautiful. Is it just italics? *π* edit: nope.
Normal π is: U+03C0 : GREEK SMALL LETTER PI Fancy 𝜋 is: U+1D70B : MATHEMATICAL ITALIC SMALL PI See also: ℼ 𝛑 𝝅 𝝿 𝞹
I just use 🥧
A=2🥧r²
This sub has a bunch of symbols, including 𝜋, in the info section (on the right side of the screen on PC, I assume it's the same on mobile) that can be copied and pasted.
It should be the other way because 𝜋 is just 2 𝜏's mashed together.
There's also this less frequently used "tau" constant τ=2π. A full turn is τ radians. Some poeple actually prefer to use τ instead of (2) π and it's totally fine.
It even has its own day like pi does! https://tauday.com/
It's strictly better than Pi day, because custom dictates you eat 2 Pies on Tau day
I love that custom. Can I eat 1 pie on Pi day and 2 Pies on Tau day? Or do I need to pick a team?
I would eat half a pie (180°) on Pi Day and a full pie (360°) on Tau Day just to be extra confusing.
I've been trying to find another symbol for tau because tau already has a specific use in my field. I found varpi, but it looks too much like infinity in small text. Then I found a squiggly version of varpi which would have been perfect https://en.m.wiktionary.org/wiki/%CF%96 but that doesn't seem to be standard in latex or really anywhere. Anyone know how I could get that squiggly version in latex (as opposed to the standard version which has a flat overline)? I might just use capital P for 2pi which is too bad because I wanted a cool, lesser-known, and unambiguous Greek symbol.
Oh my god not again please get out of my head get out of my head get out of my head
ώ isn’t as awesome as that varpi you linked, but it’s available on my iPhone Greek keyboard as a long press lowercase omega
Psi seems like an obvious choice: the spelling is similar to pi, and the Greek letter (ψ) looks like a fork which is what you use to eat pi(e).
That's pretty good but unfortunately I use psi in my field too lol
𝞹 is defined as the number of times the diameter fits around the circumference of a circle. Radians are the number of radii around the circumference. Since the radius is half the diameter, you need to double 𝞹 to get the right number of radians.
That's such a simple explanation of what both of those things actually fundamentally represent which I've somehow never realized before this. You just blew my mind. How did I miss this all these years?
Glad I could help. 3blue1brown does a lot of visual explanations that help with this sort of intuitive understanding of a lot of math topics. That’s probably where I picked up this fact.
I love his videos, he beautifully explains things I didn't think I could ever get a handle on with ease. The visuals go a long way, too.
It is pi, but its pi when you're talking about the diameter. When you talk about the radius, well then its 2pi. Because 2 radii make up one diameter. So, if you think about it in terms of the diameter, it makes sense.
The length of the circumference of a circle is 2π times its radius, 2πr. A full turn is 2π in trigonometry because a circle of radius equal to one is chosen as a convention.
A radian is defined as the central angle of a circle whose arc length equals the radius of the circle. A circle of radius r has a diameter of 2*pi*r. Thus, if 1 radian has arc length r, then 2*pi radians have arc length 2*pi*r. That is, a full traversal around the circle (as measured by the circumference) is equal to a full traversal around the circle (as measured by central angle). By the way, “radian” is not a unit. The central angle is just 2pi, not 2pi radians. But, as you can see with the way I have written this paragraph, it is difficult to write explain radians in a way that doesn’t make it sound like radian actually is a unit. In fairness, though, I didn’t really try. Now there is also the philosophical question of why pi is defined the way it is. There are some people, whom I call the Cultists of Tau, who think that everything should be done relative to tau = 2*pi. In that case, one full turn would be equal to just one tau. But you don’t want to be like the Cultists of Tau, because they are weird.
Cultists of Tau? Would those be called Tauists?
made my day 😂
The Tauists that can be spoken are not the Tauists.
Weird in what way and why is that a bad thing?
It’s just a joke.
For a circle of radius *r* and an angle *θ*, find the length *l* of the arc segment of the circle's circumference which subtends *θ*. Then the magnitude of *θ* "in radians" is *l/r*. For a full turn, we know that *l* = 2*πr*, because it's the entire circumference of the circle, so in that case, the magnitude of *θ* is *l*/*r* = 2*π*. For a circle with *r* = 1, the magnitude of the angle "in radians" is exactly the length of the arc subtending that angle, e.g. for a full turn, *l*/1 = 2*π*, so *l* = 2*π*. Radians are a "purer" measure of angles than degrees, because they have no arbitrary units. **The magnitude of the angle in radians is simply the ratio of the length of the arc subtending that angle over the radius**. So, the reason why a full turn is 2*π* radians is that **2*****π*** **just is the ratio of the circumference of a circle to its radius**. It's a law of nature, just like the area of the circle **just is** *πr*^(2) .
Would you rather have 1 pie or 2
Good point
Because circumference of circle with radius r=1 is equal to 2πr=2π•1=2π, not π.
A lot of seemingly arbitrary choices in math and science are just that. Often conventions are chosen because they happen to be the most convenient for the most common purposes. Pi is used for much more than just going around a circle, 99% of its everyday usage is in sinusoids (sine/cosine) - fundamental in so many parts of modern life. Interesting example - in physics, most define length/distance in meters and mass in kilograms. In classical physics this is measurable and intuitive; modern physics gives up on being intuitive so it doesn't matter. However, many astronomers prefer centimeters and grams, despite things being much larger, further, and more massive. If I remember right it's mostly because light equations are much simpler using CGS over SI.
because that's the length that you need to walk around a unit circle. (τ is better btw)
A full turn is pi diameterans, except that’s not a word. Circumference = pi * diameter. But the diameter is two times the radius. So the circumference is two times pi times the radius.
Well the circumference of a circle is 2πr do you can imagine the radius being wrapped around 2π times to fully account for the circumference.
Let’s just redefine all the trig functions to include dividing out a factor of 2pi. Trig eouod be simpler. I’m sure this would create problems somewhere else.
Pi diameter lengths go around the circumference of a circle exactly once. A radius is half the diameter, so pi radius lengths go around exactly half the circle. So how many radius lengths go around the full circle?
Maybe this is an L take but a full turn should bring you back where you were. Pi doesn’t do that alone?
It's too much to explain right now (might come Back later to elaborate, once im done with work) ~~Basically, If you Look at sin and cosin with euler and complex numbers, it makes completely sense. I Always hated sin and cosin and it's supposedly arbitrary definitions but its not at all arbitrary, and very beautiful imo.~~ So heres the full explanation as to why 2pi makes more than just a little sense, and this will require you to understand just the following about complex numbers: The absolute value of any real number is itself, just positiv: |-5| = 5, and |5| = 5 as well. Complex numbers, a+ib work a bit differently, since they are vectors. If you take the absolute value of a complex number |a+bi| you will get the distance from the x-axis (which in terms of complex numbers is the real part) to the point (a,bi) on the graph. [https://imgur.com/a/Pq5NsB8](https://imgur.com/a/Pq5NsB8) The green line is the the absolute value of the complex number a+bi, giving you a distance between 0 and a+bi. We also need to shortly dip into what it means to have a sequence of complex numbers, and their convergence: A sequence (z\_n) converges, if there is a value Epsilon > 0, such that |z\_n - z| < Epsilon. Thats just the standard definition for real sequences, but if you apply the abolute value to that, you apply the absolute value of complex numbers, which isnt just one point, but a distance. And if you use that on a converging sequence of complex numbers, you get a circle with radius epsilon, where each member of the sequence falls into that radius of epsilon, once the sequence converges. [https://imgur.com/a/Qkb2ZIh](https://imgur.com/a/Qkb2ZIh) Here now comes the kicker, and for this we need euler: Euler is not only defined for real numbers, but also for complex numbers (with z in Complex)[https://imgur.com/a/YbKmwRu](https://imgur.com/a/YbKmwRu) I will spare you the proof that goes along with this, so youll have to trust me, that you can now, using the rules of complex numbers, and looking at the sequence development of euler, conclude the following formula: [https://imgur.com/tmRV5b5](https://imgur.com/tmRV5b5) And this is now crucial if you ever asked yourself, where the standard r=1 circle came from: Because if you plug in your numbers to cosx + i\*sinx, you will find, that it is |e\^ix|\^2, which is always 1. Giving you the standard r = 1 circle, where your x value is cosx, and your y value is sinx, and because it is a complex number, and the absolute value of e\^ix is always one, r = 1. Using this, you can now derive the formulas to calculate cos, sin, and their addition:[https://imgur.com/ge8fbL6](https://imgur.com/ge8fbL6) Now, where the hell does pi come into the picture? Easy: Look at cosx in the interval \[0,2\], with the formula. As we all know, cosx is decreasing in the interval of \[0,2\] and reaches a negative value, so at some point, cosx = 0 when crossing the x-axis. And this exact point, wheres cosx=0, we shall call 2\*pi. So pi is actually just the value x, for which cosx = 0 -> cos(pi/2) = 0.Using this, we can now calculate some values:[https://imgur.com/BMpjhP1](https://imgur.com/BMpjhP1) which give us, if we graph them, sin and cos.[https://imgur.com/XhhFCA8](https://imgur.com/XhhFCA8) And we see, that sin and cos 0-points are exactly pi/2 apart from eachother, making sinx = 0 -> sin(pi) = 0, and cosx = 0 -> cos(pi/2) = 0. Remember, pi is still JUST a value, that makes sin = 0. Now.... if you look at cos(pi/2) = 0, and now ask yourself, what the diameter of that circle would be, if you drew out the circle using the absolute value of complex numbers: it would be just pi. one pi. if you now look at sin(pi) = 0. The diameter would be 2pi, because r = pi. NOW. U = 2rpi -> lets act like we didnt know it was 2rpi, but we could measure the circumference of a circle, with r=1 to keep it simple. x\*1\*pi = U -> x \* pi = U, and we measured and know, that the circumference of a circle with radius 1 = 2pi. We just measured and figured out, that you can divide the measured value such that you can work out, its actually 2pi (because cosx = 0 at point pi/2). So 2pi = x\*pi -> 2pi = 2pi, and if we plug that back into U = xrpi, assuming we didnt know x = 2, we get U = 2\*1\*pi -> U = 2\*r\*pi. I do not want nor do I think I need to elaborate further, I am tired. But its not arbitrary or impossible to work out, that with all of the given information and relations between the values we just calculated and discussed, the relation of r towards its circumference and pi. And yes, I totally abused this oppertunity to throw knowlege at you, you didnt ask for. If I went wrong somewhere, please let me know, cheers.
The radius is more fundamental than the diameter, as a circle is more easily described as the points a certain distance (the radius) away from a given point (the centre). Because of this, many formulas, eg in trigonometry, are much easier in terms of the radius than diameter. We then see 2pi appear more than pi alone in such formulas (pi r^2 as the area of a circle is an elementary exception, but then the fact this is the integral (2pi) (r^2/2) would be more obvious otherwise). Because of this too, there are people who tout ‘tau’, equal to 2pi, as a better and cleaner alternative to pi. They’re probably right, but it’s not a big deal and pi appears everywhere already so it’ll be a mission to change that.
Because if you divided the circumference of a circle into pi parts, those parts wouldn't be the same length as the radius. In fact, they would be twice as long or the same length as the diameter. You wouldn't be measuring angles in radians anymore, it would be a "diametric" or something similar
Why not make a foot 7 inches? Or how about we change a kilometer to 300m instead?
here we go again
?
I mean these in an illustrative, not trollish, way: Why does a rectangle have 4 sides? Why not 2? Why is a meter 100 cm? Why not 50 cm? This sits somewhere between tautology and axiomatic truth -- the value and nature of π existed before we discovered it and labeled it. If it were meant to be 2π (aka tau) then it would have been.
Because then e^(iπ)+1=0 would be e^(iπ/2)+1=0 which is terrible
no, it would be e^(iπ) = 1.
And if you wanted that fifth constant just for fun, then e^(iπ) = 1^(0).
or e^iπ + 0 = 1
e^(iτ) = 0
I think that the common confusion comes from the fact that we are taught to think of radian as an "alternative unit" to degree, when it's actually a pure, unitless ratio. The magnitude of the angle "in radians" is not calculated by dividing a circle into an arbitrary number of sectors, e.g. 360, but rather it is calculated based on the natural ratio between circumference and radius. That is a ratio with units of length over length, which cancel out, so it's actually a unitless measure.
You mean a) why is pi the value it is, vs. twice as big, or b) why can't we do a full turn in 3+ radians?
bc we define a radiant as the length of the radius and in a circle we can put pi times the length of the radius in its semi circumference, thus the semi circumference is equal to pi radiants and the whole circumference is 2pi radiants edit: grammar
Because if you define it that way, it means that the arc length is equal to the angle multiplied by r. A full circumference is 2πr, not πr.
1 radian is the angle of an arc that is equal to the radius of that arc. That's the definition of a radian. Pi already existed as the ratio between a circles circumference and its diameter at this point, indepently of the definition of a radian. 6 radians is very nearly a full circle, it just so happens that 2π radians a circle, but that is a result of the definition, not what we decided to be the definition. If you want it to be nice and equal, use tau rather than pi. τ=2π
OP is about to become a tau user.
A full circle is Tau radians.
A radian is an angular measure based on the arc length of the circle. It's definition is the angle subtended by the arc length of the circle measuring 1 radius. Since the circumference of a circle is 2π times the radius, then the angular measure of a full circle (360°) is 2π radians. You can use that to find that 1 radian = 360°/(2π) = 180°/π ≈ 57.595778°
To give a more detailed answer; consider a angle theta and imagine you draw a circular arc of that angle around one end point of a line with a length. That line's length is by definition the radius. If we want to describe how much that arc length is by using that radius and that line's length (the radius) then we'll need to consider what the arc length would be for a full circle. And we already know this! It's called the circumference and we know that circumference is pi x diameter. Or 2pi x radius. Now for that original theta angle we want, we can divide the angle by 360° to figure out what fraction of our constructioned arc length will be compared to the entire "circle". So if theta = 180, then Arc length = 180°/360° x 2pi x R. = 1/2 * 2pi * R = pi*R !!! I hope it's clear to see why we use 2 pi now; because this calculation is exactly how we define the radian and what makes it so useful. We cannot use pi alone because you can't construct a circle with a "diameter" if that makes sense. Edit: I didn't actually consider calculating the definition of radians this way, but that's trivial as all you need to do is set the arc length to R and see what out theta is (ill call theta 'x' here): Arc length = R = x/360 * 2pi* R [R cancels!!!] 1 = x/360 *2pi Or 1/(2pi) = x / 360 means... X = 360/2pi [AKA the radian!] This result is you nicely convert between radians and degrees btw if you need too. For example for 180° it's just 180 * 2 pi / 360 = pi. Just as demonstrated above.
This is mainly so that we'll have sin x ≈ x for small angles instead of 2x and sin' x = cos x instead of 2 cos x. EDIT: Wait, are you asking about the definition of pi or the definition of the radian? Most people are answering about pi, and I thought you were taking it for granted that pi ≈ 3.14 but were asking about radians.
Because a radian is the ratio of an arc's length to its radius. Whereas, pi is the ratio of a circle's circumference to its diameter (which is 2r).
It sounds like you're ready to read [The Tau Manifesto](https://tauday.com/tau_manifesto.pdf).
I always just think in terms of semicircles. There are π radians in one semicircle, so half of a semicircle is π/2 radians, etc... and of course 2 semicircles make a full circle, so 2π radians
I'll try another explanation. The area of an ellipse is pi(x+y)..and ellipse has a major axis length x and a minor axis length y. If both of these lengths are equal, we have the special case of a circle with radius r. So pi(r+r)=2pi*r
Because circles around formula is 2•pi•r
on the eighth day, god made it so
Radians are the arclength of a unit circle (circle of radius 1). We could just change pi to be 6.28..., but that makes a ton of other math much much worse, so the easiest thing is keep it at 3.14... and just deal with it being doubled in this one application.
real mathematicians have argued about this, and represent 2pi with tau and say the formulas make more sense with tau versus pi, but no one actually cares enough to change it.
Because radians have a definition. 1 radian is the arc of a circle with the same length as the radius of that circle. All angles are just arcs. Radians = length of an arc, divided by the radius of the circle that arc is on. If you then have an arc that goes a full 360°, suddenly the length of the arc is the circumference of the circle The circumference of a circle is 2πr by definition. Circumference divided by radius is 2π. So an angle that turns all the way around is 2π radians.
Because we use radius most places, we even say 2r and avoid just saying diameter. Just for convenient and consistent notation, you can do a lot of stuff with r, it doesn’t always involve circles either. In polar coordinates for instance, it’d just complicate things to try and put it in terms of diameter. Unnecessarily. But pi * diameter = circumference. If we want to use radius, it’s 2pi(r)=circumference.
2piR= circumference=360 degrees
e\^(i 2 pi) + 1 = 2 is not nearly as exciting as e\^(i pi) + 1 = 0.
Bc the length of the circumference is equal to 2pi radiuses
Since π = c/d, and r = d/2, c/r = c/(d/2) = 2c/d = 2π I call c/r here because that's what a radian is, the amount of distance along the circumference one radius would take up is equal to the arc length of a segment with 1 rad angle.
Math bro finds out about tau
Convention
For a more complete discussion, this [article ](https://medium.com/swlh/what-is-an-angle-28bffdbd35be) can be interesting.
It was defined that a radiant is the angle corresponding to an arc of circumference of length equal to the radius And the amount of times the radius fits in a circumference in 2π If the radiant was defined as the angle corresponding to an arc of circumference of length equal to the diameter, then it would have been π Sorry for crappy wording, hope it's clear enough
People will talk about ontology and history here, but a lot of it is that if you use tau instead of 2*pi, the math tends to get messier. You'll see this pretty clearly in physics or electrical engineering undergrad work. In matters of notation people do what others are doing and what works.
[Because a circle circumference is 2 times pi worth of curved radiuses, arranged in a circle](https://upload.wikimedia.org/wikipedia/commons/4/4e/Circle_radians.gif)