The binary angular measurement isn't used in things like computer graphics typically if you were wondering, since this integer overflow is a unique property of how binary data is stored in computer memory, and is specific to your resolution. 256 works for a 1 byte variable, 65,536 works for a 16 bit/2 byte variable, 4,294,967,296 works for a 32 bit/4 byte variable (Such as 32 bit floats), and so forth. It's specific to the type of variable, the number of bits it uses, and how the computer handles integer overflow (Not all systems are happy with it, microcontrollers despise integer overflow) A useful piece of knowledge, but entirely obsolete in the modern world where typically compute memory buses are significantly larger than just 1 byte. Was useful back in the day of 8 bit processors though
It's more common in older software that didn't have as reliable and performant floating point hardware. That said, were floating point not optimized to hell and back, fixed point arithmetic would actually be more suitable for many things we typically use floats for, including but not limited to angle measure.
Yeah now we have fast floating point hardware and trigonometric functions, binary angles are rare. Their use was also connected to the use of lookup tables for sin and cos, as power of two sizes are easier to work with. Not necessarily in 8 or 16 bit units though, sometimes say 12 bits depending on accuracy required and memory/cache size. You can always use bitwise AND to get any power of two size you want.
@@igxniisan6996 The solid angle subtended by a hemisphere at its center measures τ = 2π ≈ 6.28 steradians. You'd need about 28.65 hemispheres' worth of steradians to get to 180.
I hear "clock directions" used often ("on my 6", "cybertruck at 11 o'clock"). Way more than binary angles (bangles?) or even grads. True, we never use arithmetic on them, but they are a measure of angle in common use. To a lesser DEGREE, same goes for the compass rose. "North by West Northwest"
You gave tr as a unit for a turn, but there's another, more common, unit for the turn: τ. Yes, the very same τ used as a constant for radians, because it actually doesn't matter whether you treat it as a constant number of radians in a turn or a a unit for a turn itself; it's the exact same angle either way. In a way, every unit is just a distinct fraction of a turn: 1, 360, τ, 400, and 2^n for some choice of n, usually a power of 2 multiple of 8. One thing to note is that while you mentioned highly composite numbers when talking about degrees, 1, the number of turns in a turn, is also a "highly composite number" (despite not being a composite number), and 2, which the number of binary angle units is always a power of, is not just highly composite, but a "superior highly composite number" like 360 (despite being *_prime)._* P.S. I am biased as a programmer, but with halving being such an important operation, degrees, despite being to divide into fifths and thirds, honestly disappoint me by not being to halve them more than 3 times before the result in no longer an integer. On the other hand, halving is the single thing that binary units (angle or otherwise) do best. Turns and radians completely ignore this and don't even bother trying to represent fractions of a circle as integers and instead leave them as, well, fractions.
But only in radians does τ = 2π, which is a real (specifically transcendental) number. In turns, 1 τ means 1 revolution around the circle. Which, don't get me wrong, is amazing and useful for simply expressing stuff, but what about calculations? If you tried to use the Taylor series for sin(x) you can't plug in 1 for 1 τ, it doesn't work. Also, while on the topic of τ, IF it stands for "turn", that's weird because "turn" in Greek doesn't start with a "τ". My prediction: 1. it does stand for "turn" but people used the Greek equivelant of "t", or 2. it doesn't stand for turn, but either way it's a very similar letter to "π".
@@DeJay7 tornos, from Etymonline: "lathe, tool for drawing circles". The English word did pass through Latin and French before arriving in English, with the exact meaning shifting along the way. I can't find if the word or a derivative is used in Modern Greek, so it could be that it only survived in other languages.
i'd say 300 bc they tried to pull what we babylonian-degree people would call a 180, but then they somehow went off in a completely different direction
Not sure why there is so much emphasis on tau in this channel. I’m not a hater, but pi is more commonly used in higher mathematics, scientific papers, etc. In other words, you’ll always see pi but you won’t always see tau. Thus, if your aim is to introduce these concepts to a younger audience and make them more excited about mathematics, it would make much more sense to use pi versus tau. Pi is common, while tau is novelty.
@@JaybeePenaflor Well, it's fair to assume that someone watching these videos is already acquainted with π. The idea of introducing τ is clarity, and radian measures are far and away the best place to do this. People are often confused by radians when written in terms of π, but then they get it in terms of τ; it's just much more natural for us to think of a full turn instead of a half-turn. If someone ever encounters a radian measure in terms of π later on, they can simply use the handy conversion τ = 2π to switch between them, based on what they find easier to work with.
It doesn't really matter what constant you denote, whether it's length divided by radius, diameter, or half length by diameter. Euler himself used pi interchangably as whole turn, half turn or quarter turn. Similarly, using tau instead of pi is just reinventing the wheel. We already have 2π. It's just trying to use whymsical notation to make something look deeper or more complex than it is, which isn't what math is about
@@dapcuber7225 It's not just for no reason. You don't see anyone assigning a specific letter to something like 322π. If j = 322π, how many radians are in 1/4 turn? j/644. If τ = 2π, how many radians are in 1/4 turn? τ/4. One of these is much clearer than the other.
I never heard of binary angle before. I do program so I know all about the annoyance of overflow. But I never considered looking at that as a circular event. I'm truly stunned by how much stuff in real life is connected to pi through circular cycles.
By the way, angle is a dimensionless quantity, which is why it is bounded in a certain area. A radian is define to be the angle of 1m length formed by a 1m radius. 6:04 Gradians can also use the % symbol, and when someone says that an incline is of a 10% grade, it uses gradians. 7:52 Binary angular measurement can be used to find the angular velocity of a given vector-valued function using complex numbers. There is also a unit called solid angle, and there is the roll, which is the solid angle that forms a sphere, followed by the square degree, the steradian, the stergradian, and the binary solid angular measurement. 1 roll = 41,252 sd = 4π sr = 800% sg = 1 B
No, percent grade is derived from the tangent of an angle, not the angle itself. 45 degrees is a 100 percent grade, because the tangent of 45 degrees is 1.
No, when inclines are expressed as a percentage this is read as a slope (dy/dx) which correlates to the tangent of the angle, not the angle itself. A "100%" incline is not 100 grads (i.e. vertical) but a rather 50 grads (or 45 degrees).
You see Gradians on the road (on hills) where they give a "% grade". A "10% grade" is very steep and means it is 10% of the way between flat and vertical, or 10 grad or 9 degrees.
Oh yeah, that's actually a different thing. It's a measure of slope, not angle. A 10% grade means that for every 100 units you go forward, you go up by 10 units. I can definitely understand how you got confused here; unfortunate naming choices have led us to this point. If you want to figure out the angle measure associated with a slope, you have to apply the arctangent function. 10% is the same as 1/10, so we're looking for arctan(1/10). We get that this is about 0.0997 radians, which is extremely close to 1/10 radian. This is explained by the tangent small-angle approximation: For small angle measure θ in radians: tan(θ) ≈ θ θ ≈ arctan(θ) Anyway, we can convert our angle measure to other units as well: it's about 5.7 degrees, or about 6.3 gradians.
Turns don't get enough love, they are the only real unit of angles. Your mission, if you choose to accept it, add measurements in turns to a few Wikipedia articles today.
Me, looking at the formula for area of a circular sector at 5:46 but with the knowledge that θ=s/r: Dude!!! That r² is like that because it's compensating for a hidden r included in θ! Once you do A=1/2*(s/r)*r²=1/2*sr, it's just the formula for the area of a isosceles triangle! You can even show this by drawing a funny "quadrilateral" that has two parallel circular edges of length s and two parallel straight edges of length r and using the area formula for a regular rectangle! Then halve that, because the triangle is half a rectangle (one of the radius segments of this figure bisects the quadrilateral in half). But wait a minute, an isosceles triangle with the equal sides of length r doesn't have a height of r, but you have to do some Pythagoras magic to obtain h, and then its area is 1/2*sh (s is the length of the base). Well, here's the magic of the circle sector "triangle": because the base edge is an arc centred at the tip of the "triangle" (the vertex of the two straight edges), its height is r! Now, I'm sure some proper mathematicians have either opened a bottle of hard liquor, or they are sharpening their knives, but I find this enlightening.
Good intuition! The equality s = rθ is often used throughout mathematics and physics. However, note that the formula A = (1/2)bh works for any triangle, not just isosceles ones. Also, I don't think "parallel" is the right word there, since it doesn't seem to fit any of our conceptions of parallel lines; I'd just call them opposite edges. If you want, you can think of the area of a circular sector by slicing it up into ring sectors, as if you took the circular sector from an onion. Then, you can flatten these pieces out on top of each other. The resulting shape is approximately a triangle; the thinner the slices, the better the approximation. Its base length is the arc length, and its height is the radius. This gives the same formula of A = (1/2)sr, which can be rewritten as A = (1/2)θr^2, as previously mentioned. The disk can be seen as a special case of a circular sector, where the arc length is the entire circumference, and where the central angle measure is τ. This means the formula for the area of a disk is A = (1/2)Cr, or A = (1/2)τr^2. In this way, the geometry of the disk is deeply connected to that of the triangle. Rest assured, no hard liquor or knives will be necessary.
You missed the percentage which is for example: 10% angle means for every 100 units of distance traveled the elevation rises with 10 units it is used in engineering.
@skelet8337 Assuming you mean distance travelled along the slope, that's the sine of the angle from the horizontal. Apply the arcsine function to convert the sine to an angle of arc. For example, 10% is 0.1 and arcsine(0.1) is about 5.74 degrees. (If you meant distance travelled horizontally, it would be the tangent of the angle from the horizontal and you would need to apply the arctangent function to convert it to the angle. For example, arctangent(0.1) is about 5.71 degrees.)
When describing the various measurements for 2-D angles, there is a missed opportunity for one more type: the steradian, used to measure solid 3-D angles.
Radians is the only angle measurement that natively works on trigonometric functions. If you wanted to calculate the output of a trigonometric function by hand, you would need to convert to radians first.
@@RCHobbyist463 I'm not sure if this is a well-defined concept. You could definitely rewrite the Taylor series for sine to accommodate, say, turns; you'd have to append some constants, but you could manage. Or maybe you think that that would be "building in" the conversions, therefore qualifying as cheating. Well, sine is opposite over hypotenuse, so that's another method of calculation. But I doubt even more that you'd accept that, since this entails being directly given only lengths, not the angle measure. But hey, I can't be sure what you're thinking. I'm not a mind reader.
Regarding radians, as someone deeply in love with mathematics, they are such a natural unit of measurement that they are just the base unit for angles. In fact, the unit of measurement rad is very often written, almost always to denote that we are talking about units, but it means nothing, 1 rad = 1, the only reason we don't throw a rad everywhere is because it would be confusing. But really, think about it, if we "assinged" a value of 2π rad for 1 full revolution, so it's technically arbitrary, then one wouls argue that the trigonometric functions having a period of 2π is also arbitrary, but then look at the Taylor series of sin(x), it's just as clean and natural as the one for e^x, no coefficients for correction, no nothing. In many ways, 2π is the BEST value for one full revolution, and I still don't know why. I guess π is not a random transcendental number, it is indeed special as the ratio between a circle's diameter and circumference, but why is that enough? I want to know.
@@DeJay7 I think this might be another case of π causing unnecessary confusion. After all, this topic has nothing to do with the diameter of a circle; instead, it has everything to do with the radius. Aside from that, the primary concern is that you seemingly don't know how the radian is actually defined, instead assuming that it was arbitrarily chosen so that 2π radians is one turn, which is a huge problem. The definition of the radian is stated in the video, so I would recommend taking a look at that. If you still don't get it, I can try clearing up any confusion you may have.
@@isavenewspapers8890 No not at all, I totally understand radians. An angle θ rad on a circle's center [verb which I seriously can't remember] an arc of r*θ length. I know it's not arbitrary whatsoever, 2π is the circumference of a unit circle, therefore it's the angle of a full revolution. The actual definition and use does not confuse me, the applications, however, and specifically the naturalness of it is what I don't truly understand. Mathematically, it's as natural as the natural logarithm, only difference being one was defined to be the natural and the other came from circles. Why does the sum, for n from 0 to infinity, of (-1)^n/(2n+1)! * x^(2n+1), equal exactly 1 for x = 2kπ + π/2? I've not yet found why the, mind you, transcendental number π, which comes from circles, have this insane property that is deeply connected "pure" mathematics. And there are many other reasons I think radians are such a natural unit for angles, but WHAT is the deeper mathematical connection?
@@DeJay7 Regarding the mystery verb, I think I can help you out: "An angle measuring θ radians with its vertex at the center of a circle of radius r [encloses/intercepts] an arc of length rθ on the circle." But I get the feeling that that's not what you meant, as there's a special verb used within this general area of discussion: subtend. However, according to the conventional definition, you would've gotten it backward: a circular arc subtends an angle, not the other way around. If you want to remember this word, it helps to know where it comes from. It's related to the word "extend". It contains the sub- prefix, meaning "under", like in "subway" or "subtitle". So, "subtend" means "extend under". The circular arc extends under the angle. It's like how a bright patch of ground extends under a street light. About the definition of the natural logarithm, there are actually a few different definitions you could use. For instance, since the derivative of ln(x) is 1/x, you can integrate 1/t dt from 1 to x to get ln(x). This is a fairly common definition. Alternatively, using the typical way of defining a logarithm as the inverse of an exponential function, it can be defined as the inverse of e^x, which can itself be defined as the unique exponential function equal to its own derivative. To answer your first question, it's because what you gave is the Taylor series of sin(x), which is precisely equal to sin(x), and the points where the sine function attains a value of 1 are readily apparent. Well, how do we know that the Taylor series of sin(x) is equal to sin(x) itself? It is of course something we can prove-to use math words, we must show that sine is an analytic function-but the answer turns out to be pretty complicated, and I don't think I can do it justice here. So far, though, I get the impression that I haven't told you anything you don't already know. Maybe this is just something you find unintuitive, and you're looking for a way to make it intuitive. To tell you the truth, such a way might not exist. There are some things in math that are simply too weird to fit within human comfort levels, and we just have to live with it. I just want to make sure that you can tell the difference between a genuine mathematical question and an unanswerable riddle.
I'm surprised there was no mention of grade. It is simple the tangent of the angle expressed as a percentage. Ie a 45° angle creates a slope with a grade of 100% because tan(45)=1
@@isavenewspapers8890 Basically tau/revolutions, but with pi (so the unit circle has a diameter of 1, not a radius) and 1 revolution around the circle=pi darians, where pi=3.14...
@@peterchan6082 Really, nobody? Hmm, must be some other reason τ was proposed in the first place, and why it's gained so much attention from calculators and programming languages. Maybe people wanted to be able to write Stirling's approximation slightly faster.
@@isavenewspapers8890 U r taking it too seriously although he is not that wrong. Tau could be mathematically more convenient but I can guarantee you not many hs students use tau
@@praiseboggy Look, I understand that people can use figurative language sometimes, and not everything should be taken literally. But when I hear someone say "nobody", if they don't actually mean nobody, I would at least expect them to be talking about an extremely small group of people. But given the significant rise of τ in a multitude of different places, I just don't think that's fair to say.
Wait, what? Do all English native speakers pronounce centi- the way it is done in this video at 0:48? That is, the "i" in "centi" is pronounced more or less like the "e" in "centi". I thought it was a short "ee" sound, just as it the case with the final "i" in "milli".
@@Zurich_for_Beginners I see. I couldn't find anything about this online, but if there are 6400 of them in a turn, then this unit is equivalent to a NATO mil. Switzerland is a member of the Partnership for Peace, which is a NATO program, in case that information is relevant here.
@@alex-wl4sb It *has* happened. The major proposal that got the ball rolling was in 2010, and since then, it has spread among many different circles. It's in several online calculators and a whole bunch of programming languages.
All you have to understand is how a turn works. If you know what a quarter-turn is, then you know what τ/4 means. Based on what I've seen, many people find that intuitive, so using τ makes sense.
@@isavenewspapers8890 Let's take a look at the literature, shall we? If we can find it being used in journal articles or textbooks, then we might say it's being used. But if not, then the fact that Vi Hart likes it doesn't really mean anything. Ironically, your response applies to you in the inverse: That you know someone who does use it does not comprise everyone.
@ I'm sorry, is one of us hallucinating right now? Certainly, you would seem to believe that your initial comment actually reads, "Nobody who writes journal articles or textbooks uses τ," but that added condition is not what my own eyes read. After all, if Vi Hart uses τ, who cares about the journal or textbook authors? Vi Hart may not be such a person, but they're definitely not nobody. I may not be such a person, but I'm definitely not nobody. Programmers may not be such people, but they're definitely not nobody. Yes, the only possible explanation is that one of us has perhaps suffered a concussion or some other ailment of the brain, thereby failing to read the same comment. No other explanation would suffice. The problem would appear to extend even further, for you have seemingly presumed me to claim that EVERYONE uses τ. Unfortunately, I don't recall having claimed such a thing throughout the course of my natural life. How utterly bizarre. We shall both require our heads medically examined.
@@geoffstrickler Well, if you just wanted definitions and nothing else, then sure, you could do it much faster. But that's not what this video is about.
Yes, because ten is a terrible number to use as a base, but metric was built off it anyways. The fact that units have consistent proportions between them is nice, but it would be even better if that proportion wasn't ten (or technically a thousand in SI).
The name of the letter is written as "tau" using the Latin alphabet. As for pronunciation, it's conventional throughout academic disciplines to pronounce it how it's written, even if it doesn't correspond with the modern-day Greek pronunciation. As long as people can understand each other and communicate effectively about what letter they're talking about, there's no issue.
The day I realised that radians were a measure of the arc on a unit circle was a happy day as so much suddenly made sense.
For me, arc length = r·φ for arbitrary radii clicked some time during vector calculus or electromagnetism at uni.
that's actually how my precalc teacher introduced them to our class
Wait, you weren’t introduced to the definition of radian when it’s first mentioned?
@Joker-fj8hg Not that I remember. It was 20+ years ago, though.
The binary angular measurement isn't used in things like computer graphics typically if you were wondering, since this integer overflow is a unique property of how binary data is stored in computer memory, and is specific to your resolution. 256 works for a 1 byte variable, 65,536 works for a 16 bit/2 byte variable, 4,294,967,296 works for a 32 bit/4 byte variable (Such as 32 bit floats), and so forth. It's specific to the type of variable, the number of bits it uses, and how the computer handles integer overflow (Not all systems are happy with it, microcontrollers despise integer overflow)
A useful piece of knowledge, but entirely obsolete in the modern world where typically compute memory buses are significantly larger than just 1 byte. Was useful back in the day of 8 bit processors though
Floats (32b or other) don't work like ints and microcontroller ALUs aren't fundamentally different to computer ALUs, but otherwise yes.
It's more common in older software that didn't have as reliable and performant floating point hardware. That said, were floating point not optimized to hell and back, fixed point arithmetic would actually be more suitable for many things we typically use floats for, including but not limited to angle measure.
Super Mario 64 computes its angles in units of 16-bit fractions of a turn.
Yeah now we have fast floating point hardware and trigonometric functions, binary angles are rare. Their use was also connected to the use of lookup tables for sin and cos, as power of two sizes are easier to work with. Not necessarily in 8 or 16 bit units though, sometimes say 12 bits depending on accuracy required and memory/cache size. You can always use bitwise AND to get any power of two size you want.
thank you, now i know how to eat a triangle
aka dorito
I eat hemispheres measuring 180 steradians
@@igxniisan6996 The solid angle subtended by a hemisphere at its center measures τ = 2π ≈ 6.28 steradians. You'd need about 28.65 hemispheres' worth of steradians to get to 180.
@@igxniisan6996 A hemisphere subtends a central solid angle measuring τ = 2π ≈ 6.28 steradians, so you're eating about 28.65 hemispheres.
@@igxniisan6996 It seems TH-cam will not let me post a reply analyzing the math involved here. Too bad.
I love how you used Tau to introduce radians corresponding to degrees
Steve Mould will be proud
I hear "clock directions" used often ("on my 6", "cybertruck at 11 o'clock"). Way more than binary angles (bangles?) or even grads. True, we never use arithmetic on them, but they are a measure of angle in common use.
To a lesser DEGREE, same goes for the compass rose. "North by West Northwest"
1 hour here is (of course "straight ahead" is "12:00") equivalent to 30⁰ clockwise from 12:00- like a clock!
You gave tr as a unit for a turn, but there's another, more common, unit for the turn: τ. Yes, the very same τ used as a constant for radians, because it actually doesn't matter whether you treat it as a constant number of radians in a turn or a a unit for a turn itself; it's the exact same angle either way.
In a way, every unit is just a distinct fraction of a turn: 1, 360, τ, 400, and 2^n for some choice of n, usually a power of 2 multiple of 8. One thing to note is that while you mentioned highly composite numbers when talking about degrees, 1, the number of turns in a turn, is also a "highly composite number" (despite not being a composite number), and 2, which the number of binary angle units is always a power of, is not just highly composite, but a "superior highly composite number" like 360 (despite being *_prime)._*
P.S. I am biased as a programmer, but with halving being such an important operation, degrees, despite being to divide into fifths and thirds, honestly disappoint me by not being to halve them more than 3 times before the result in no longer an integer. On the other hand, halving is the single thing that binary units (angle or otherwise) do best. Turns and radians completely ignore this and don't even bother trying to represent fractions of a circle as integers and instead leave them as, well, fractions.
But only in radians does τ = 2π, which is a real (specifically transcendental) number. In turns, 1 τ means 1 revolution around the circle. Which, don't get me wrong, is amazing and useful for simply expressing stuff, but what about calculations? If you tried to use the Taylor series for sin(x) you can't plug in 1 for 1 τ, it doesn't work.
Also, while on the topic of τ, IF it stands for "turn", that's weird because "turn" in Greek doesn't start with a "τ". My prediction: 1. it does stand for "turn" but people used the Greek equivelant of "t", or 2. it doesn't stand for turn, but either way it's a very similar letter to "π".
@DeJay7 Technically, it stands for the Greek word for "lathe": "τόρνος", which is the origin of the English word "turn".
@angeldude101 What the hell is τόρνος? I'm Greek.
@@DeJay7 every intangible unit is basically a number multiplied by a constant. What makes turn any different?
@@DeJay7 tornos, from Etymonline: "lathe, tool for drawing circles". The English word did pass through Latin and French before arriving in English, with the exact meaning shifting along the way. I can't find if the word or a derivative is used in Modern Greek, so it could be that it only survived in other languages.
Binary angle measurement also aligns itself perfectly to the cardinal directions.
As does the gradian -- the hundreds digit directly aligns with whatever quadrant that angle falls in.
@@Stratelier I think the person means ALL of the cardinal directions, including the sub-cardinals like North-East, East-North-East, etc.
6:42 okay then, french revolution is how many gradians?
@@askcaralice Uh... 200 times the number of overthrows of government? I dunno.
Also, nice profile picture.
i'd say 300 bc they tried to pull what we babylonian-degree people would call a 180, but then they somehow went off in a completely different direction
1 gradian per 100 kilometers.
I hope more educators start introducing tau as it makes learning trigonometry so much more intuitive.
but then no pie day in math class :(
@@galacticdragon9841why not a tau day?
@@Oxzone. June 28 would probably fall within summer break for many schools.
@@isavenewspapers8890 June 28 is a schoolday where I went to school, but we write it as 28.6, so it doesn't really work as a tau day here either.
the gradian is long gon
It gradually turned rad
Seeing Earth bouncing around that elliptical orbit at 2:30 - couldn't help but giggle a little at that animation! ^^
Not sure why there is so much emphasis on tau in this channel. I’m not a hater, but pi is more commonly used in higher mathematics, scientific papers, etc. In other words, you’ll always see pi but you won’t always see tau. Thus, if your aim is to introduce these concepts to a younger audience and make them more excited about mathematics, it would make much more sense to use pi versus tau. Pi is common, while tau is novelty.
tau just seems pointless in my opinion, I feel like mathematicians like to make up constants for the sake of it, 2pi just makes more sense to me
@@JaybeePenaflor Well, it's fair to assume that someone watching these videos is already acquainted with π. The idea of introducing τ is clarity, and radian measures are far and away the best place to do this. People are often confused by radians when written in terms of π, but then they get it in terms of τ; it's just much more natural for us to think of a full turn instead of a half-turn. If someone ever encounters a radian measure in terms of π later on, they can simply use the handy conversion τ = 2π to switch between them, based on what they find easier to work with.
It doesn't really matter what constant you denote, whether it's length divided by radius, diameter, or half length by diameter. Euler himself used pi interchangably as whole turn, half turn or quarter turn. Similarly, using tau instead of pi is just reinventing the wheel. We already have 2π. It's just trying to use whymsical notation to make something look deeper or more complex than it is, which isn't what math is about
@@dapcuber7225mathematicians don't do it, youtubers do...
@@dapcuber7225 It's not just for no reason. You don't see anyone assigning a specific letter to something like 322π. If j = 322π, how many radians are in 1/4 turn? j/644. If τ = 2π, how many radians are in 1/4 turn? τ/4. One of these is much clearer than the other.
I never heard of binary angle before. I do program so I know all about the annoyance of overflow. But I never considered looking at that as a circular event. I'm truly stunned by how much stuff in real life is connected to pi through circular cycles.
Like how you always include tau in these videos
Tau supporter I assume?
anybody that uses tau is a tau supporter.
note: I used tau on this message.
τ
tau
I mean it makes sense in this context, generally if you're given the radius you use tau, if given the diameter you use pi
i say we also need a three- and a four-legged variant of τ
Video suggestion: every temperature scale explained. Because besides Fahrenheit, Celsius and Kelvin there are a bunch of weird, obsolete ones.
5:45 i think the visuals are a bit messed up here
Making a pun on the word "gradual" and then missing the fact you said degree a few seconds later is criminal. 7:18
By the way, angle is a dimensionless quantity, which is why it is bounded in a certain area. A radian is define to be the angle of 1m length formed by a 1m radius.
6:04 Gradians can also use the % symbol, and when someone says that an incline is of a 10% grade, it uses gradians.
7:52 Binary angular measurement can be used to find the angular velocity of a given vector-valued function using complex numbers.
There is also a unit called solid angle, and there is the roll, which is the solid angle that forms a sphere, followed by the square degree, the steradian, the stergradian, and the binary solid angular measurement. 1 roll = 41,252 sd = 4π sr = 800% sg = 1 B
No, percent grade is derived from the tangent of an angle, not the angle itself. 45 degrees is a 100 percent grade, because the tangent of 45 degrees is 1.
No, when inclines are expressed as a percentage this is read as a slope (dy/dx) which correlates to the tangent of the angle, not the angle itself. A "100%" incline is not 100 grads (i.e. vertical) but a rather 50 grads (or 45 degrees).
You see Gradians on the road (on hills) where they give a "% grade". A "10% grade" is very steep and means it is 10% of the way between flat and vertical, or 10 grad or 9 degrees.
Oh yeah, that's actually a different thing. It's a measure of slope, not angle. A 10% grade means that for every 100 units you go forward, you go up by 10 units. I can definitely understand how you got confused here; unfortunate naming choices have led us to this point.
If you want to figure out the angle measure associated with a slope, you have to apply the arctangent function. 10% is the same as 1/10, so we're looking for arctan(1/10). We get that this is about 0.0997 radians, which is extremely close to 1/10 radian. This is explained by the tangent small-angle approximation:
For small angle measure θ in radians:
tan(θ) ≈ θ
θ ≈ arctan(θ)
Anyway, we can convert our angle measure to other units as well: it's about 5.7 degrees, or about 6.3 gradians.
Turns don't get enough love, they are the only real unit of angles. Your mission, if you choose to accept it, add measurements in turns to a few Wikipedia articles today.
"real"?
Please do not frivolously edit Wikipedia articles. Also radians while not being as nice as turns are way better for mathematics and science.
Please do frivilously edit Wikipedia articles.
Use "τ" as the unit instead of "tr" and nobody would be able to realise that you're not using radians.
@bigfennec it's not frivolous, turns are solidly better
Wonderful explanation, thanks!
Best part of binary degrees? You can keep dividing it in half, and you'll keep getting integer amounts (until you get below 1).
9:16 this is also why people thought that Y2K was gonna be a big problem.
Me, looking at the formula for area of a circular sector at 5:46 but with the knowledge that θ=s/r: Dude!!! That r² is like that because it's compensating for a hidden r included in θ! Once you do A=1/2*(s/r)*r²=1/2*sr, it's just the formula for the area of a isosceles triangle! You can even show this by drawing a funny "quadrilateral" that has two parallel circular edges of length s and two parallel straight edges of length r and using the area formula for a regular rectangle! Then halve that, because the triangle is half a rectangle (one of the radius segments of this figure bisects the quadrilateral in half).
But wait a minute, an isosceles triangle with the equal sides of length r doesn't have a height of r, but you have to do some Pythagoras magic to obtain h, and then its area is 1/2*sh (s is the length of the base). Well, here's the magic of the circle sector "triangle": because the base edge is an arc centred at the tip of the "triangle" (the vertex of the two straight edges), its height is r!
Now, I'm sure some proper mathematicians have either opened a bottle of hard liquor, or they are sharpening their knives, but I find this enlightening.
Good intuition! The equality s = rθ is often used throughout mathematics and physics. However, note that the formula A = (1/2)bh works for any triangle, not just isosceles ones. Also, I don't think "parallel" is the right word there, since it doesn't seem to fit any of our conceptions of parallel lines; I'd just call them opposite edges.
If you want, you can think of the area of a circular sector by slicing it up into ring sectors, as if you took the circular sector from an onion. Then, you can flatten these pieces out on top of each other. The resulting shape is approximately a triangle; the thinner the slices, the better the approximation. Its base length is the arc length, and its height is the radius. This gives the same formula of A = (1/2)sr, which can be rewritten as A = (1/2)θr^2, as previously mentioned. The disk can be seen as a special case of a circular sector, where the arc length is the entire circumference, and where the central angle measure is τ. This means the formula for the area of a disk is A = (1/2)Cr, or A = (1/2)τr^2. In this way, the geometry of the disk is deeply connected to that of the triangle.
Rest assured, no hard liquor or knives will be necessary.
i learn more from this then i did in school
You missed the percentage which is for example:
10% angle means for every 100 units of distance traveled the elevation rises with 10 units it is used in engineering.
@@skelet8337 Does a measure of slope really count?
@@skelet8337 Yes, you can convert from slope to angle with the arctangent function, but it doesn't exactly cover every possible angle.
@@isavenewspapers8890Unless you go above 100% grade, the angle from horizontal is always 45⁰ or less.
@@wyattstevens8574 Does one not usually go above 100%?
@skelet8337 Assuming you mean distance travelled along the slope, that's the sine of the angle from the horizontal. Apply the arcsine function to convert the sine to an angle of arc. For example, 10% is 0.1 and arcsine(0.1) is about 5.74 degrees. (If you meant distance travelled horizontally, it would be the tangent of the angle from the horizontal and you would need to apply the arctangent function to convert it to the angle. For example, arctangent(0.1) is about 5.71 degrees.)
I like the binary degree quite a lot.
6:40 you mean the french full rotation?
This channel appears to be very good. (I'll need to watch a few more videos to be certain, and I won't be doing that right now.)
When describing the various measurements for 2-D angles, there is a missed opportunity for one more type: the steradian, used to measure solid 3-D angles.
@@Aero_Yuki Wouldn't really fit this video.
6:16 so sad that its gone 😢 i really wanted to know it.
Radians is the only angle measurement that natively works on trigonometric functions. If you wanted to calculate the output of a trigonometric function by hand, you would need to convert to radians first.
@@RCHobbyist463 I'm not sure if this is a well-defined concept. You could definitely rewrite the Taylor series for sine to accommodate, say, turns; you'd have to append some constants, but you could manage.
Or maybe you think that that would be "building in" the conversions, therefore qualifying as cheating. Well, sine is opposite over hypotenuse, so that's another method of calculation. But I doubt even more that you'd accept that, since this entails being directly given only lengths, not the angle measure. But hey, I can't be sure what you're thinking. I'm not a mind reader.
Constant expansion,Time Dialation,that one second was true at that moment 👍
Regarding radians, as someone deeply in love with mathematics, they are such a natural unit of measurement that they are just the base unit for angles. In fact, the unit of measurement rad is very often written, almost always to denote that we are talking about units, but it means nothing, 1 rad = 1, the only reason we don't throw a rad everywhere is because it would be confusing. But really, think about it, if we "assinged" a value of 2π rad for 1 full revolution, so it's technically arbitrary, then one wouls argue that the trigonometric functions having a period of 2π is also arbitrary, but then look at the Taylor series of sin(x), it's just as clean and natural as the one for e^x, no coefficients for correction, no nothing. In many ways, 2π is the BEST value for one full revolution, and I still don't know why. I guess π is not a random transcendental number, it is indeed special as the ratio between a circle's diameter and circumference, but why is that enough? I want to know.
@@DeJay7 I think this might be another case of π causing unnecessary confusion. After all, this topic has nothing to do with the diameter of a circle; instead, it has everything to do with the radius.
Aside from that, the primary concern is that you seemingly don't know how the radian is actually defined, instead assuming that it was arbitrarily chosen so that 2π radians is one turn, which is a huge problem. The definition of the radian is stated in the video, so I would recommend taking a look at that. If you still don't get it, I can try clearing up any confusion you may have.
@@isavenewspapers8890 No not at all, I totally understand radians. An angle θ rad on a circle's center [verb which I seriously can't remember] an arc of r*θ length. I know it's not arbitrary whatsoever, 2π is the circumference of a unit circle, therefore it's the angle of a full revolution.
The actual definition and use does not confuse me, the applications, however, and specifically the naturalness of it is what I don't truly understand. Mathematically, it's as natural as the natural logarithm, only difference being one was defined to be the natural and the other came from circles.
Why does the sum, for n from 0 to infinity, of (-1)^n/(2n+1)! * x^(2n+1), equal exactly 1 for x = 2kπ + π/2? I've not yet found why the, mind you, transcendental number π, which comes from circles, have this insane property that is deeply connected "pure" mathematics.
And there are many other reasons I think radians are such a natural unit for angles, but WHAT is the deeper mathematical connection?
@@DeJay7 Regarding the mystery verb, I think I can help you out:
"An angle measuring θ radians with its vertex at the center of a circle of radius r [encloses/intercepts] an arc of length rθ on the circle."
But I get the feeling that that's not what you meant, as there's a special verb used within this general area of discussion: subtend. However, according to the conventional definition, you would've gotten it backward: a circular arc subtends an angle, not the other way around.
If you want to remember this word, it helps to know where it comes from. It's related to the word "extend". It contains the sub- prefix, meaning "under", like in "subway" or "subtitle". So, "subtend" means "extend under". The circular arc extends under the angle. It's like how a bright patch of ground extends under a street light.
About the definition of the natural logarithm, there are actually a few different definitions you could use. For instance, since the derivative of ln(x) is 1/x, you can integrate 1/t dt from 1 to x to get ln(x). This is a fairly common definition. Alternatively, using the typical way of defining a logarithm as the inverse of an exponential function, it can be defined as the inverse of e^x, which can itself be defined as the unique exponential function equal to its own derivative.
To answer your first question, it's because what you gave is the Taylor series of sin(x), which is precisely equal to sin(x), and the points where the sine function attains a value of 1 are readily apparent. Well, how do we know that the Taylor series of sin(x) is equal to sin(x) itself? It is of course something we can prove-to use math words, we must show that sine is an analytic function-but the answer turns out to be pretty complicated, and I don't think I can do it justice here.
So far, though, I get the impression that I haven't told you anything you don't already know. Maybe this is just something you find unintuitive, and you're looking for a way to make it intuitive. To tell you the truth, such a way might not exist. There are some things in math that are simply too weird to fit within human comfort levels, and we just have to live with it. I just want to make sure that you can tell the difference between a genuine mathematical question and an unanswerable riddle.
@@isavenewspapers8890 Amazing reply, thank you.
@@DeJay7 No problem :)
SM64 uses binary minutes in that sense, 65536 possible values before it wraps around
WHAT IS GREAT is that you use Tao in redian
I forget the name, but there's another one that emerges from trigonometry.
0⁰ = 0
45⁰ = 1
90⁰ = ∞
@@billpg Sounds like grade, as in slope. That's what you get from applying the tangent function.
7:20 degree of popularity? Or dare you say, gradian of popularity
instructions unclear, i ate the pie and exploded 3.1 radians later.
milliturn -> mtr -> meter
Science.
Steradians - solid angle of a sphere.
I'm surprised there was no mention of grade. It is simple the tangent of the angle expressed as a percentage. Ie a 45° angle creates a slope with a grade of 100% because tan(45)=1
I only knew about the existence of degrees and radians.
last time I heard about gradians was on my protractor in secondary school (over 25 years ago). 😅
It is used in some professions where it makes some calculations easier but pretty much extinct elsewhere
A grad is just 1/4 of a centiturn
You forgot right angle bro, the unit proposed by euclid
You missed one. The “gradient” or “grade” which is rise/run often represented as a %
And darians, too but what can you expect.
@@thebestthingbeforeslicedbr8562 Darians?
@@isavenewspapers8890 Basically tau/revolutions, but with pi (so the unit circle has a diameter of 1, not a radius) and 1 revolution around the circle=pi darians, where pi=3.14...
@ Does anyone use this, and if so, for what?
@@isavenewspapers8890 Its used in the rail industry and in bicycling to talk about how steep a climb will be.
What about Akhnam & Zam? Or that weird percentage thing I keep seeing on traffic signs?
Based and taupilled
Statue of Liberty
2:22 sundials exist. Don’t they?
It has a complete answer, azmuth,6'=00
Tau gang rise up!
5:46 misaligned Thetas.
My stock knowledge is starving
whats the font used in the video, looks really nice
Looks like Comic Sans.
Roman calendar had 13 months of 30 days
In the end, that are all angles
what is nanoturn written as
Nobody answer this.
@@isavenewspapers8890bro is a hero
@@isavenewspapers8890 why?
@@DimaMuskindJust don't. You will regret
brooooo😭😭😭
popularity is not important. gradian is almost always used in high surveying activities.
Don't you just love Nanoturns?
@@TeaTime0300 Ah, the third time a comment like this has been posted.
Practically, nobody uses tau in radians. Everybody uses pi.
So pi radians = 180°
@@peterchan6082 Really, nobody? Hmm, must be some other reason τ was proposed in the first place, and why it's gained so much attention from calculators and programming languages. Maybe people wanted to be able to write Stirling's approximation slightly faster.
@@isavenewspapers8890 Objectively he is correct lol pi is used more
@@praiseboggy Yeah, but everybody? That's a bit of an exaggeration.
@@isavenewspapers8890 U r taking it too seriously although he is not that wrong. Tau could be mathematically more convenient but I can guarantee you not many hs students use tau
@@praiseboggy Look, I understand that people can use figurative language sometimes, and not everything should be taken literally. But when I hear someone say "nobody", if they don't actually mean nobody, I would at least expect them to be talking about an extremely small group of people. But given the significant rise of τ in a multitude of different places, I just don't think that's fair to say.
2:36 mt city ❤❤❤
So I can call a year a decidecade, centicentury or millimillennia
@@kanck7909 Pretty sure "millimillennium" is the singular and "millimillennia" the plural.
Pi seconds = 1 nanoCentury
The difference between ase and metric
How much of a turn is a Joan of Arc?
Wait, what? Do all English native speakers pronounce centi- the way it is done in this video at 0:48?
That is, the "i" in "centi" is pronounced more or less like the "e" in "centi". I thought it was a short "ee" sound, just as it the case with the final "i" in "milli".
What about deciturn?
I learn the mil as A‰ but the full circle has 6400 A‰.
I believe you're thinking of NATO mils.
@@isavenewspapers8890 It is actually Swiss Artillery‰.
@@Zurich_for_Beginners I see. I couldn't find anything about this online, but if there are 6400 of them in a turn, then this unit is equivalent to a NATO mil. Switzerland is a member of the Partnership for Peace, which is a NATO program, in case that information is relevant here.
25ths of a binary degree?!
The modern abbreviation for grade is gone? Where did it go? 😉
The ending🤣
Gregory Munck adjustment 12;months daylight savings time, leapyear to adjustment
what about the steradian?
That's a unit of solid angle. This video is about plane angles.
Blud is trying to make tau happen
@@alex-wl4sb It *has* happened. The major proposal that got the ball rolling was in 2010, and since then, it has spread among many different circles. It's in several online calculators and a whole bunch of programming languages.
[n3]=00) binary quote,=OO')×π=Azmuth
Okay,X,2x+5=8')
What would be the abbreviation for 1 billionth of a turn
Not again.
1000 ptr
@ Outstanding.
1 ptr
@@isavenewspapers8890 Only if you use the short scale! :)
7:22 So Estonia can into Nordics?
Nifty
How about steradians?
Steradian is not a unit of a planar angle like the ones on the video, it's a unit of solid angle.
Curveature of the Earth, longitude latitude, azmuth
Bro forgot quaternions
Is this ai?
dumb question
nope
No, lets not switch to τ. I have enough trouble with math as it is without having to completely shift my understanding of angles and trigonometry.
Skissue.
All you have to understand is how a turn works. If you know what a quarter-turn is, then you know what τ/4 means. Based on what I've seen, many people find that intuitive, so using τ makes sense.
[n3] =00
Up charge
Leapyear, daylight savings time, difference bye time
[n]
0=0
Nobody uses tau
Instead, 2π
@@dhwyll Plenty of people use τ. The set of people that you personally know does not comprise everyone.
in high school, tau isn’t common
@@fireboytheone True, but it's gaining traction.
@@isavenewspapers8890 Let's take a look at the literature, shall we? If we can find it being used in journal articles or textbooks, then we might say it's being used.
But if not, then the fact that Vi Hart likes it doesn't really mean anything.
Ironically, your response applies to you in the inverse: That you know someone who does use it does not comprise everyone.
@ I'm sorry, is one of us hallucinating right now? Certainly, you would seem to believe that your initial comment actually reads, "Nobody who writes journal articles or textbooks uses τ," but that added condition is not what my own eyes read. After all, if Vi Hart uses τ, who cares about the journal or textbook authors? Vi Hart may not be such a person, but they're definitely not nobody. I may not be such a person, but I'm definitely not nobody. Programmers may not be such people, but they're definitely not nobody. Yes, the only possible explanation is that one of us has perhaps suffered a concussion or some other ailment of the brain, thereby failing to read the same comment. No other explanation would suffice.
The problem would appear to extend even further, for you have seemingly presumed me to claim that EVERYONE uses τ. Unfortunately, I don't recall having claimed such a thing throughout the course of my natural life. How utterly bizarre. We shall both require our heads medically examined.
10 minutes to explain this seems excessive.
@@geoffstrickler Well, if you just wanted definitions and nothing else, then sure, you could do it much faster. But that's not what this video is about.
Early
This video is tau propaganda
Well, I'd not use a word with such negative connotations. I prefer to think of it as education.
Metric is flawed
Yes, because ten is a terrible number to use as a base, but metric was built off it anyways. The fact that units have consistent proportions between them is nice, but it would be even better if that proportion wasn't ten (or technically a thousand in SI).
most definitely, but less flawed than imperial for sure.
Nobody cares about Tau
Well, that seems a bit unfair.
Nobody cares about you and what you think :^)
"tau"... its called TAF... Not tau...
in.. modern greek. Which isn't relevant to the mathematical symbols, which follow an anglicization of ancient greek pronunciation.
no
The name of the letter is written as "tau" using the Latin alphabet. As for pronunciation, it's conventional throughout academic disciplines to pronounce it how it's written, even if it doesn't correspond with the modern-day Greek pronunciation. As long as people can understand each other and communicate effectively about what letter they're talking about, there's no issue.
and the town in Germany isn't called Cologne but Köln.
101 1001 10001 10000099001