Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.
That's actually exactly the question I asked my math teacher when I first learned about radians. I mean, I learnt degrees when I was very little, at an age when one tended not to question why, but I learned radians at an age old enough to question why. The answer I received was about making trigonometric identities cleaner: the derivative of sine becomes "just" cosine rather than a hypothetical turn-based sine (called usin by the article) having a derivative of a turn-based cosine multiplied by 2pi.
But this article seems to do a good job explaining that a lot of those 2pi factors appear when you deal with differentiation. So it seems useful to have both turn-based trigonometric functions and this new differentiation operator.
Due to the fact differential operators are linear and the nature of the accumulation of constants of integration it’s pretty easy to prove that the differential operator proposed is in fact equivalent to the standard derivative operator.
Source: used to tutor calculus and differential equations in college.
Generally speaking this is not a useful trick.
There are _plenty_ of amazing ways to leverage Euler’s identity but I fail to see how this is one of them.
I too fail to see a problem. There is a certain degree of trickiness in all walks of life. Sometimes there is a decent hack that simplifies things and sometimes there is a notion that is just as complex to deploy as the current state of the art and hence isn't worth pursuing.
Pi has a definition for a good reason. Sometimes a discipline has to put up with perceived oddities until the real, deep problem is surfaced, grappled with and kicked in the nuts until it gives up and allows a paper or two to emerge without ridicule. With luck it might really show some ankle and a Nobel heaves into view 8)
Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere.
Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!
This is not true. Angles very much have units and it's why you can express the same concept with different numbers. Pi equals 180 degrees equals 0.5 turns.
1 radian has different units than 1 steradian and if they didn't there wouldn't be a need for two different words to denote them.
The quantity is a ratio of two lengths, and the length measure does "drop out". But it's not just any ratio, it's a very particular ratio, and the unit defines the particularness of that ratio.
The reason it is confusing is because an angle measure is a kind of logarithm of a rotation, and logarithms (sort of) have a unit: the base.
The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ, which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivector) but is unitless.
Such a rotation z can be thought of as the ratio of two vectors of the same magnitude: z = u / v satisfies zv = u, i.e. is the object by which you can multiply v on the left to obtain u. Whatever original units your vectors u and v had gets divided away.
This is similar to the way the "ten" in "scale by ten" is unitless, but if you take the logarithm you get "scale by 10 decibels" or "go up by 3 octaves and 3.9 semitones", which have the base of the logarithm as a kind of unit.
I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose).
But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.
This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space
The logarithmic base for an angle is something like "degrees", "radians" or "turns".
This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system.
Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.
> 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space
Just as 1 meter straight ahead divided by 2 meters straight ahead is the unitless scalar number 0.5, we can likewise treat angles (i.e. rotations) as ratios: 1 meter straight ahead divided by 1 meter to the right has the unitless bivector-valued ratio i, oriented like the ground you are standing on. You can multiply this bivector by some other coplanar vector to rotate it a quarter turn. For example, you can multiply it by the vector «3 inches due North» to get the new vector «3 inches due West»; notice how the units do not change because our bivector i is unitless.
I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain
Edit: another example difference. I can't measure an octave or dB. I can measure a degree
Edit2: we've reached reply limit but I concede you can measure a decibel. Point about dB degrees still stands though
You absolutely can measure an octave or decibel. It's just a relative quantity, in just the same way any quantification of orientation is relative.
(For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.)
You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good company. Most working engineers, scientists, and mathematicians don't have or need a particularly clear philosophical understanding of angles.)
I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you
A) multiply two degrees together
B) multiply two dBm values together
The output "units" in A change but do not in B. dB and angles are very different.
Edit: the units in B do change, but the dB part doesn't. Tired.
You can compute solid angle (a.k.a. spherical excess, normalized spherical surface area) by taking a surface integral, but the units are not "square degrees" or "square radians", but instead an entirely new type, usually just measured in radians ("steradians"). Some people have defined https://en.wikipedia.org/wiki/Square_degree but that is a stupid unit.
While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).
The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.
Yeah, units that algebraically reduce to 1 are always very interesting to me. Consider a chart showing how many CPU seconds you're consuming per second. The unit is seconds/second, which is equal to 1, but it is still a distinct concept from radians.
Jeez, I've really kicked off quite the heated discussion in the replies... I don't want to get bogged down in lengthy arguments, but I feel I should explain my reasoning in more detail.
Essentially, I think that whatever angles are, they are not like other dimensionful physical quantities. I have two arguments.
The first: Someone mentioned symmetries in a reply. I wanted to mention them too but didn't have time to structure my thoughts into a coherent argument. But the gist of it is that dimensionality is just a kind of scale invariance, and the scale invariance of angles is fundamentally different from that of linear quantities due to their periodicity — to apply a unit transformation, you have to scale the quantity _and the period_.
The second: Consider units from a "type theory" perspective instead. If you are considering exclusively linear trigonometry (no arcs), it's trivial to assign a dimensional type structure to expressions (e.g. cos takes angle type and maps it to dimensionless type). But as soon as you allow arc lengths, it becomes cumbersome to type common expressions.
I think these distinctions form the crux of the disagreement. Ultimately, it depends on your intuitive notion of what "dimensionality" actually means, and how it ought generalise to other kinds of quantities.
Here is an example to highlight my point. Let there be a circle C of centre O and radius r. Let A be a point on the circle. Let there be a point M outside the circle such that (AM) is tangent to C. Let B be the intersection of C and [OM]. Let s be the arc length along C from A to B. Then we want to write AM = r tan(s/r).
How does one get s/r to resolve to an angular dimension? Ought we instead ascribe s dimensions of length-angle? Imagine, then, that the circle is in fact a pulley, and we wish to measure a change x in length of rope as the pulley rotates through the angle of the arc from A to B. We would want to write x = s. But this is now dimensionally inconsistent.
It's certainly possible to make all these expressions correctly typed by introducting appropriate conversion constants. But this seems to me to be cumbersome. Since in physics, arc and linear lengths can convert freely into one another, it seems more economical to just let angles be dimensionless.
The solution to your quandary is to realize the division operator is massively overloaded in your expression. What you actually want is to specify s and r as true line segments and define a new operator which takes two segments and outputs the angle between them. This operator happens to reduce to division of magnitudes in certain circumstances.
Edit: in other words, you've encoded tons of information in the problem statement about the relationship between r and s and you aren't properly encoding that in your type system allowing s/r to output an angle
Angles aren't dimensionless any more than lengths are dimensionless (feet per second makes just as much sense as rpm). It's just that angles have symmetries that lengths don't, which is where 2 pi comes in. Do you want units where your symmetries are expressed in multiples of 1, 2, or 2 pi (for turns, half-turns, and radians, respectively)?
Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...
Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.
The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.
In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:
sin(x) is the sinus function with the angle implicitly divided by 1 radian
sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle
sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn
sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree
It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.
> sin(x) is the sinus function with the angle implicitly divided by 1 radian
This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.
Units are not the same as dimensions, something can have a dimension of 1 (which is what we usually mean by "dimensionless") and still have different units, just as something can have a dimension of length but still be measured in meters or feet.
As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.
Actually the dimensions are by definition the same as the fundamental units, i.e. the units that are chosen freely, independently of all other units.
A dimensional formula of a quantity just writes its unit as a function of the fundamental units.
In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.
A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.
There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.
This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.
Sure, but that's orthogonal to the "angles don't really have units" assertion and the "it does make sense to add an angle to its cube" assertion, which are the ones I'm responding to.
As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.
you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed
The argument of exp does have to be dimensionless, exactly because adding seconds to seconds squared doesn't work. If t has units of time, there has to be another factor with units of inverse time, for example continuous compound interest is exp(rate*time).
If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim
> Angles aren't dimensionless any more than lengths are dimensionless
They are dimensionless, but they still have units. The concepts are orthogonal.
As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.
> They are dimensionless, but they still have units. The concepts are orthogonal.
The concepts are not orthogonal, they are incompatible.
A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.
When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.
There is no way to choose a unit for a dimensionless quantity in the usual sense.
At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.
Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.
For example you claim that angles are not dimensionless, but that dimensionless quantities are formed by the ratio of two quantities with the same unit. Since the angle subtended by an arc in a circle is the ratio of the arc length and radius, it would seem that these two claims contradict each other.
I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.
No, even if you have reproduced a definition of the plane angle that is encountered in many textbooks "the angle subtended by an arc in a circle is the ratio of the arc length and radius", this definition is very incorrect.
This very wrong definition forced upon many students is the root of all misconceptions about plane angles.
The reason why this definition is wrong is because some words are missing from it and after they are added it becomes obvious that its meaning is different from what many teachers claim.
First there is no relationship whatsoever between the magnitude of a radius and the magnitude of an angle. What that definition intended to say was:
"the angle subtended by an arc in a circle is the ratio of the arc length and of the length of an arc whose length is equal to the radius".
By definition, a radian is defined as the angle subtended by an arc whose length is equal to the radius.
To explain how plane angles are really defined would take more space, but the only thing that matters is that the characteristic property of plane angles is that the ratio between two plane angles subtended by two arcs of a circle is equal to the ratio of the lengths of the two arcs.
Introducing this characteristic property of the angles in the so-called definition from above reduces it into the sentence "the angle is measured in radians". This is either a trivially true sentence when the angle is indeed measured in radians, or it is a trivially false sentence when the angle is measured e.g. in degrees. It certainly is not a definition.
If that had been the definition of plane angle, that would have meant that the plane angle was discovered only in the second half of the 19th century, together with the radian, while in reality plane angles have been used and measured with various units for millennia.
To measure plane angles, it is necessary to first choose an arbitrary angle as the unit angle, for instance an angle of one degree.
Then you can measure any other angle by measuring both the length of the corresponding arc and the length of the arc corresponding to the chosen unit angle. Then the two lengths are divided, giving the numeric value of the measure of the angle.
I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.
A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.
The claim that plane angle, solid angle and logarithms are dimensionless quantities is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration without ever stopping to think whether this claim can be proved.
I will discuss only the plane angle, because it is the most important, but the situation is the same for solid angle and logarithms.
The justification commonly given is that the plane angle is dimensionless because it is the ratio of two lengths, the length of the corresponding arc and the length of the radius. This justification is stupid, because that is not the definition of the plane angle, but it already includes the choice of a particular unit.
As formulated. this justification only states the trivial truth that the numeric value of any physical quantity is the ratio between that quantity and its unit. By the same wrong justification, length is dimensionless, because it is the ratio between the measured length and the length of a ruler that is one meter long.
Correct is to say that the plane angle is a physical quantity that has the property that the ratio between two plane angles is equal to the ratio between the lengths of the corresponding arcs.
This is a property of the same nature like the property of voltage that the ratio of two voltages across a linear resistor is equal to the ratio of the electric currents passing through the resistor. This kind of properties are frequently used in the measurement of physical quantities, because few of them are measured directly but in most cases ratios of the quantities of interest are converted in ratios of quantities that are easier to measure.
This property of the plane angle allows the measurement of plane angles, but only after an arbitrary unit is chosen for the plane angle. Because the choice of the unit is completely free, i.e. completely independent of the units chosen for the other physical quantities, the unit of plane angle is by definition a fundamental unit, not a derived unit.
The freedom of choice for the unit of plane angle is amply demonstrated by the large number of units that have been used or are still used for plane angle, e.g. right angle (the unit used by Euclid), sexagesimal degree, centesimal degree, cycle a.k.a. turn, radian.
The fundamental units of plane angle, solid angle and logarithms must never be omitted from the dimensional formulae of the quantities, otherwise serious mistakes are frequent & such mistakes have delayed the progress of physics with many years (e.g. due to confusions between angular momentum & action; the Planck constant is an angular momentum, not an action, as frequently but wrongly claimed). This is a problem especially for the unit of plane angle, which enters in the correct dimensional formulae of a great number of quantities, including some where this is not at all obvious (e.g. magnetic flux).
> is a horrendous mistake and many generations of physicists have been brainwashed by being taught this aberration
The synathroesmic writing style and inapt analogies obfuscates the claims along with distracting and detering others from refuting them, but does not make them true.
Angles are defined in the abstract realm of mathematics not physics, which use them to describe physical phenomenon. In Cartesian space, the basis dimensions are defined using fixed perpendicular oriented lines, called coordinate lines on coordinate axes. Angles are not a basis dimension and as defined in Cartesian space are dimensionless. Angles are a dependent measure. Arc length is measured based on the units of the system. x radians or x degrees cannot be physically measured in and of themselves. In coordinate systems using an angle as a basis like spherical coordinates, angles exist as a dimension and the unit for them is free to chosen. Once the units are chosen, then the angles can be physically measured.
This is more or less "the point" for those of us who argue for tau instead of pi.
I should note that using this "trick" of prescaling rotations by 2pi so that they are in the 0..1 range is de rigueur in computer graphics programming.
You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.
It is possible to completely remove the e^x function from mathematics without losing anything.
It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x.
Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns).
The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.
In my opinion the simplifications that are introduced everywhere else are more important than this disadvantage.
The real and complex function e^x was preferable in the 19th century, when numeric computations were avoided as too difficult and simple problems were solved by symbolic computation done with pen and paper.
Now, when anything difficult is done with a computer, both the function e^x and associated units like the radian and the neper are obsolete.
When programming computations on a computer, using 2^x and 1^x results in simpler and more accurate computations.
Moreover, when doing real physical measurements it is possible to obtain highly accurate values when the units are cycle and octave, but not when they are radian and neper.
The function 1^x is constant everywhere. You can write (-1)^x to get the kind of effect you are looking for, but doing this with 1^x is madness.
>The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.
There is more to it than that. The exponential function is not really e^x, it is lim_{N->∞} (1 + x/N)^N. The advantage of this definition is that, since it is valid everywhere on the complex plane, it allows the rigorous definition of real powers without recourse to inverse functions. We still teach students to prove things.
But also, isn't this a little contradictory? The goal of the original blog post was to simplify various differential equations. If you aren't simplifying symbolic differentiation, then what are you simplifying?
All mathematical formulas can be inter-converted based on the identity:
e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi)
Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae.
In most applications, more equations are simplified than those which become more complicated.
A very important advantage of 2^x and 1^x versus e^x is that for the former the reductions of the argument to the principal range where the function is approximated by a polynomial can be done with perfect accuracy and very quickly, unlike for the latter. Moreover, for the former it is easy to verify the accuracy of any approximation, because for any argument that is represented as a binary number the functions 2^x and 1^x can be computed with a finite number of sqrt invocations (based on the formulae for half angle) and sqrt can be computed with any number of desired digits. Computing e^x with an arbitrary precision is trickier, because it requires criteria for truncation of an infinite series.
It should be noted that 1^1.0 = 1, 1^0.5 = -1, 1^0.25 = i, 1^0.75 = -i
You have misunderstood my point. I have not said anything about polynomials.
I have said that the so called "natural" exponential, normally written as e^x or exp(x) of either real or complex argument and its inverse, the hyperbolic a.k.a. natural logarithm, and any other functions derived from it are neither needed nor useful when computations are done by computers, as opposed to computations done with pen and paper.
In all traditional formulae where the "natural" exponential function or functions derived from it occur, all occurrences can be replaced using a pair of functions of real argument, the function 2^x with real value and the function 1^x with complex value, either directly or with functions derived from this pair, e.g. the binary logarithm.
In computer programs this substitution results in both higher accuracy and higher speed and it has as a side effect that the units radian and neper are never needed.
It should be noted that even in the 19th century, when the "natural" exponential and logarithm and the trigonometric functions with argument in radians were useful for symbolic computations done by hand, they were never used for practical numeric computations.
All practical numeric computations were done using the function 10^x and the trigonometric functions with argument in degrees and their inverses, by using mathematical tables where the values of these functions were tabulated (or equivalently, by using slide rules).
The use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians for practical computations has become widespread only after the development of the electronic computers, after programming languages like Fortran have included them as standard functions.
I consider that this has been a mistake, similar to the use of decimal numbers in some computers. Both the use of decimal numbers and the use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians are sub-optimal in all their possible applications.
The simpler derivation formula was important when such symbolic computation was done by hand.
Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important.
The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation.
In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.
If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.
For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm).
The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler.
Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.
