[WastingMyTime89]

The writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1.

Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explain the lack of familiarity with radian I see here. Am I wrong?

[kortilla]

> Am I wrong?

Yes. Trigonometry is extensively taught in the US. People forget this stuff if they don’t use it.

Ask some 30 year old chef in whatever country you fantasize teaches properly to compare and contrast turns vs radians and you’ll get similar responses.

[js2]

I'm a 50 yo programmer. I have a CS degree. I don't even remember my college calculus much less my high school trig. I just haven't had cause to use it in my career, not as a sysadmin, not as a programmer. My son is taking calc 3 and I knew I happened to have my calc 3 notes from the mid-90s, so I pulled them out of the filing cabinet and my very carefully taken notes, my proofs, my hand drawn graphs, it was all gibberish to me. That was stuff I knew like the back of my hand when I graduated but it quickly faded away.

[jerf]

By far the most annoying myth I face when trying to discuss the pros and cons of various education techniques is the pervasive idea that everybody is a magical knowledge sponge and will go to their grave still remembering how to integrate by parts and every detail about some particular battle they covered in seventh grade, and therefore, if we slightly tweak a curriculum plan to drop something that was included on theirs we'll be stealing that knowledge from all the 70 year olds who will eventually have been on that plan.

Where this idea comes from I have no idea. Personally looking around in school itself it was plainly obvious this was all going in one ear and out the other for the majority of students even at the time. The better students retained it long enough to spew it out on the test but that was already above average performance. That doesn't mean there isn't still a certain amount of value in that in terms of what that knowledge may do to their brain during the brief period of time it is lodged in there. (I think there's a lot of value in just learning the "shape" of all this stuff, and perhaps having some index of what might be valuable to know.) But the idea that we can spend 15 minutes and a one-page homework assignment on something and expect that to last 60+ years is just nonsensical.

I mean, honestly, anyone over the age of 22 or so ought to be able to notice a distinctly sub-100% retention rate simply by looking inside themselves.

Yes, to a first approximation everyone with a normal education in the US has been present while some sort of trig was discussed. Not all of them, but still quite a lot of them, were present for the Taylor expansion discussion. The vast bulk of them have had it decay by 25, and there simply isn't anything to be done about that if you're talking about humans and not some homo educationous who mythically retain all knowledge they were exposed to even for 30 seconds just as the mythical as homo economicus perfectly rationally conducts all their economic business at all times. Perhaps they're actually the same species.

[simiones]

I remember being amused by this same observation when my own country decided to reduce mandatory education from k+12 to k+10 (cutting two years of high-school). They immediately began re-arranging the curriculum in high-school, for example to move organic chemistry from 11th grade to 10th grade, on the basis that it's important for students who only finish the mandatory 10th grade to know some organic chemistry as well, instead of the old curriculum which would have only taught them inorganic chemistry after 10th grade (this has the bonus of making the chemistry curriculum inorganic I -> organic I -> inorganic II -> organic II, for maximum confusion).

To me, even though I was barely out of high-school at the time, this was obviously absurd - expecting especially someone who wants to drop out of high-school early to retain any notion of organic chemistry taught in a school year, that they couldn't learn on the job if it was really required, seems so obviously nonsense that I couldn't help but laugh. Especially since the same thing was done to basically every other subject as well, with the same intentions.

One note: in my country, the curriculum is completely centralized; there is some small amount of choice, but it amounts to, at most, 1-2 classes per semester; everything else is fixed.

[PKop]

Think it's part of the equality/blank-slate myth that everyone is the same and has the same potential and natural abilities.

[nightpool]

Actually, it's almost entirely the opposite—the idea that students are a "sponge" that can soak up knowledge perfectly is then taken directly to mean that some students are better at soaking up / retaining knowledge then others, and that the "smart" kids who do the best on the tests are the ones who are going to retain the knowledge the best. And then the ones that were the best knowledge-sponges will eventually go on to become the next generation of teachers, since they know the most information. Whereas for most kids it's completely the opposite—they memorize the information in their short-term memory without understanding the fundamentals, they do great on the tests, and then they forget all of it immediately. But they stand out from their peers as better students, because they're able to play the "game" of school better and optimize for being a knowledge-sponge that will absorb the most information as possible and forget it as quickly as possible.

[ngc248]

>>> simply isn't anything to be done about that if you're talking about humans and not some homo educationous who mythically retain all knowledge they were exposed to even for 30 seconds just as the mythical as homo economicus perfectly rationally conducts all their economic business at all times. Perhaps they're actually the same species.

Yeah they belong to the genus homo mythicus

[airstrike]

I just wanted to say I deeply appreciate the eloquence of this comment. Thank you

[fisherjeff]

On a related note, it bothers me that there’s so much urgency to teach younger kids more and more advanced math. I use more and higher math on a day-to-day basis than practically anyone I know, but it’s very rarely even calculus, and even then it’s typically just discrete integrals or derivatives.

There’s just an absolute ton of math being taught that’s going completely to waste, and it’s at the expense of the humanities.

[Swizec]

My biggest “Screw everything” moment about math was the first lecture of my numerical methods class in college when the professor said: “All that calculus you’ve been learning your whole lives? It’s useless. Carefully curated set of a few dozen problems that are doable by hand. Here’s how it’s really done for anything remotely practical”

And then we learned a bunch of algorithms that spit out approximate answers to almost anything. And a bunch of ways to verify that the algorithm doesn’t have a bug and spat out an approximately correct answer. It was amazing.

But the most long-term useful math class (beyond arithmetic and percentages) has been the semester on probabilities and the semester on stats. I don’t remember the formulae anymore, but it gave me a great “feel” for thinking about the real world. We should be teaching that earlier.

[JackFr]

When I took 400 level Real Analysis: “All that calculus you’ve been learning your whole life? It’s a lie. Those epsilon delta proofs? They were fake - none of you were smart enough to challenge us on ‘limits’. And now we’re gonna do it all again only this time it’s really gonna be rigorous.”

[tsimionescu]

Is there any somewhat simple explanation of what are the limitations of the epsilon-delta definition of limits that make it non-rigorous? I've been trying to find some information about your comment, but have so far come up empty.

[BeFlatXIII]

100% agreement on teaching stats as the "pinnacle" of high school mathematics. Those are what directly rule the lives of the average non-engineer.

[MichaelCollins]

I think much of math 'education' is constructed as a filter to identify a small handful of math prodigies. The general population suffering anxiety and youth lost in the filter is seen as an acceptable sacrifice for the greater good of finding the math prodigies so those can be given a real math education.

[fisherjeff]

Yes, this is a very good point. In my experience from, uh, several decades ago, it also felt like a lot of math educators watched (and showed in class...) Stand and Deliver way too many times and the only message they took away was "we should teach everyone calculus!"

[BeFlatXIII]

I doubt the students would actually learn humanities in the extra time allotted if it's not used for math. I remember a distinct refusal to internalize, especially in my male peers, during "English" classes.

[MichaelCollins]

Forget humanities. The hours a week after school that highschool students spend on calculus homework would probably be better spent socializing with their friends. They'll never be young again, wasting the time of a teenager with unproductive busywork is a horrible thing to do.

[cjbgkagh]

I’m the opposite, Im 15 years into my career of applied research which for me is like an extension of university. I tend to lean on Mathematica to do my calculus though. I think high school curriculum was optimized to expose a lot of people to things they won’t need on the off chance that a few will end up as researchers of some sort. It would be more efficient to identify such people earlier and split them off. I think historically that was the idea but there has been an egalitarian push to broaden the pool.

[schindlabua]

I think the point of high school is to make kids' brains do work, and what you are learning is secondary.

People love to hate on their school curriculum and all the useless knowledge they had to acquire but I'm positive it makes you a smarter person overall, and the body of high school knowledge makes learning more specialized knowledge easier (even if that's baking bread or whatever)

(People also love to talk about how little they remember from school, yes the brain is a muscle and you stopped working out, congratulations.)

[mathieuh]

I'm 27, educated in the UK, all I remember about trigonometry is SOHCAHTOA.

[_0nsg]

USA here, same acro. I still start off solving by writing it off and drawing slashes through O/H A/H O/A for reference.

Came to use trig functions quite frequently while playing video games, and that was a big surprise to me. Not to assume you've played it, but I've recently discovered that Stormworks is a programmer's game - you can write microcontroller code in LUA for your vehicle designs. And, wow, does it ever use my trig knowledge everywhere.

Realized the transponder beeps can be triangulated, tick being 1/60th of a sec and that's a distance estimate resolution of up to 5-10 km. And that's when cos and sin came back to be useful because you can do intersection of circles and figure out where to do a sea rescue more precisely. So video games, trig. Who would've thought?

[mcv]

The Dutch version was SOSCASTOA, with a picture of a ship called the Castoa sending out an SOS because it was sinking. That picture really helped.

And I even remember what it means:

SOS: sine = opposing side divided by diagonal (schuine) side

CAS: cosine = adjacent divided by diagonal

TOA: tan = opposing divided by adjacent.

I don't think I've ever used it for anything practical, but I can still reproduce it after all this time (I'm 47 now).

[js2]

> I don't think I've ever used it for anything practical.

Just the other day I wanted to compute viewing angle and did it by hand even though there are plenty of calculators like this out there:

http://www.hometheaterengineering.com/viewingdistancecalcula...

I’d say I use it for something practical/random like that a few times a year?

Another example was placing some ceiling speakers whose tweeters had a 15° angle so that they were pointed directly at a seating position below. How far did I need to place them in front of the seating position from directly overhead.

I would guess any sort of construction you’re using it fairly often.

[dylan604]

The one that I still use is the 3-4-5 rule to ensure a right angle. Still use that one to chalk off sporting fields of play.

[mcv]

I use that one all the time, but that's Pythagoras, not sin and cos.

[dllthomas]

It was by no means uncommon when I was taught in the US but I somehow missed it, instead just internalizing the various relationships directly, and was briefly confused when classmates started talking about SOHCAHTOA working together in college math courses.

What I remember from trig is to draw a unit circle. Most of the rest falls out of that.

[js2]

I’m handy outside of work and use sohcahtoa often enough to remember it. Triangles are everywhere and sometimes you need to compute angles and lengths of sides.

Statistics is also useful and applicable to everyday life, but I didn’t learn that till college as best I can recall.

I don’t regret having spent time learning calc, or physics or chemistry or biology for that matter. If you asked me to come up with a curriculum I’d have a really hard time prioritizing. Maybe the one thing I’d like to see kids learn better is how to be self-directed learners. I’m still fairly surprised at the number of colleagues I have who seem unable to problem solve and figure something the fuck out. Even knowing when and how to ask for help.

[mech422]

I'm a 50+ year old American, of british decent... I never managed to remember the 'american' mnemomic, but my dad taught me one the used to use in England around WWII: Percy has a bald head, poor boy

Perpendicular/hypotanuse = Sin

Base / Hypotanuse: CoSin

Perpendicular / Base: Tan

edit: try to fix the HN god awful formatting

[duncan-donuts]

Ha I also remember the mnemonic but I don’t have a clue how to use it.

[krallja]

it's to remember the ratios for trigonometric functions on a right triangle:

Sine: Opposite over Hypotenuse

Cosine: Adjacent over Hypotenuse

Tangent: Opposite over Adjacent

[dev_tty01]

Otto had a heap of apples.

[dekhn]

There's lots of stuff I knew well and then forgot, but can re-learn quickly. For example, nearly all of calculus (useful when dealing with machine learning). Other bits I've retained and never forgotten, such as everything I've seen involving matrices. There are even things which I had conveniently completely "forgotten" but later emerged as suppressed latent memories- for example, set theory. I was so unhappy with the lead-up to Russell's paradox that I actively suppressed thinking about sets, groups, rings, and fields for several decades.

There are even other bits that I was shown, never incorporated into my brain at all, but later recognized as truly important (Taylor series expansions, the central limit theorem, the prime number theorem, etc).

[bee_rider]

Informally, big-O and limits have a similar smell to them, calc might have helped get some wheels turning in your head for that.

I do recall taking a "probability in CS" as an electrical engineering student -- it was pretty mind-blowing to me the extent to which the CS students did not like to talk about any continuous math. It makes sense, though, these are different specialties after all.

[HideousKojima]

Honestly the typical developer needs a solid understanding of algebra, but not much beyond that. Though any time I get into game dev stuff I start ripping my yair out over quaternions

[sbf501]

Thankfully you didn't publish a blog post claiming you discovered a new way to represent coordinates, and then assert that programmers should switch.

[LeifCarrotson]

I'd argue it's not so much taught in the US as it is tested. The common core standards say [0] that students should:

> Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

> Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

and so on. However, in practice, this means that students need to be able to answer "C" when presented with the question:

> One radian is:

> A) Another word for degree.

> B) Half the diameter.

> C) The angle subtended on a unit circle by an arc of length 1.

> D) Equal to the square root of 2.

A surprising number of students can get through without ever really comprehending what a radian is. They might just choose the longest answer (which works way too often), identify trick answers and obviously wrong answers, and eventually guess the teacher's password from a lineup by association of the word salad of "radians" and "subtended."

They might not even have a clue what the word "subtended" means, but they know it's got something to do with "radian" and that's enough. It is more important for the school that the students answer (C) than that they understand what a radian is.

[0] https://web.archive.org/web/20220112000314/http://www.corest...

[kortilla]

No, you’re missing the point. I went to school in the 80s and learned this stuff without multiple choice and fully understood it all of the way through undergrad where I took up through calc 3 and differential equations. Then I spent nearly 30 years as a SWE not using it and forgot nearly all of the details within maybe 15 years.

This happens with very basic things like human languages. Bilingual people can forget an entire secondary language if they don’t use it for a decade+.

[mlyle]

It's taught extensively in the US, but what's never done is showing how terrible many of those identities or integrals are in degrees.

Derivative of sin(x) is cos(x). Many people probably think this works for degrees, but it's actually some abomination like pi cos(pi x/180)/180.

Of course, turns are very reasonable units sometimes for sure.

[tzs]

> Derivative of sin(x) is cos(x). Many people probably think this works for degrees, but it's actually some abomination like pi cos(pi x/180)/180.

That's what it would be if you are using sin in degrees and cos in radians. But if you are using degrees for both then the derivative of sin(x) is pi/180 cos(x).

[mlyle]

Yup, I knew I'd mess it up. :D

[zugi]

Like most here, I've learned and forgotton lots of trig and calculus.

However, I still remember that "eureka!" moment of realizing that radians were special, that the small angle approximation of sin(x) = x, and many related math rules, work only when x is expressed in radians. I guess that's a credit to my math teacher, who basically led the class in deriving mathematical formulas rather than just presenting them to us.

I think the article is still valid and interesting, as "turns" in some use cases might improve performance and accuracy. But radians aren't at all "arbitrary" - if we ever encounter technologically advanced aliens, they certainly won't use degrees, but they will understand radians.

[jeltz]

Sure, but are people here chefs? I would expect most programmers in mu country to remember this, but most chefs to have forgot it.

[bathtub365]

I would only really expect programmers who work with angles regularly (those working in 3D) to remember it. Even then, you’re likely just smashing quaternions together anyway.

[kortilla]

Effectively yes. A tiny subset of programmers need trig.

[loup-vaillant]

> The writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1.

First, I would be cautious about suspecting someone of Casey Muratori's calibre didn't consider something just because he didn't directly addressed it.

Second, the choice of unit is kind of arbitrary, even if the unit itself is not. Radiants are nice because the length of a 1 radiant arc is the same as the length of the radius. But turns are also nice because angles expressed in turns are congruent modulo 1 instead of modulo 2π.

Third, he talks in the context of video games. Such games use code, that have to be read by humans and executed by the CPU. And that's the main point of his article: in this context, expressing stuff in terms of (half) turns reduces the amount of code you have to write & read, reduces the number of multiplications & divisions the CPU has to make, and makes some common operations exact where they were previously approximated.

Do we even care at this point whether the definition of radians is arbitrary or not? I love the elegance of radiants, but for game engine code I'm willing to accept they're just the wrong unit for the job.

[dahart]

> radian is not an arbitrary unit but a dimensionless one

This is by convention, but has been and is still being debated because calling it dimensionless causes some problems. https://en.wikipedia.org/wiki/Radian#Dimensional_analysis

Furthermore, the whole reason to treat radians as dimensionless, the problem, is with angles, not with radians specifically. Degrees are also considered dimensionless. So, a turn could be treated as dimensionless too, with a conversion constant to radians & degrees, just like between degrees and radians.

Of course, the declared dimensionlessness of angles like radians isn’t something generally discussed in pre-college trig courses, that’s a subtle subject that matters more in physics. In my high school trig, we all understood radians to be a unit of angle and never pondered whether angles had dimension.

Also subtle point, but dimensionless doesn’t mean unitless. It’s another separate convention to drop the units when working with radians.

[insightcheck]

> "I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explain the lack of familiarity with radian I see here. Am I wrong?"

It varies by school, but overall I think this prediction is incorrect. Trigonometry was an important subject in high school — for all of the math, physics, and possibly chemistry courses — and then if you take calculus in university, it's very, very important to learn trigonometry well (or you'll really struggle as a student).

So, even on the off-chance that trigonometry is not taught in high school (which I predict is rare), a first-year student taking calculus in university must learn it on their own time. Good calculus textbooks (e.g. Thomas Calculus) even account for this, having fairly comprehensive textbook sections on what you need to know about trigonometry to succeed in the calculus course.

Most students who therefore took math to pre-calculus or calculus (or physics and possibly chemistry), should therefore have a good exposure to the definition of the radian.

[genewitch]

I was a bad student through 8th grade, but managed to get selected for a STEM magnet school. I was supposed to enter 9th grade with Geometry, then algebra II, trig, Calc for the 4 years. But they discovered i'd never passed algebra prior, they put me in algebra, which means i would have finished in trig. Due to a crazy 3.5 years, i never got a high school math education. Calculus makes my eyes glaze over, and all i know about triangles is sohcahtoa.

Every couple of years i try to get some higher math education, but nothing makes sense. It's one of the reasons i [think] i suck at programming - i should note that another reason is i first learned BASIC, then qbasic, then fortran, and then C never made sense to me. At least i can putter around with python and R.

however i can do "basic" math things that generally everyone else has to dig out a calculator app for in my head, percentages, fractions, moving decimals, "making change". Since i suck at higher math, i'm only able to help my kids with basic math, and i try to ensure that they know it fairly well.

[weinzierl]

"[..] and then C never made sense to me"

I had my fair share of higher math, but C never made really sense to me either. It's not us, it's C that's to blame.

[PaulHoule]

C puts the C in Cthulhu.

[aaaaaaaaaaab]

So which programming language makes sense to you? Dare I say… Rust?

[weinzierl]

Yes, Rust does indeed and a long time before that it was Pascal. I really love Pascal's syntax, it makes a lot of sense when you approach it with a math background.

- '=' is for equality only

- assignment is ':=' which is the next best symbol you can find in math for that purpose

- numeric data types are 'integer' and 'real', no single/double nonsense

- 'functions' are for returning values, 'procedures' for side effects

- Function and procedure definitions can be nested. I can't tell you what shock it was for me to discover that's not a thing in C.

- There is a native 'set' type

- It has product types (records) and sum types (variants).

- Range Types! Love'em! You need a number between 0 and 360? You can easily express that in Pascal's type system.

- Array indexing is your choice. Start at 0? Start at 1? Start at 100? It's up to you.

- To switch between call-by-value and call-by -reference all you have to do is change your function/procedure signature. No changes at the call sites or inside the function/procedure body. Another bummer for me when I learned C.

Pascal wasn't perfect but I really wish modern languages had syntax based on Wirth's languages instead of being based on BCPL, B and C.

[KerrAvon]

It sounds like you just don’t like C — a perfectly reasonable position — not that you don’t understand it.

[insightcheck]

It's time-consuming, but there are great resources to learn high school math to a very high level (likely much more effectively in many cases, than actually taking a high school math course, due to thoughtful exercises and more control over the pace of learning).

I learned a lot from the Art of Problem Solving book series because they're highly focused on the reader solving problems to learn, versus giving explanations. Even if you don't finish all of it, you can strengthen any problem areas.

For a less-comprehensive but still great introduction to precalculus (with a great section on trigonometry in particular from memory), Simmons' Precalculus in a Nutshell has a great introduction to this. Then you can read a book like Thomas Calculus, which has a great introduction to trigonometry in the first review chapter.

I would even say that you would be better off working through the books above than if you had the high school classes; the best math students probably took the same approach too (working through books instead of focusing just on the class material). The main obstacle is time, because it's hard to find time when you have work and children to take care of.

[bombcar]

Calculus made a lot of it “click” for me but I somehow got through high school and college without ever doing or understanding trigonometry.

[weberer]

>I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA

Basic trig is taught in middle school, but exclusively using degrees. Advanced trig is optional in high school if you take the "hard math" track.

[rockostrich]

> Advanced trig is optional in high school if you take the "hard math" track.

This depends on the state. NYS absolutely requires everyone to learn "advanced trig" in high school.

[mod]

I think I would have put the quotes around "learn."

I don't think every high school student can master advanced trig.

[retrocryptid]

Wow. You people went to crap schools. We got the derivation of modern trig functions w/ maclauren/Taylor series in 9th grade (though yeah... that was the "hard core math track".) And a year of proofs and derivations in 11th grade. Quaternions and their application in physics was 12th grade.

[phkahler]

>> The writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1.

It's been a while, but I used to have an argument that rad should be a unit. This even plays well in physics where it allows torque to not have the same units as a joule.

[gugagore]

I don't see how radians come into the discussion of torque and energy, both of which are N*m in SI.

That discussion has to do with the failure of SI to notate the directions of vectors. When it's torque, the N and the m are at a right angle. When it's work, they are both in the same direction.

[phkahler]

>> That discussion has to do with the failure of SI to notate the directions of vectors.

Well if radian is a unit then torque become Nm/r and is no longer Nm like energy. Then when multiplied by an angle in radians you get energy. It was *something like that*.

[acidburnNSA]

rad is a unit already. https://en.wikipedia.org/wiki/Rad_(unit)

[bluGill]

Trig is generally called per-calculus in US high schools. It is not a required course, but it is one of the courses everyone on the college track is expected to take.

Though most people haven't used any of that since college and so don't know it very well anymore. I smelled BS when I read the blog, but couldn't put my finger on why - the comment you replied to explained what I knew was the case but couldn't remember.

[happimess]

    s/per-calculus/pre-calculus/g

[kortex]

The percalculus ion is just calculus in its highest oxidation state.

[hansvm]

> I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA

Education quality and quantity vary greatly across the country. Many schools don't require trig at all or lump it in with other classes. I memorized SOH CAH TOA and brute forced a CLEP test (the state of MN is required to allow you to test out of classes and to write a test if one doesn't exist; usually AP and CLEP tests are accepted, and they don't count for/against your GPA).

It's also culturally accepted to "be bad at math," with undertones of defeat and that it's the world doing that to you and not something you can change (maybe the blame lies elsewhere like with how math is taught as a sequence of dependencies and bombing one course makes the rest substantially more difficult). I don't know how many people scrape by a D in trig and subsequently forget it all, but I'd wager it's a lot.

[adrian_b]

No, like many others you have been confused by the incapacity of those who vote the modifications of the International system of units to decide what kind of units are the units for plane angle and for solid angle: base units or derived units.

A base measurement unit is a unit that is chosen arbitrarily.

A derived measurement unit is one that is determined from the base units by using some relationship between the physical quantity that is measured and the physical quantities for which base units have been chosen.

While there are constraints for the possible choices, the division of the units into base units and derived units is a matter of convention.

Whenever there are relationships between physical quantities where so-called universal constants appear, you can decide that the universal constant must be equal to one and that it shall be no longer written, in which case some base unit becomes a derived unit by using that relationship.

The reverse is also possible, by adding a constant to a relationship, you can then modify its value from 1 to an arbitrary value, which will cause a derived unit to become a base unit for which you can choose whatever unit you like, e.g. a foot or a gallon, adjusting correspondingly the constant from the relationship.

There are 3 mathematical quantities that appear frequently in physics, logarithms, plane angles and solid angles (corresponding to the 1-dimensional space, 2-dimensional space and 3-dimensional space). All 3 enter in a large number of relationships between physical quantities, exactly like any physical quantity.

For each of these 3 quantities it is possible to choose a completely arbitrary measurement unit. Like for any other quantities, the value of a logarithm, plane angle or solid angle will be a multiple of the chosen base unit.

For logarithms, the 3 main choices for a measurement unit are the Neper (corresponding to the hyperbolic a.k.a. natural logarithms), the octave (corresponding to the binary logarithms) and the decade (corrsponding to decimal logarithms).

Like for any physical quantities, converting between logarithms expressed in different measurement units, e.g. between natural logarithms and binary logarithms is done by a multiplication or division with the ratio between their measurement units.

The same happens for the plane angle and the solid angle, for which arbitrary base units can be chosen.

What has confused the physicists is that while for physical quantities like the length, choosing a base unit was done by choosing a physical object, e.g. a platinum ruler, and declaring its length as the unit, for the 3 mathematical quantities the choice of a unit is made by a convention unrelated to a physical artifact.

Nevertheless, the choices of base units for these 3 quantities have the same consequences as the choices of any other base quantities for the values of any other quantities.

Whenever you change the value of a measurement unit you obtain a new system of units and all the values of the quantities expressed in the old system of units must be converted to be correct in the new system of units.

The fact that the plane angle is not usually written in the dimensional equations of the physical quantities in the International System of Units, because of the wrong claim that it is an "adimensional" quantity, is extremely unfortunate.

(To say that the plane angle is adimensional because it is a ratio between arc length and radius length is a serious logical error. You can equally well define the plane angle to be the ratio between the arc length and the length of the arc corresponding to a right angle, which results in a different plane angle unit. In reality the value of a plane angle expressed in radians is the ratio between the measured angle and the unit angle. The radian unit angle is defined as an angle where the corresponding arc length equals the radius length. In general, the values of any physical quantity are adimensional, because they are the ratio between 2 quantities of the same kind, the measured quantity and its unit of measurement. The physical quantities themselves and their units are dimensional.)

In reality, the correct dimensional equations for a very large number of physical quantities, much larger than expected at the first glance, contain the plane angle. If the unit for the plane angle is changed, then a lot of kinds of physical quantity values must be converted.

To add to the confusion, in practice several base units of the 3 mathematical quantities are used simultaneously, so the International System of Units as actually used is not coherent. E.g. the frequency and the angular velocity are measured in both Hertz and radian per second, the rate of an exponential decay can be expressed using the decay constant (corresponding to Nepers) or by the half-life (corresponding to octaves), and so on.

[sacrosancty]

Thanks for writing that. While I don't automatically believe it all, I think it's important to see what's arbitrary and what's natural in our units. I've struggled with the Hz vs rad/s before and I think I resolved it by including the cycle as a quanitity, so Hz = cycle/s and rad/s = 1/s. You don't seem to agree and I'm not confident of my decision, but it's now part of a big technical debt :P

A clear sign of how wrong people can be about the naturalness of units is Avogadro's constant which was recently demoted from a measured value to an exact arbitrary value. Chemists often believe that N_A, moles, atomic mass units, etc. are all somehow important or fundamental and don't realize that it's all based on a needlessly complicated constant with an (until 2019) needlessly complicated definition that could have just been a simple power of 10 if history had gone differently. Luckily the people defining SI have finally moved away from the old two independent mass units to just the kg that can now be exactly converted to atomic mass units by definition.

[jasonlotito]

> in the USA

While it might be something you are now realizing, the US is not a single entity in many ways. Rather, it's some 50 states that form a country. Each state has it's own laws and ways of doing things. While there are many similar ways of doing things, none are exactly the same. On top of that, even within the state you'll have different school systems with different policies.

And we aren't even going to discuss going to American schools in Europe.

[nkrisc]

Use it or lose it. Most people have no reason to need knowledge of trigonometry, so even if they’re taught it they quickly forget it.

I never really learned trigonometry until I started doing game programming in my spare time when suddenly that knowledge and linear algebra became necessary to understand. They only way I learned it was by needing to know it.

In fact, I regularly forget knowledge I don’t need to know. The stuff I do need to know remains fresh in my mind.

[11235813213455]

any angle unit is dimensionless, radian is no exception

[bonzini]

But they're not equal to 1, for example a degree is 0.01745...

[fluoridation]

1° is actually 0.0174... radians. 1 radian is 57.295...°. The choice of unit to specify an angle is arbitrary.

[bonzini]

But the angle is an adimensional unit (it's the ratio of two distances, one along the circumference and one along the radius) so 1 rad = 1. Therefore 1 degree is 0.0174... radians but it is also just 0.0174.

[fluoridation]

No, you're describing one particular way to measure angles. Radians express such a ratio, but degrees don't. 1° is not a ratio between distance along a circumference and radius, it's a ratio between amount rotated and complete revolution. 1° actually stands for 1/360 (of a revolution).

Which is why it's important to add the unit after the measurement. If someone tells you an angle measures 1, can you tell whether it's 1/360 of a revolution or the angle that would be formed by traveling along a circumference a distance equal to the radius of the circle?

[bonzini]

Angles in the SI are a ratio of two lengths (and solid angles are a ratio of two surfaces), so degrees are also a ratio of two lengths. 1 degree is a ratio of pi/180=0.01745, which happens to be 1/360th of a revolution; and you have to write down the unit to indicate the multiplicative factor. But writing down radians is just for clarity.

[ttoinou]

Correct, radians are a "fake" unit made up to understand better formulas (the same way we use types in programming languages)

[bluGill]

While it is a fake unit, it was made to make the math easy. You could call the origin of everything the place where I'm standing - but good luck calculating a path for the mars rovers to travel if I happen to walk to the bathroom.

[foobarbecue]

I drive a mars rover and this cracked me up. Understanding reference frames is indeed a big part of the job. We do have to deal with "site frame updates" based on rover observations of the sun -- important but annoying. I will bring your person-centered frame suggestion to the team :-)

[dtparr]

Speaking of reference frames, I deal with quite a few for Earth-bound things, and the primary ones we use are ECEF (Earth-Centered, Earth-Fixed) and ECI (Earth-Centered, Inertial), which then we will often move to a relative local frame for whatever object matters.

Is the equivalent set available for Martian Nav (MCMF/MCI, I guess), or do you have different/specialized/etc. frames based on something unique to Mars.

[foobarbecue]

Yes... there are similar frames for Mars and other bodies. A good intro: https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pd...

For the rover, we're pretty much always dealing in local coordinate systems based on reference frames defined using the rover's observations of the sun and alignment of local imagery with orbital imagery. The two frames used most frequently are called RNAV (centered on the rover) and SITE (centered on where we last did a sun observation). But then there is a tree of frame transformations for knowing the location and orientation of each part of the rover with a lot of named frames (especially important for operating the robotic arm, which I also do).

[ttoinou]

I don't understand your comment. What I meant is that we can use the numeric values of radians without ever writing the radians unit, it is indeed dimensionless (it is length / length = 1, no unit)

[_the_inflator]

I agree with you. People confuse their edge case with Math as a whole.

To sum it up: "Computer Science has nothing to do with Math!" ;)

[moffkalast]

Well it's not exactly surprising, the US is fundamentally built on arbitrary baseless measurement units so getting out of that mindset is probably difficult.

A unit that could be inherently defined by math itself and not a farmer looking at their hands and feet? Preposterous!

[shoxidizer]

> A unit that could be inherently defined by math itself and not a farmer looking at their hands and feet?

Where do you think base ten comes from?

[moffkalast]

Base 2, 10, 3, 8, hex, radians stay radians.

[InitialLastName]

Radians aren't a unit, and degrees have nothing to do with peoples' hands or feet.

Imagine using an angle measurement for physical things such that angles with rational measurements stack to a full circle. Preposterous.