8 ms·

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 sub

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?

ttoinou 4y ago
Correct, radians are a "fake" unit made up to understand better formulas (the same way we use types in programming languages)
- bluGill 4y ago
  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 4y ago
    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 4y ago
      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 4y ago
        Yes... there are similar frames for Mars and other bodies. A good intro: https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pdf/individual_docs/17_frames_and_coordinate_systems.pdf 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 4y ago
    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)
11235813213455 4y ago
any angle unit is dimensionless, radian is no exception
- bonzini 4y ago
  But they're not equal to 1, for example a degree is 0.01745...
  - fluoridation 4y ago
    1° is actually 0.0174... radians. 1 radian is 57.295...°. The choice of unit to specify an angle is arbitrary.
    - bonzini 4y ago
      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 4y ago
        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 4y ago
        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.
genewitch 4y ago
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.
- bombcar 4y ago
  Calculus made a lot of it “click” for me but I somehow got through high school and college without ever doing or understanding trigonometry.
- weinzierl 4y ago
  "[..] 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.
  - aaaaaaaaaaab 4y ago
    So which programming language makes sense to you? Dare I say… Rust?
    - deleted 4y ago
      [deleted]
    - weinzierl 4y ago
      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.
weberer 4y ago
>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 4y ago
  > 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 4y ago
    I think I would have put the quotes around "learn." I don't think every high school student can master advanced trig.
- retrocryptid 4y ago
  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.
adrian_b 4y ago
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.
moffkalast 4y ago
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 4y ago
  > 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 4y ago
    Base 2, 10, 3, 8, hex, radians stay radians.
    - InitialLastName 4y ago
      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.
bluGill 4y ago
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 4y ago
  s/per-calculus/pre-calculus/g
  - kortex 4y ago
    The percalculus ion is just calculus in its highest oxidation state.
insightcheck 4y ago
> "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.
kortilla 4y ago
> 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 4y ago
  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.
  - mathieuh 4y ago
    I'm 27, educated in the UK, all I remember about trigonometry is SOHCAHTOA.
    - duncan-donuts 4y ago
      Ha I also remember the mnemonic but I don’t have a clue how to use it.
      - krallja 4y ago
        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 4y ago
      Otto had a heap of apples.
    - mcv 4y ago
hansvm 4y ago
> 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.
_the_inflator 4y ago
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!" ;)
phkahler 4y ago
>> 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.
- acidburnNSA 4y ago
  rad is a unit already. https://en.wikipedia.org/wiki/Rad_(unit) https://en.wikipedia.org/wiki/Rad_(unit)
- gugagore 4y ago
  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 4y ago
    >> 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*.
deleted 4y ago
[deleted]
dahart 4y ago
> 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 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.
jasonlotito 4y ago
> 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 4y ago
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.
loup-vaillant 4y ago
> 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.

ttoinou 4y ago

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

bluGill 4y ago
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 4y ago
  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 4y ago
    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 4y ago
      Yes... there are similar frames for Mars and other bodies. A good intro: https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pdf/individual_docs/17_frames_and_coordinate_systems.pdf 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 4y ago
  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)

11235813213455 4y ago

any angle unit is dimensionless, radian is no exception

bonzini 4y ago
But they're not equal to 1, for example a degree is 0.01745...
- fluoridation 4y ago
  1° is actually 0.0174... radians. 1 radian is 57.295...°. The choice of unit to specify an angle is arbitrary.
  - bonzini 4y ago
    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 4y ago
      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 4y ago
        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.

genewitch 4y ago

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.

bombcar 4y ago
Calculus made a lot of it “click” for me but I somehow got through high school and college without ever doing or understanding trigonometry.
weinzierl 4y ago
"[..] 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.
- aaaaaaaaaaab 4y ago
  So which programming language makes sense to you? Dare I say… Rust?
  - deleted 4y ago
    [deleted]
  - weinzierl 4y ago
    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.

weberer 4y ago

>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 4y ago
> 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 4y ago
  I think I would have put the quotes around "learn." I don't think every high school student can master advanced trig.
retrocryptid 4y ago
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.

adrian_b 4y ago

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.

moffkalast 4y ago

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 4y ago
> 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 4y ago
  Base 2, 10, 3, 8, hex, radians stay radians.
  - InitialLastName 4y ago
    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.

bluGill 4y ago

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 4y ago
s/per-calculus/pre-calculus/g
- kortex 4y ago
  The percalculus ion is just calculus in its highest oxidation state.

insightcheck 4y ago

> "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.

kortilla 4y ago

> 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 4y ago
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.
- mathieuh 4y ago
  I'm 27, educated in the UK, all I remember about trigonometry is SOHCAHTOA.
  - duncan-donuts 4y ago
    Ha I also remember the mnemonic but I don’t have a clue how to use it.
    - krallja 4y ago
      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 4y ago
    Otto had a heap of apples.
  - mcv 4y ago

hansvm 4y ago

> 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.

_the_inflator 4y ago

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!" ;)

phkahler 4y ago

>> 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.

acidburnNSA 4y ago
rad is a unit already. https://en.wikipedia.org/wiki/Rad_(unit) https://en.wikipedia.org/wiki/Rad_(unit)
gugagore 4y ago
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 4y ago
  >> 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*.

deleted 4y ago

[deleted]

dahart 4y ago

> 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 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.

jasonlotito 4y ago

> 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 4y ago

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.

loup-vaillant 4y ago

> 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.