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Why do people even measure rotations in radians? It’s a stupid unit that’s never ever written out but always expressed as either “pi radians” or “tau radians”,

Why do people even measure rotations in radians? It’s a stupid unit that’s never ever written out but always expressed as either “pi radians” or “tau radians”, and just like with degrees, or the measure which has 200 to a circle you end up spending time teaching students how to convert to the sensible unit we all default to, which is just plain rotations. Everyone understands intuitively half a rotation. A full rotation or a quarter rotation. And when you tell a student 30 degrees, they all spend a bit of mental effort thinking before they go “Ah yes, 1/12 of a full rotation” and if you tell them 0.52 radians they’ll just look at you blankly stating they need a calculator before they know what are you are on about.

bronzeage 5y ago
because sin and cos have simple Taylor series, but only when measuring angles in radians. sin(x)/x at zero is one only in radians. Just multiplying by the angle is a good approximation of sin for low angles, again only in radians. I bet even an alien mathematician will describe angles in radians.
- deleted 5y ago
  [deleted]
- boublepop 5y ago
  The equivalent if sin taking revolutions (let’s say sinr) is just sinr(x)=sin(x2pi) there is no fundamental mathematical relation broken by this, that would indicate math was broken at the core. And the Taylor expansion isn’t any less simple, you simply state the pi’s in the expansion rather than having them implied in your input variable. It’s amazing people assume math will break somehow if you use a different unit for angles, when most should at least be familiar with using both 360 and ~6.283 and some using 200 as the scale for one rotation. And when trying to reason about it every single person always translates this into ratios of rotations or percentages of rotations.
jonsen 5y ago
Hz, rpm, no radius nor pi involved.
rsj_hn 5y ago
We don't name things based on what students who know nothing about the topic find most intuitive based on zero background knowledge. We name things based on what people working in a field find most convenient for their work and the job of the students is to learn the field. The unit of measurement you propose will be just a new radian - basically 2pi = 1, or pi (old radian)= 1/2 (new radian) except you want to call it a "revolution" because for the circle, this length is the circumference (screw learning other shapes, right?). Then the power series for sine, cosine, etc would be rescaled by this term. This means the derivative of the sine is no longer the cosine, there is this ugly term added. Thus neither is the derivative of the cosine the negative sign. Fourrier series formulas now change. The wave equation now changes as it's not true that taking the derivative twice and adding to the original function is zero -- trig functions no longer satisfy the wave equation! So all these other formulas now get this scaling term inserted into them as well, it just keeps multiplying. Moreover students would discover when they turn to ellipses and other shapes, that for their formulas to generalize, a "revolution" stops being a "revolution" but merely becomes a distance along an arc and they have to go back and really understand what they were doing all this time was working in terms of arc length rather than revolutions, except that for a unit circle and this special scale, there was an illusion created that it was revolutions that was going into the formulas rather than arc length. That would then be even more confusing and is kinda the definition of a bad education. So the advantage of rescaling the radian to this new scale and calling it a "revolution" is a net negative to anyone who actually learns the field - which is hopefully the student. The key concept, the more generalizable and useful concept, is "distance along an arc", not "numbers of revolutions". These two concepts are not the same, and only one of them is at the heart of trigonometry and calculus. We have moved past the understanding the Babylonians had and now we have a deeper understanding of unifying concepts. It is these concepts that we should teach students, even if they are a bit more subtle than "revolution". As the saying goes, "make everything as simple as possible. But not simpler."
- boublepop 5y ago
  > We name things based on what people working in a field find most convenient for their work My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. However they do it by squeezing together 2Pi (or very few by using tau) as a pseudo-unit. Imagine if we stopped using meters and instead used a unit we called meterians which was 1/sqrt(2) meter, yet everyone always in every single equation or table listing wrote out “72 sqrt(2) meterians” would that not be stupid? As for redefining sin and cos, there really is no need you almost always see equations using cos(x pi) and the “annoying factors” you get in derivations still pop out, just not from the function itself, but instead from the factor you’ve put in place so you can keep dealing with rotations or half rotations while pretending to use radians. As for concerns about other shapes it’s literally the case that the opposite of what you proclaim is true. If you rotate any other shape than a circle then there is no connection between radians as a length along the perimeter and rotation. A unit square requires 8 perimeter length units to rotate. By any kid even before they learn to multiply will be able to show you exactly that rotating a square once completely is one rotation. The fact that you might personally say “no that’s not intuitive! We should say that it’s 1 times (8 radial perimeter units) just shows that you’ve disconnected the math from what’s intuitive. And changing how we deal with trigonometric formulas to drop the constant reference to pi, even though we’re multiplying it in and deciding net it away, doesn’t change anything, the fact that it shows up everywhere in formulas is just a concealed even of mathematical robustness, if we decided instead to measure angles in units of 1/200*(4pi^2)/tau as is done in land measurements doesn’t make the math simpler or harder, it just changes the scaling factor. But there is the benefit that people immediately and intuitively understand the measurements written down without need of a calculator, though that would also be the case with simple fractions and direct percentages.
kevin_thibedeau 5y ago
e^(iπ)+1 = 0 This is a law of the universe. It doesn't know anything about degrees or sexagesimal number systems. You can't give this to students straight away because they don't have the background to deal with it. Hence the diversions into triangles, imaginary numbers, and radians.
ilyagr 5y ago
I think this is the real point of confusion. It is true, but far from obvious, that radians are the One True Way of measuring angles (for understanding the universe, not necessarily for communicating to human beings). Once you believe in radians, the full angle is tau=2pi, so that is clearly the more fundamental number than pi. The reason why radians are the One True Way is the power series expansions for trig functions. Measuring the angle in radians makes e^ix = cos x + i * sin x work out where e^t = 1 + t + t^2/2 + t^3/3! + t^4/4! + ... (Exercise: derive the power series expansion for sin and cos from this) This formula connects geometry of circles to algebra and calculus in a fundamental way - there is no geometry needed to calculate e^ix, sin, or cos with power series (a computer could do it!), while cos and sin are all about angles and circles. Measuring angle in anything other than radians makes the formula horrendous (try it).

bronzeage 5y ago

because sin and cos have simple Taylor series, but only when measuring angles in radians. sin(x)/x at zero is one only in radians. Just multiplying by the angle is a good approximation of sin for low angles, again only in radians. I bet even an alien mathematician will describe angles in radians.

deleted 5y ago
[deleted]
boublepop 5y ago
The equivalent if sin taking revolutions (let’s say sinr) is just sinr(x)=sin(x2pi) there is no fundamental mathematical relation broken by this, that would indicate math was broken at the core. And the Taylor expansion isn’t any less simple, you simply state the pi’s in the expansion rather than having them implied in your input variable. It’s amazing people assume math will break somehow if you use a different unit for angles, when most should at least be familiar with using both 360 and ~6.283 and some using 200 as the scale for one rotation. And when trying to reason about it every single person always translates this into ratios of rotations or percentages of rotations.

jonsen 5y ago

Hz, rpm, no radius nor pi involved.

rsj_hn 5y ago

We don't name things based on what students who know nothing about the topic find most intuitive based on zero background knowledge. We name things based on what people working in a field find most convenient for their work and the job of the students is to learn the field. The unit of measurement you propose will be just a new radian - basically 2pi = 1, or pi (old radian)= 1/2 (new radian) except you want to call it a "revolution" because for the circle, this length is the circumference (screw learning other shapes, right?). Then the power series for sine, cosine, etc would be rescaled by this term. This means the derivative of the sine is no longer the cosine, there is this ugly term added. Thus neither is the derivative of the cosine the negative sign. Fourrier series formulas now change. The wave equation now changes as it's not true that taking the derivative twice and adding to the original function is zero -- trig functions no longer satisfy the wave equation! So all these other formulas now get this scaling term inserted into them as well, it just keeps multiplying. Moreover students would discover when they turn to ellipses and other shapes, that for their formulas to generalize, a "revolution" stops being a "revolution" but merely becomes a distance along an arc and they have to go back and really understand what they were doing all this time was working in terms of arc length rather than revolutions, except that for a unit circle and this special scale, there was an illusion created that it was revolutions that was going into the formulas rather than arc length. That would then be even more confusing and is kinda the definition of a bad education. So the advantage of rescaling the radian to this new scale and calling it a "revolution" is a net negative to anyone who actually learns the field - which is hopefully the student. The key concept, the more generalizable and useful concept, is "distance along an arc", not "numbers of revolutions". These two concepts are not the same, and only one of them is at the heart of trigonometry and calculus. We have moved past the understanding the Babylonians had and now we have a deeper understanding of unifying concepts. It is these concepts that we should teach students, even if they are a bit more subtle than "revolution". As the saying goes, "make everything as simple as possible. But not simpler."

boublepop 5y ago
> We name things based on what people working in a field find most convenient for their work My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. However they do it by squeezing together 2Pi (or very few by using tau) as a pseudo-unit. Imagine if we stopped using meters and instead used a unit we called meterians which was 1/sqrt(2) meter, yet everyone always in every single equation or table listing wrote out “72 sqrt(2) meterians” would that not be stupid? As for redefining sin and cos, there really is no need you almost always see equations using cos(x pi) and the “annoying factors” you get in derivations still pop out, just not from the function itself, but instead from the factor you’ve put in place so you can keep dealing with rotations or half rotations while pretending to use radians. As for concerns about other shapes it’s literally the case that the opposite of what you proclaim is true. If you rotate any other shape than a circle then there is no connection between radians as a length along the perimeter and rotation. A unit square requires 8 perimeter length units to rotate. By any kid even before they learn to multiply will be able to show you exactly that rotating a square once completely is one rotation. The fact that you might personally say “no that’s not intuitive! We should say that it’s 1 times (8 radial perimeter units) just shows that you’ve disconnected the math from what’s intuitive. And changing how we deal with trigonometric formulas to drop the constant reference to pi, even though we’re multiplying it in and deciding net it away, doesn’t change anything, the fact that it shows up everywhere in formulas is just a concealed even of mathematical robustness, if we decided instead to measure angles in units of 1/200*(4pi^2)/tau as is done in land measurements doesn’t make the math simpler or harder, it just changes the scaling factor. But there is the benefit that people immediately and intuitively understand the measurements written down without need of a calculator, though that would also be the case with simple fractions and direct percentages.

kevin_thibedeau 5y ago

e^(iπ)+1 = 0 This is a law of the universe. It doesn't know anything about degrees or sexagesimal number systems. You can't give this to students straight away because they don't have the background to deal with it. Hence the diversions into triangles, imaginary numbers, and radians.

ilyagr 5y ago

I think this is the real point of confusion. It is true, but far from obvious, that radians are the One True Way of measuring angles (for understanding the universe, not necessarily for communicating to human beings). Once you believe in radians, the full angle is tau=2pi, so that is clearly the more fundamental number than pi. The reason why radians are the One True Way is the power series expansions for trig functions. Measuring the angle in radians makes e^ix = cos x + i * sin x work out where e^t = 1 + t + t^2/2 + t^3/3! + t^4/4! + ... (Exercise: derive the power series expansion for sin and cos from this) This formula connects geometry of circles to algebra and calculus in a fundamental way - there is no geometry needed to calculate e^ix, sin, or cos with power series (a computer could do it!), while cos and sin are all about angles and circles. Measuring angle in anything other than radians makes the formula horrendous (try it).