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Turns are better than radians

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Re: Turns are better than radians

#41

Post by layer8 »

adrian_b wrote: post #26

While I agree that "turns" are much more convenient in some applications than radians, there is no need to invent a new terminology. For a long time, including the 19th century, the plane angle measurement unit corresponding with 4 right angles, i.e. a complete rotation around a point, has been named "cycle". That is why in many old physics or engineering books one will find wave numbers measured in "cycles per meter…

> there is no need to invent a new terminology.

It's not a new terminology: https://en.wikipedia.org/wiki/Turn_(angle)

Re: Turns are better than radians

#42

Post by qsort »

Double_a_92 wrote: post #34
mjw_byrne wrote: post #25

"math never decreed that sine and cosine have to take radian arguments!" This is at best questionable and at worst false. If you only want to use sin and cos as functions for doing trigonometry, it is true that you can choose whatever angle unit you like and stick with it and it will be fine. For most other stuff, e.g. differential equations, complex analysis, signal processing and mechanics, it's pretty much inescap…

Why do they need to be at multiple integers of pi, instead of at half turns?

Because - numeric precision arguments aside, there's an excellent comment explaining that problem in this thread - it's the only unit of measurement that makes sense for angles.

It provides an easy way to connect the complex exponential with trigonometric functions (and everything you get from that, i.e. Taylor series, nice behavior in diffeqs). You can do the same in degrees as well, but you end up with weird conversion factors with pi in the denominator, a strong indication you should have multiplied by pi to begin with.

Re: Turns are better than radians

#43

Post by ncmncm »

adrian_b wrote: post #26

While I agree that "turns" are much more convenient in some applications than radians, there is no need to invent a new terminology. For a long time, including the 19th century, the plane angle measurement unit corresponding with 4 right angles, i.e. a complete rotation around a point, has been named "cycle". That is why in many old physics or engineering books one will find wave numbers measured in "cycles per meter…

The difference between representation in units of right angles (1/4 cycles), and cycles is just a matter of bumping the exponent up or down two clicks. So cycles / turns and radians suffice.

I suppose it is in solutions to differential equations where radians become important.

Re: Turns are better than radians

#44

Post by trashtester »

jjgreen wrote: post #3
ClassyJacket wrote: post #2

I'm so glad someone else finally said this. This article takes the approach of simplicity of code, which I do agree with, but additionally I've been thinking turns would have to be more intuitive and easier to teach (particularly to disinterested teenagers) for YEARS, and I finally feel vindicated at least one person agrees with me. Turns are really the most neutral way to count an angle. We don't use them for everyd…

... and the derivative of sin(x) is ... ?

Re(exp(ix)).

Using degrees, turns, etc instead of radians, is like using 10^y instead of e^x (where y=x/ln(10)). Useful for many practical things, but useless for a lot of math applications, especially involving differential equations, complex numbers etc.

Re: Turns are better than radians

#45

Post by gfv »

krajzeg wrote: post #16
lordnacho wrote: post #8

That was quite convincing actually. I guess we all have this realization at some point in early math education. Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?) Then at some point you get shown radians, which relates the arc length to the radius. That som…

The 400 system is the grads or gradians, indeed originating from the French revolution. Nowadays I don't think they're used as the principal unit in any country. Wikipedia does mention it gets some use in specialized fields such as surveying, mining and geology.

They are used indirectly through distance. At the time, the meter was defined as one ten millionth of the distance between the north pole and equator through the Paris meridian. That means that the meter corresponds to 1/100000 of a grad of latitude -- which is better read as "a kilometre is 1/100 of a grad".

This is symmetrical to the nautical mile, which is one minute of arc.

Re: Turns are better than radians

#46

Post by ncmncm »

severak_cz wrote: post #20
lordnacho wrote: post #8

That was quite convincing actually. I guess we all have this realization at some point in early math education. Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?) Then at some point you get shown radians, which relates the arc length to the radius. That som…

It's 360 degrees because of stars. One degree is how much stars move by one day.

Or, as close as made no nevermind, oncet.

Re: Turns are better than radians

#48

Post by the_other »

I'm interested in this from the perspective of learning maths, rather than writing code (for now).

I've wondered for a few years now whether teaching angles and trig using turns, rather than degrees or radians, would be better from the very beginning. Degrees are arbitrary and based on the numeric preferences of a dead culture, rather than on what's happening on the page or in 3d space. Radians seem better because the units are related to a property of the circle, but they're hard to visualise and reason about because they don't fit a circle in whole numbers. Surely turns are the most clear.

I'm rusty and don't practice maths much. If I did I'd probably have the skills of 14y/o me, for anything outside set theory. I'd definitely do a trigonometry course based on turns if I could find one.

Re: Turns are better than radians

#49

Post by sambeau »

adrian_b wrote: post #26

While I agree that "turns" are much more convenient in some applications than radians, there is no need to invent a new terminology. For a long time, including the 19th century, the plane angle measurement unit corresponding with 4 right angles, i.e. a complete rotation around a point, has been named "cycle". That is why in many old physics or engineering books one will find wave numbers measured in "cycles per meter…

This isn't new terminology.

https://en.wikipedia.org/wiki/Turn_(angle)

Re: Turns are better than radians

#50

Post by cillian64 »

Double_a_92 wrote: post #34
mjw_byrne wrote: post #25

"math never decreed that sine and cosine have to take radian arguments!" This is at best questionable and at worst false. If you only want to use sin and cos as functions for doing trigonometry, it is true that you can choose whatever angle unit you like and stick with it and it will be fine. For most other stuff, e.g. differential equations, complex analysis, signal processing and mechanics, it's pretty much inescap…

Why do they need to be at multiple integers of pi, instead of at half turns?

If you differentiate sin(x) with respect to x then you get cos(x), but only if your trig functions are using radians. Any other unit results in an extra coefficient appearing. That’s not an insurmountable problem, but radians are the fundamental unit here, not just an arbitrary choice.
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