Someone explain radians to me

I like degrees, they're a nice whole number, and they go all the way to 360, perfect for a nice and smooth circle.

I've noticed some graphing calculators (and the python turtle) don't use degrees though, they use radians I searched up 360 degrees in radians to figure out what the maximum angle was, and it is apparently 6.28318531
Screenshot 2024-04-08 15.31.53
That is weird, it's small and a fraction, no really smooth circles are alright but despite the six you can't even make a hexagon with them.

I already built scripts for converting to and from radians (import the stage pic) but I just wanna know real applications for them, that degrees can't really be used for.

Well, radians are actually based on the circumference of a circle. You can think of it as taking a part of the circle's circumference, and dividing out the r. So, One whole Circle's circumference is $$\tau r$$, right? You divide out the $$r$$, and you are left with $$\tau\approx6.28318531$$. (By the way, $$\pi=\frac{\tau}{2}$$)


degrees radians
$$0$$° $$0$$ radians
$$90$$° $$\frac{\tau}{4}$$ radians
$$180$$° $$\frac{\tau}{2}$$ radians
$$270$$° $$\frac{3\tau}{4}$$ radians
$$360$$° $${\tau}$$ radians

360 degrees = 2 * pi radians. It's as simple as multiplying, so 180 degrees = pi radians, 90 degrees = pi/2 radians, etc. Idk really why we use them in computing because computers are bad at irrationals.

It'll make more sense if you think of it as 2𝜋. So, the circumference of the unit circle. So, if you imagine the angle between two radii of the circle, the angle in radians is equal to the length of the arc (the portion of the circle itself) between those radii.

A lot of the math comes out nicer when angle measure equals arc length measure. But also, the series for numerical computation of trig functions are simpler in radians; that's why python (and pretty much all programming languages) uses them. If you want as-precise-as-possible trig functions of angles in degrees, you shouldn't just take the angle, multiply by 𝜋/180, and call the trig function in the math library of your programming language. Instead you are supposed to get the angle into the range [0, 45°] by taking advantage of symmetries in the trig functions, and then multiplying by 𝜋/180. (This is the only subtle thing I know about numerical analysis, and I know it only because I directly asked Prof. Kahan, the one who invented the IEEE floating point format.)

Radians is how many radii it would take to get to the exact Circumference measure.
A few years ago I made a Degrees to Radians Calculator and it obviously failed because I was 12 or 13 back then and didn't know any better. Still don't now but I do understand what Radians are.

270° = 2τ/4 radians = τ/2 radians = 180°


Mathematicians aren't so interested in drawing polygons, but are interested in trig functions more for the sake of Fourier series, a way to represent and analyze periodic functions, e.g., sound waves. (See also Fourier analysis.) The Wikipedia articles are tough going, but if you just read each one up to the table of contents, you'll get the idea. This bit of mathematics just makes much, much more sense if you take the trig functions as operating in radians.

Many things in life would be simpler without irrationals. Notoriously, this is why the Indiana House of Representatives once declared 𝜋=3.2. (Luckily the bill was shot down in their Senate.) This is also why the Pythagoreans are said to have killed their colleague who proved that √2 is irrational, the first known-to-be-irrational number. (Sadly, Wikipedia seems to think it didn't really happen.)

Alas, reality is what it is. (Philosophers of science argue about whether mathematics is invented or discovered, but irrationals are a good argument for discovery, since nobody would willingly invent them.)

When we want to get things exactly right, we don't do numerical computation at all; we just keep 𝜋 and friends in the formulas as such. But when we need a numerical approximation, we get a very good one because of the efforts of a small bunch of numerical analysis experts, especially Prof. Kahan, who invented the floating point notation everyone now uses. (He didn't invent floating point, which is just the same scientific notation people used back in the days of slide rules. But before IEEE floating point, every computer manufacturer made up their own version, all of which had errors in the hardware that would lead to very wrong answers in some situations.)

Having said that, I always tell my own students "Do not attempt to write your own floating point code. Hire a numerical analysis specialist." (The other thing like that is security code, also beyond the ken of ordinary mortals.) But Prof. Kahan gets mad at me for that; he wants everyone to learn how to do at least some computations on real numbers correctly.