Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi:
e^(i*tau) = 1
I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).
I've been posting the manifesto to friends and colleagues every tau day for the past ten years. Let's keep chipping away at it and eventually we won't obfuscate radians for our kids anymore.
Oh, pi has its place: in engineering, for example, it's much easier to measure the diameter of a pipe than its radius: just put calipers around the widest point (outside or inside depending) and you have the diameter. In fact, you probably wouldn't ever measure the radius; in places where you need the radius, you'd just measure the diameter and divide by 2.
But for teaching trig? Explaining radians should definitely be tau-based.
Yes, though more broadly my point was that the radius is the natural measurement of the circle for most things since most things are center-based. But for some physical measurements, mostly based around pipes, "what is the width of this pipe" is the question you need answering, and that is diameter-based. And pi is circumference/diameter, while tau is circumference/radius.
But yes, if the world switched to tau then you wouldn't need pi anymore, you'd just write tau/2 in the rare cases where having the circumference/diameter ratio handy is useful.
I wonder how many places we have in modern math symbols which we use for historical reasons, rather than because it's most convenient overall. I guess we are balancing things here.
Which one of those is preferable? It seems to me that they are both historically based. 10 x 10 is also 100 in base-12 (it's only in base-10 that it looks like 144).
IMHO, in a modern setting base-16 would be the most convenient. Then I maybe wouldn't struggle to remember that the CIDR range C0.A8.0.0/18 (192.168.0.0/24) consists of 10 (16) blocks of size 10 (16).
There’s nothing particularly convenient about base-ten; for real-world uses base-twelve would be preferable thanks to its large number of divisors (and even larger number of divisors of its multiples like 60). Which is exactly why 12 and 60 historically appear in many contexts.
A number theorist would probably want a prime base, so that N (mod 10) would be a field.
A power-of-two base wouldn’t be particularly convenient to anyone except a small minority consisting mostly of hardware and software engineers.
The gain is pedagogical: giving kids a good intuition about angles is so much easier when the constant you're working with represents an entire turn around the circle (360°) rather than a half-turn of 180°. The advantage of using tau instead of pi is much smaller in other situations, but when it comes to measuring angles in radians, it's huge. And kids who have a better understanding of angles and trigonometry are just a little bit more likely to become good engineers. So persuading math teachers that there's a better way to teach trig is an investment in the future whose potential payoff is 20-30 years (or more) down the road.
I'd really be curious to see any substantial proof for that claim.
The first time pupils encounter pi isn't when measuring angles. At least over here, that's still done in degrees, which is much easier to explain, and also latches onto common cultural practice (e.g. a turn of 180 degrees). So I suppose that already makes them good engineers.
But the first time pupils encounter pi is when computing the circumference and surface of a circle. While the former would look easier with the radius (tau * r), it looks just as weird when using diameter or when using it for the surface.
I don't know of any studies yet comparing the two approaches, but https://www.tauday.com/a-tau-testimonial is the story of one student who finally "got it" when using tau instead of pi. I strongly suspect she's not unique.
If there's more data available, I don't yet know where to find it.
P.S. Yes, angles are first presented in degrees in most contexts, and understanding sines and cosines is easier when given the degree units you're familiar with. But radians do need to get introduced at some point during trig, and it's exactly the study of radians which should be done using tau (the equivalent of 360°) rather than pi (180°). Because a right angle, 90°, is a quarter of the way around the circle, and that's tau/4. A 45° angle is tau/8, one-eighth of the way around the circle. There's no need to memorize formulas when you do it this way, it's just straight-up intuitive (whereas 45° = pi/4 is not intuitive the same way).
The one place where radians are more convenient is when you are at the centre of the circle. Then something which is as wide (or tall) as it is far away subtends one radian in your view. (And correspondingly, if it subtends half a radian it is half as wide as it is far away, etc.)
This happens to be the most common situation in which I measure angles.
More convenient than degrees. This is unrelated to pi vs tau (using tau or pi doesn't change the meaning of radians, the properties you mention are not affected). What OP is getting at is that the same number of radians, e.g. 1.57 (quarter turn) is more naturally expressed as tau/4 than pi/2.
Disagree. This is not so much about epistemological correctness as it is about what's useful and convenient. math.tau is an easier and more intuitive constant to work with.
Math didactics is all about making math more digestible for the next generation, even if it breaks with history.
For now, I’ve just explicitly written exp(2πiν) etc instead of exp(iπν) in my work; explicitly writing out 2π and treating it as effectively one symbol does have similar conceptual benefits as working with τ.
e^(i*tau) = 1
I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).