e^(i * tau) = 1 is the more "honest" formula anyway, because it says exactly what it ought to: rotating by tau takes you all the way around a circle. 0i is also arguably not cheating there, since...
e^(i * tau) = 1 is the more "honest" formula anyway, because it says exactly what it ought to: rotating by tau takes you all the way around a circle.
0i is also arguably not cheating there, since the setting is complex analysis, so every number does have a real and an imaginary component.
The "regular" version is saying, "a turn of the ratio of the circumference to the radius traces the circle halfway" The tau version says, "a turn of the ratio of the circumference to the diameter...
The "regular" version is saying, "a turn of the ratio of the circumference to the radius traces the circle halfway"
The tau version says, "a turn of the ratio of the circumference to the diameter traces the circle exactly once"
My unpopular math opinion is that degrees and radians are both bad angle units. The common angle unit that normal people use should have been a full turn of a circle. A right angle is just a...
My unpopular math opinion is that degrees and radians are both bad angle units. The common angle unit that normal people use should have been a full turn of a circle. A right angle is just a quarter of a turn.
I like tau as a circle constant for the sole reason that you can kind of use it this way if you write radians as multiples of tau and pretend it's a unit.
The beauty of the radian is that it isn't really a unit at all. A radian is, by definition, just the angle subtended when the arc of a circle equals its radius. But since we know that we can...
The beauty of the radian is that it isn't really a unit at all. A radian is, by definition, just the angle subtended when the arc of a circle equals its radius. But since we know that we can measure the arc length of a circle using s = θ r, that means that when s = r (i.e., when the angle is 1 radian), it is also the case that θ = 1 (per SI 2019, no units necessary!).
This allows us to sensibly define Taylor series of functions that have angular arguments without needing a bunch of ad-hoc conversion factors, e.g. the tried and true
Yep same, the closer an implementation can be to its underlying intuition, the better. Like understanding "measurement of an angle has to do with how far around the circle you go" is the...
Yep same, the closer an implementation can be to its underlying intuition, the better. Like understanding "measurement of an angle has to do with how far around the circle you go" is the intuition, and so the implementation should be that angles are the same as fractions, with one whole turn being one. So one needs to be tau, not pi.
Tau vs Pi is such a silly argument. Obviously we should be using Pau as a compromise. Also if anyone is interested Pi won the debate 11 years ago because Matt is better at maths than Steve.
Tau vs Pi is such a silly argument. Obviously we should be using Pau as a compromise.
I dunno,
e^(tau/2)i +1 = 0
doesn't have the same cachet.
The tau version is e^(i * tau) = 1
or as the author points out, e^(i * tau) = 1 + 0i
0i feels like cheating to me. I guess there’s an argument that the tau version without 0 at all is more succinct.
e^(i * tau) = 1 is the more "honest" formula anyway, because it says exactly what it ought to: rotating by tau takes you all the way around a circle.
0i is also arguably not cheating there, since the setting is complex analysis, so every number does have a real and an imaginary component.
But then we don't have our additive identity 😭
e^iτ - 1 = 0
close enough.
e^(tau/2)i = -1 makes perfect sense though - you're going halfway around the circle, so of course there's a "/2" in there.
It just feels less aesthetically pleasing.
The "regular" version is saying, "a turn of the ratio of the circumference to the radius traces the circle halfway"
The tau version says, "a turn of the ratio of the circumference to the diameter traces the circle exactly once"
To me the tau version is a lot nicer
My unpopular math opinion is that degrees and radians are both bad angle units. The common angle unit that normal people use should have been a full turn of a circle. A right angle is just a quarter of a turn.
I like tau as a circle constant for the sole reason that you can kind of use it this way if you write radians as multiples of tau and pretend it's a unit.
The beauty of the radian is that it isn't really a unit at all. A radian is, by definition, just the angle subtended when the arc of a circle equals its radius. But since we know that we can measure the arc length of a circle using s = θ r, that means that when s = r (i.e., when the angle is 1 radian), it is also the case that θ = 1 (per SI 2019, no units necessary!).
This allows us to sensibly define Taylor series of functions that have angular arguments without needing a bunch of ad-hoc conversion factors, e.g. the tried and true
instead of
or
Yep same, the closer an implementation can be to its underlying intuition, the better. Like understanding "measurement of an angle has to do with how far around the circle you go" is the intuition, and so the implementation should be that angles are the same as fractions, with one whole turn being one. So one needs to be tau, not pi.
Since this is now the official pi day not tau/2 day thread:
Tau vs Pi is such a silly argument. Obviously we should be using Pau as a compromise.
Also if anyone is interested Pi won the debate 11 years ago because Matt is better at maths than Steve.
I'm a day late to the thread, but I did enjoy a folded circle snack.