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Why Tau is better than Pi

Silly Druid

Entry posted by Silly Druid in Mathematics

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I'm writing this on Pi Day, but I think pi makes no sense as a fundamental mathematical constant, and it's used only because of force of habit.

The circle is defined as the set of points on the plane with a given distance to the center. That distance is the radius. So the "circle constant" should be defined using the radius, not diameter. It has been proposed that this constant would be denoted by the Greek letter tau. Tau is equal to 2*pi.

The full angle (360 degrees) in radians is one tau instead of two pi, which is a more simple and natural way of doing things.

Also Euler's identity is in my opinion even more beautiful when using tau instead of pi:

e^(i*tau) = 1

Edited by Silly Druid

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Splashee

Happy PI day! :Tempest:

 

Radians are stupid is all I can say :yeahno:

 

Spoiler

Please @Silly Druid ask ChatGPT why radians are better than degrees.

 

Internet is so full of texts like this:
Why use radians:

  1. Pi is great
  2. Radians make it possible to relate a linear measure and an angle measure.

While I still see a unit of Pi to be highly unstable since it is an irrational number constant. Doing a linear arithmetic operation would lead to extreme precision loss the further you wrap around the circle, not to say 2Pi is horrible already.

 

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Silly Druid

@Splashee Here is the answer from ChatGPT:

Quote

Radians are generally considered better than degrees because they are a more natural unit for measuring angles in mathematics and physics.

The main reason for this is that radians are based on the geometry of the circle. A radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. This means that if you have a circle with a radius of 1 unit, an angle of one radian will subtend an arc of length 1 unit along the circumference of the circle.

In contrast, degrees are an arbitrary unit of measurement that do not have any direct relationship to the geometry of the circle. Degrees divide a circle into 360 equal parts, with each degree being equal to 1/360th of a circle. This makes calculations involving angles more cumbersome, as conversion factors are required to convert between degrees and radians.

Another advantage of radians is that they simplify many mathematical formulas involving trigonometric functions, such as sine, cosine, and tangent. In fact, many trigonometric identities and formulas are most naturally expressed in terms of radians.

In summary, while degrees are a useful unit of measurement for everyday angles, radians are the preferred unit for mathematical and scientific calculations involving angles and trigonometry.

I can add to this that, for example, when using radians you can assume that sin x ≈ tan x ≈ x for small x.

Edited by Silly Druid
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Splashee

Oh no ChatGPT noooooo :ButtercupLaugh:

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This makes calculations involving angles more cumbersome, as conversion factors are required to convert between degrees and radians.

We are basically talking about programming languages here and the input parameter is in radians, so obviously the conversion is more cumbersome when it is mandatory in the first place. Stay radians all the way, and it is easier, except it isn't. That's why programmers always convert using PI as a constant. We all say "90 degrees field of view" when programming 3D perspective projection, and not Half PI.
It forces the programmer to specify an approximation of PI for every input to a trigonometric function, even if it expects values in the range of PI to -PI, where PI is unknown because of implementation. While (on the other hand) saying 90 degrees is exactly and always 90 degrees.

 

Quote

Another advantage of radians is that they simplify many mathematical formulas involving trigonometric functions, such as sine, cosine, and tangent. In fact, many trigonometric identities and formulas are most naturally expressed in terms of radians.

Again, because of programming languages taking radians as parameters.

 

Quote

I can add to this that, for example, when using radians you can assume that sin x ≈ tan x ≈ x for small x.

Why is radians better than degrees? The summery of ChatGPT says that it is better because it is preferred by a specific group of people (mathematicians), and that's why it is better. It is not the answer I was hoping an AI would come up with. Yes, but math is not about being close to, but to give exact results. For very small values, everything is close to 0 anyways.

 

I love ChatGPT though! Make it speak in lolcat next time :ticking:

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Sir Hugsalot
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This is actually a curious thing You just brought up, although to be fair with You I speak from perspective of a person, who's hasn't touched this subject for many years. 

I always found "pi" an incredibly weird thing. I feel like it is simply too specific and invites imperfection to math - something that should instead strive to be perfect and give exact, not estimated answers whenever possible. If radiant is an answer to that problem I certainly would love to learn more about how it's applied, how it works etc. I doubt I will ever use it, but curiosity alone makes me wonder... :fluttershy:

 

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Silly Druid

@Sir Hugsalot

Unfortunately, there will be always imperfection in math, not only with not being able to give an exact value to numbers like pi, but even on the deepest level (see "Gödel's incompleteness theorems").

Replacing pi with tau won't help in this situation, it's just more convenient than pi in many cases. We're using pi just because of the tradition that is hard to change, just like the Americans are using their overcomplicated system of measurements, for example.

As for radians, they don't help much either, because they usually make the angle some number multiplied by pi (or tau for that matter), and we can't have the exact value of that... They are the "natural" way of measuring angles though (in a similar way as the natural logarithm is natural), and do make some things simpler, as ChatGPT pointed out.

@Splashee

You asked for it...

Spoiler

Oh hai! Radians be better dan degrees cuz dey be moar purrrecise an' less confusin'. Plus, dey make da math easier an' less messy. So no more derpy conversions, jus' use radians, and u can haz da easier time doin' ur trigonometry. Trust meow!

 

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Brony Number 42

We can't change the past. Pi will always be used, electrons will always be negatively charged, and our counting system will always be base 10 (instead of a far superior hexadecimal).

Spoiler

 

 

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