# The Proper $\pi$ Manifesto

## The circle constant debate

Being the passionate mathematicians we are, we found the $\pi$ vs $\tau$ debate frustrating. Each side has their points, but every purposed change seem to break more than it fixes. After a lot of deliberating, we think we have this debate figured out, and would like to purpose a new, elegant solution.

## Understanding the problem

Radians are a unit for measuring angles. An angle in radians is the length of the arc along a circle with radius $1$. The circumference of such circle is $2\pi$, and this results in $2\pi$ being the length of a full turn. With this definition, a single turn becomes twice the circle constant. This is absurd, and evidently this factor of $2$ haunts us throughout mathematics. To fix this, some suggested that we should redefine $\pi$ to be $2\pi$, but this is not the solution.

$\pi$ is the all important circle constant. It has historical and cultural significance, and it is here to stay. Changing it is unreasonable, and most importantly unnecessary, since:

The proper way to define angles is to use the arc of a unit circle with a diameter of length $1$. The circumference of this unit circle is exactly $\pi = 3.1415926...$. An angle $\alpha$ would be defined as the length of the arc along this circle. This is a simpler and much more elegant definition for angles. One immediate consequence is a quarter of a turn being $\frac{\pi}{4}$, and the same is true for all parts of a circle.

We will call this system for measuring angles "darians". Instead of the $rad$ symbol defining angles in radians, we would use the Greek letter Digamma $\digamma$. For example, half a turn would be written as $\frac{\pi}{2} \digamma$.

Figure 1 : Some special angles, in darians.

## The diameter is the fundamental property of a circle

The reason darians are better measure for angles is that the diameter is the fundamental property of the circle. No other shape is described by distance from its center. Imagine if we were to describe a square by its radius, it would make no sense! The radius of a circle is not even a directly measurable property. The diameter is. It's a measurable property that is representative of the size of a shape. This is why the standard definition of $\pi$ is: $$\pi = \frac {Circumference}{Diameter}$$

Which holds for any circle.

Looking at the diameter clears up a whole lot of calculations. For instance, the circumference of a unit circle is exactly $\pi$. Similarly, the surface area of a unit sphere (i.e. a sphere with diameter $1$) is exactly $\pi$. Looking at the diameter clears up both definition of length and surface: $$C_{2d} = d \cdot \pi \ \ ; S_{3d} = d^2\cdot \pi$$ Where $d$ is the diameter, $C$ is the circumference of a $2d$ sphere (a circle) , and $S$ is the surface area of a $3d$ sphere.

Figure 2 : A unit sphere has a diameter $1$ and $\pi$ surface area.

## Calculus agrees

Using this definition for angles, we get a better definition for $\sin (x)$ that uses darians. For example $\sin (0)=\sin(\pi) = 0$. Additionally, this redefinition results in some beautiful calculus identities: $$\frac{d}{dx} \sin(x) = 2\cos(x)$$ $$e^{ix} = \cos \left(\frac{x}{2} \right)+i\sin\left(\frac{x}{2}\right)$$ furthermore, the new $\sin$ function also has a beautiful Taylor Series: $$\sin(x) = 2x - \frac{(2x)^3}{3!}+\frac{(2x)^5}{5!}-...= \sum_{n=0}^\infty \frac{(2x)^{2n+1}}{(2n+1)!}(-1)^n$$ Notice how these factors of 2 were hiding all along in our Radians notation. You can differentiate $e^{ix}$ and see for yourself that both definition give the same result. You would also notice that $e^{i\pi} = -1$.

## Factors of 2

Radians insistence that a full turn is $2\pi$ results in factors of 2 hiding in notation all over maths. A way to fix it is to redefine the standard exponential function to include those factors. This redefinition lets us introduce a new constant: $$e^{2(ix)} = \cos(x) + i \sin(x) = l^{ix}$$ Setting $l = e^2$. This is a useful definition, and since this constant appeared in some of the work done by the French mathematician Adrien-Marie Legendre, it is appropriate to name it Legendre's Constant. Similar to our new $\sin$ definition Legendre's constant has the property $\frac{d}{dx} l^x = 2l^x$.

## Cauchy's residue

We can use $l$ to calculate Cauchy's residue integral in the complex plain. For $z_0 \in \mathbb{C}$ we will calculate the contour integral along a unit circle (with diameter 1) centered around $z_0$. We take the integral along a path $\gamma$ setting $z(t) = z_0 + \frac{1}{2} l^{it}$, and $\frac{dz}{dt} = \frac{1}{2}2il^{it}$: $$\oint_\gamma \frac{1}{z-z_0}dz = \int_0^\pi \frac{1}{z-z_0}\frac{dz}{dt}dt =\int_0^\pi \frac{\frac{1}{2} 2il^{it}}{\frac{1}{2}l^{it}}dt = \int_0^\pi 2i dt = 2i\pi$$ This result connects $3$ fundamental constants of nature: $2$, $i$ and $\pi$.

## Gaussian redemption

A standard Gaussian distribution is usually written as $\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}$. The reader should notice the horrendous $\sqrt{2\pi}$ factor. The Gaussian distribution has the property that for every real $\sigma$ the integral of the distribution over the real number line equals $1$. We would derive this result using darians, and see that the $2 \pi$ factor shouldn't have been there in the first place.

We begin by finding the value of $\int_{-\infty}^{\infty} e^{-x^2} dx$. This is done by squaring the integral: $$\left(\int_{-\infty}^{\infty} e^{-x^2} dx \right)^2 = \int_{-\infty}^{\infty} e^{-x^2} dx \cdot \int_{-\infty}^{\infty} e^{-y^2} dy = \iint_\mathbb{R^2} e^{-x^2 -y^2} dA$$

Using the proper darian Jacobian $dA=2rdrd\theta$ and simple substitution $u=r^2$ with $du=2rdr$ we get: $$ֿ\int_0^{\infty}\int_0^\pi e^{-r^2}2rdrd\theta=ֿ\int_0^{\infty} e^{-u}du \int_0^\pi d\theta = \pi$$

This result encourages us to define the standard Gaussian as $\frac{1}{\sqrt{\pi\sigma^2}}e^{-\frac{x^2}{\sigma^2}}$ fixing the historical fascination with $\sqrt{2\pi}$ and resulting in: $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{\pi \sigma^2}} e^{-\frac{x^2}{\sigma^2}} dx = 1$$

A possible alternative definition uses Legendre's Constant that was mentioned above to finally arrive at a standard Gaussian distribution: $$\frac{1}{\sqrt{2\pi\sigma^2}}l^{\frac{-x^2}{(2\sigma)^2}}$$

## Hyperbolic trigonometric functions

In the darian system, the hyperbolic trigonometric functions will, of course, be defined with Legendre's constant: $$\sinh(x) = \frac{l^x-l^{-x}}{2}$$ This function has an intriguing property: $$\frac{d^2}{dx^2}\sinh(x) = 4\sinh(x)$$ We aren't sure what this property represents, but maths is about exploring the unknown. We encourage our readers to look into these functions and find interesting patterns. In case you find anything of note, you are more than welcome to contact us.

## Closing points

As we have shown, darians clear up our understanding of circles, angles, trigonometry and calculus. The use of darian is superior to radians even in complex analysis and probability theory, and this is just the tip of the iceberg.

But don't worry, radians, $2 \pi$ and $e$ aren't going away. You can still use them, and they will probably keep appearing in text books in the foreseeable future. But darians are the better way to go, and you should hop on board and try it for yourself.

Figure 3 : In darians, an eighth of a pie is $\frac{\pi}{8}$.