$\large\tau$ is defined as $2\pi$ and if you don't know what $\pi$ is, its not the sweet treat, I can assure you.

$\pi$ is defined as the ratio of the circumference of a circle to its diameter. This means - $\pi = \dfrac{\text{circumference}}{\text{diameter}}$

$\pi$ Visualized :)

Now, back to $\large\tau$, it is $2\pi$. This means - $\tau = \dfrac{2 \times \text{circumference}}{\text{diameter}} \ \text{or} \ \dfrac{\text{circumference}}{\text{radius}}$

The first $10$ digits of $\pi$(sometimes written as 'pi' and pronounced 'pie') go like this - $3.141592653 \ldots$

And the first $10$ digits of $\large\tau$(sometimes written as 'tau' and pronounced 'tau') are twice that - $6.283185307 \ldots$

Since $\large\tau$ is $2 \times$ an irrational number($\pi$), it is also irrational.

But here comes the big question, why waste another Greek Letter, to define it as $2$ times an already existing number?

We tried to be considerate towards the people who dislike pies?

WRONG

It was because we being humans, wanted everything to be easier. Once we started doing Trigonometry, $\pi$ started to get annoying. $\pi$ Radians was only $180^{\circ}$. But what's the point of trying to use something that's only half of the circles angle, when a full $360^{\circ}$ would be much easier. The point is, we needed a new number that made Trigonometry with unit circles, easy and we decided to use yet another greek letter for it - $\large\tau$

Since $\large\tau$ Radians would be $2 \times 180^{\circ} = 360^{\circ}$, it made all our calculations, graphs and Trigonometry as a whole, easier.

Fun Fact : Before $360^{\circ}$ in Radians was given the name Tau, it was called a Turn, until $2010$, when Micheal Hartl proposed to call it $\large\tau$.

Let's talk some more about $\large\tau$

$\hspace{1px}$

More about $\large\tau$

$\large\tau$ is the $19$th letter of the Greek Alphabeet and has a value of $300$ in Greek Numerals.

Hi, I'm Tau and this is Capital Tau. I am $2\pi$ even though I look like half of $\pi$. Capital Tau looks like T, but don't tell him that, he has very sensitive feelings :)

The Greek Letter is used in many aspects of Math and Science, some of which are listed below -

• Used to represent the Golden ratio (1.618...), although $\phi$(phi) is more common.

• Represents Divisor function in number theory, also denoted $\text{d}$ or $\sigma_{0}$.

• Tau is an elementary particle in particle physics.

• Tau in astronomy is a measure of optical depth, or how much sunlight cannot penetrate the atmosphere.

• It is the symbol for tortuosity in hydro-geology.

$\hspace{1px}$

Advantages of $\large\tau$

$\pi$ is found in many of the famous formula's and identity. But sometimes, $\pi$ is preceded by a $2$, and we all know by now, that $2\pi = \large\tau$.

Here are the well known formula's where $2\pi$ can be replaced with $\large\tau$, making the formula simpler and more elegant -

Integral over space in polar coordinates -

• $\displaystyle\int_{\color{#D61F06}{2\pi}}^{0} \int_{\infty}^{0} f(r,θ) \ r \ dr \ dθ$

Gaussian (normal) Distribution -

• $\dfrac{1}{\sqrt{\color{#D61F06}{2\pi}}\sigma} \text{e}^{\dfrac{(x-\mu)^{2}}{2 \sigma^{2}}}$

$N^{th}$ roots of unity -

A picture of Tau celebrating its many advantages

• $z^{n} = 1 ⇒ z = e^{\dfrac{\color{#D61F06}{2\pi} \color{#333333}i}{n}}$

Cauchy's Integral Formula -

• $f(a) = \dfrac{1}{\color{#D61F06}{2\pi} \color{#333333} i} \displaystyle\oint_{\gamma} \dfrac{f(z)}{z-a} dz$

Fourier Transform -

• $f(x) = \displaystyle\int_{- \infty}^{\infty} F(k) e^{\color{#D61F06}{2\pi} \color{#333333} i k x} dk$

• $F(k) = \displaystyle\int_{- \infty}^{\infty} f(x) e^{- \color{#D61F06}{2\pi} \color{#333333} i k x} dx$

Riemann Zeta Function -

• $\varsigma (2n) = \displaystyle\sum_{k = 1}^{\infty} \dfrac{1}{2^{kn}} \\ \\ \ \text{ } \ \ \ \ \ \ \ \ = \dfrac{|B_{2n}|}{2(2n)!} (\color{#D61F06}{2π} \color{#333333})^{2n}, \text{ } n = 1,2,3,......$

The Kronecker limit formulas -

• $E(\tau ,s)={\pi \over s-1}+\color{#D61F06} 2\pi \color{#333333} (\gamma -\log(2)-\log({\sqrt {y}}|\eta (\tau )|^{2}))+O(s-1)$

• $E_{{u,v}}(\tau ,1)=-\color{#D61F06} 2\pi \color{#333333} \log |f(u-v\tau ;\tau )q^{{v^{2}/2}}|$

I have highlighted the $2\pi$'s in red. If replaced with $\large\tau$, these formula's could be simpler. This is only one of the advantages of $\large\tau$.

$\hspace{1px}$

Another advantage that is very useful is in Trigonometry.

When dealing with Unit Circles in Trigonometry, we usually use Radians as our angle measure. As already stated, $\pi$ radians is a $180^{\circ}$ while $\large\tau$ radians is a full turn or $360^{\circ}$. Due to this, using $\large\tau$ is much easier.

When we use $\pi$, $\dfrac{1}{12}$ of the unit circle is actually $\dfrac{\pi}{6}$ Radians, as shown in the picture below. This is cause for major confusion.

But if we use $\large\tau$ all our problems vanish, and everything makes sense again. $\dfrac{1}{12}$ of the unit circle is $\dfrac{\large\tau}{\normalsize 12}$ radians, as shown in the picture below.

This really shows that $\large\tau$ might actually be better to use than $\pi$. Yet, we all have our weaknesses, so let's look at the Disadvantages of $\large\tau$ next.

$\hspace{1px}$

Representing $\large\tau$ using Limits

• $\large\tau \normalsize = 32 \displaystyle\lim_{n\to\infty} (n+1) \prod_{k=1}^{n} \dfrac{k^{2}}{(2k + 1)^{2}}$

$\large \text{Comment more representations that I can add here!}$

$\hspace{1px}$

$\large\tau$(Tau) vs $\pi$(Pi) Conflict

The Pi vs Tau Conflict is a very long conflict between people who love $\pi$ and people who love $\large\tau$.

It is a battle to find which of the $2$ irrational numbers is better and makes our calculations easier.

An illustration of the epic battle :)

The basic arguments given by both teams were as follows -

$\hspace{1px}$

$\large \underline{\text{Team Tau against }\pi}$

$\pi = \dfrac{\text{Circumference}}{\text{Diameter}}$ while $\large\tau = \normalsize \dfrac{\text{Circumference}}{\text{Radius}}$

A circle is usually defined by its radius and the radius is something that mathematicians are generally more interested in than the diameter.

The formula for a circle's circumference is also simplified with $\large\tau$

$C = \large\tau\normalsize r$

$\large \tau$ Radians is equal to $360^{\circ}$, while $\pi$ Radians is equal to $180^{\circ}$. This value of Pi makes it a bit confusing in dealing with Trigonometry using Unit Circles, but Tau makes it look like easy as pie (pun intended)

$\hspace{1px}$

$\large \underline{\text{Team Pi against }\large \tau}$

Pi has been around since a long time. The number Tau also has its flaws. For example - The formula for the area of a circle is made even more complex by using $\large\tau$

$A = \pi r^{2} = \dfrac{\large\tau\normalsize r^{2}}{2}$

Also, redefining the circle constant differently and replacing it will destroy the beautiful Euler Identity - $e^{i\pi} + 1 = 0$

$\hspace{1px}$

This conflict is as long as the amount of digits in Pi and Tau. I will not take sides and either number isn't necessarily better. This is only a fragment of the conflict. This conflict has been around for a few years and hasn't officially ended yet.

$\hspace{1px}$

External Links

• The Tau Manifesto

• The Pi Manifesto

• Tau on Fandom

• Why Pi matters

• Turn, now called Tau - Wikipedia

• Pi vs Tau - Khan Academy

$\hspace{1px}$

Related Wikis on Brilliant

• Pi

• circles - area

• circles - circumference

• circle geometry properties

• sigma and pi bonds

$\hspace{1px}$

Try some problems related to $\pi$ and $\large \tau$

• Geometry Circles Practice - Quizzes and Problems

• Circles and Tangent Problem by Lolly Lau

• Pi vs Tau Problem by Percy Jackson(Me)

$\hspace{1px}$

More Digits

These are the first $100$ digits of Tau, but if you want more, go here -

$6.2831853071795864769252867665590057683943387987502116419498891846156328125724179972560696506842341359 \ldots$

$\hspace{1px}$

Rate this Wiki

Thank you for your response! We are sorry for whatever bothers you, please let us know about it in the comments.

Thank you for your response! Please tell us in the comments what we can do better next time!

Thank you for your response! Please share your thoughts in the comments!

Thank you for your response! Please tell us in the comments what we can do better for that perfect rating :)

Thank you for your response! Glad you love the wiki!

Image Credits - Thanks to https://www.wikipedia.org/, for the 'Visualising Pi' image, the 'capital and small tau' image and the 'Pi vs Tau' image and to https://tauday.com/tau-manifesto for the Unit Circle images in 'Advantages of Tau'.

Cite as : Tau, Brilliant.org. Retrieved from https://brilliant.org/discussions/thread/tau/