$\large\tau$ 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 - $\pi = \dfrac{\text{circumference}}{\text{diameter}}$

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?

WRONGIt 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 $\large\tau$

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

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.

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$\pi$ 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 -

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

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 $\large\tau$.

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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.

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$\large \text{Coming Soon!}$

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

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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.

The basic arguments given by both teams were as follows -

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

$\pi = \dfrac{\text{Circumference}}{\text{Diameter}}$ 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)

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

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 - $e^{i\pi} + 1 = 0$

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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.

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

$6.2831853071795864769252867665590057683943387987502116419498891846156328125724179972560696506842341359 \ldots$

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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/

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## Comments

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How? Please comment this in Latex Club @Páll Márton

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so tau is double the value of pi even though the symbol is half of pi interesting

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Yes, a fun fact @Nathan Soliman :)

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what do we use tau to calculate

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Can you please update?

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I've completed Advantages of Tau. Only Disadvantages left. It'll be done by next week :)

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No offence, but when I first read your reply, it said you replied 3 weeks, 3 days ago.>:)

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@Frisk Dreemurr - I'm making a wiki for fun :)

How is it?

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@Frisk Dreemurr

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Nice @Percy Jackson, I never noticed any note on the

debate of Tau and Piuntil today :)Log in to reply

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@Frisk Dreemurr - I added a rating system LOL :)

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@Nathan Soliman

What?Log in to reply

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@Páll Márton - I was looking at your formatting guide while making this, its really useful, thanks :)

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BTW go to my feed to see my latest note on buttons. You can add that to your guide @Páll Márton

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@Yajat Shamji - Look at this :)

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I really like it. However, I am impartial to both $\pi$ and $\tau$ so I am neutral in this war/conflict.

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I wrote that I'm neutral but I'm actually a Tauist LOL. Glad you like it :)

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@Yajat Shamji - There is an astronomy point in 'More About Tau' :)

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Thanks.

P.S. $5$ stars!

P.S.S. I challenge you to find geometric sums and limits involving $\pi$ that result in $\tau$ and geometric sums and limits involving $\tau$ that result in $\pi$.

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@Yajat Shamji

I'll try :)P.S Thanks you for 5 stars!

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@Yajat Shamji, I added a few integral formula's that I could think of. I still don't have a limit though, I'll work on that :)

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$\pi$ so you'll need to multiply the entire limit by $2$):

I found some (but the result'sLog in to reply

Pi limits

I need something where I don't need to multiply actually. There are many Pi limits -Log in to reply

@Yajat Shamji - I added a limit formula that doesn't need me to multiply the limit by 2 and do some crazy math, but I don't know its name.

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no, the beautiful Euler identity would not disappear, since e^i tau - 1 = 0 (but I am a pie fan)

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Its what they argue, its the truth. It may not be correct, but its an arguement

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