$\large\tau$ 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 $\large\tau$

$\hspace{1px}$

$\large\tau$ 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.

$\hspace{1px}$

$\pi$ 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 $\large\tau$.

$\hspace{1px}$

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.

$\hspace{1px}$

$\large \text{Coming Soon!}$

$\hspace{1px}$

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

$\hspace{1px}$

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 -

$\hspace{1px}$

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

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

$\hspace{1px}$

$\hspace{1px}$

$\hspace{1px}$

$\hspace{1px}$

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

$6.2831853071795864769252867665590057683943387987502116419498891846156328125724179972560696506842341359 \ldots$

$\hspace{1px}$

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/

No vote yet

1 vote

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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

Log in to reply

Yes, a fun fact @Nathan Soliman :)

Log in to reply

what do we use tau to calculate

Log in to reply

Log in to reply

Log in to reply

How? Please comment this in Latex Club @Páll Márton

Log in to reply

Can you please update?

Log in to reply

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

Log in to reply

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

Log in to reply

Log in to reply

Also, redefining the circle constant differently and replacing it will destroy the beautiful Euler Identity e^(pi * i) +1 = 0

Actually, with tau, it becomes even better, but yea

Log in to reply

its the argument the pi-ers make. they're stupid lol

Log in to reply

Agreed 🥂

Log in to reply

'Agreed 🥂'😁

Log in to reply

Italic, I forgot my commenting standards.Agreed 🥂Log in to reply

Log in to reply

Log in to reply

Log in to reply

What conclusion does that idea lead you to though?

Log in to reply

lawl

Log in to reply

Log in to reply

Log in to reply

to do?I wouldn't be here at all 😂 What did you think I would do first? To log in in her computer? 😂😂Damn where does your mind go?

Log in to reply

Log in to reply

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

How is it?

Log in to reply

@Frisk Dreemurr

Log in to reply

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

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

Log in to reply

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

Log in to reply

Log in to reply

@Nathan Soliman

What?Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Thanks! I really need to take some time to complete this, but life isn't kind to you after 9th grade lol

Log in to reply

Log in to reply

"discussion"in the answer to life, the universe, and everythingAnd to sum it all up

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

It isn't good, but I am still laughing

Log in to reply

yeah, very sad

Log in to reply

I feel bad for anyone who experienced a divorce, even though I'm still laughing.

Log in to reply

Log in to reply

that, they turn each other their backs.As once a (smartest mthfker in the universe) man said: " Love is just chemistry that couples animals into breeding, it reaches a peak then fades away ..."

I heard he beated Zeus singlehandedly. The man's a certified god.

Log in to reply

Log in to reply

girlfriend: wait we dont have chemistry

Boyfriend: exactly

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Michael: this morning Meredith was ran over. I took her to the hospital and the doctors did everything they could and... she is going to be ok

Stanley: what is wrong with you

Another scene

Michael: during the weekend my father had a stroke

Of luck

Log in to reply

Log in to reply

Log in to reply

Erica : Why did you call me to the gym if we're just gonna sit like this?

Ryan : Because we're not working out.

'The Lazy Breakup'

--Ryan Higa 2015--

lmao

Log in to reply

Log in to reply

Log in to reply

Girlfriend: Why not

Boyfriend: Because I realised you are not right for me and that it’s time I left you

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

What grade are you in now?

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

what troubles me more is that a billion has different meanings in different countries.

Log in to reply

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

Log in to reply

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

Log in to reply

@Yajat Shamji - Look at this :)

Log in to reply

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

Log in to reply

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

Log in to reply

Log in to reply

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

Log in to reply

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 $\pi$.

Log in to reply

@Yajat Shamji

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

Log in to reply

@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 :)

DoneLog in to reply

Log in to reply

$\pi$ 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.

Log in to reply

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

Log in to reply

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

Log in to reply