Editing 1292: Pi vs. Tau

{{comic
| number    = 1292
| date      = November 18, 2013
| title     = Pi vs. Tau
| image     = pi vs tau.png
| titletext = Conveniently approximated as e+2, Pau is commonly known as the Devil's Ratio (because in the octal expansion, '666' appears four times in the first 200 digits while no other run of 3+ digits appears more than once.)
}}

==Explanation==
{{incomplete|Too complex, non Math people should also be able to understand this. Randalls mistake has to be emphasised, everything else here is still too much, it even doesn't belong to a trivia section. See the discussion page.}}

This is yet another of [[Randall]]'s [[:Category:Compromise|compromise comics]]. A few mathematicians argue as to whether to use pi, which is the ratio between a circle's circumference and its diameter, or tau, which is the ratio between a circle's circumference and its radius.

Most will know π (Pi) by the approximation 3.14, but not knowing τ (tau) which is just twice as large as pi. Randall is suggesting using "pau", which is a portmanteau of "pi" and "tau", as a number situated, appropriately enough, halfway between pi and tau. But of course his number would be inconvenient, as there are currently no commonly used formulas that involve 1.5 pi (or 0.75 tau).

Some consider pi as being the wrong convention and are in favor of using tau as ''the'' circle constant (see the [http://tauday.com/tau-manifesto Tau Manifesto], which was inspired by the article "[http://www.math.utah.edu/~palais/pi.html Pi is wrong!]" by mathematician Robert Palais). Others consider proponents of tau to be foolish and remain loyal to pi (see the [http://www.thepimanifesto.com Pi Manifesto]). Of course, regardless of which convention is used, the fundamental mathematics will remain unaltered. But the choice of pi vs tau can affect the clarity of equations, analogies between different equations, and how easy various subjects are to teach.

===Title text===

The title text is a bunch of slightly-incorrect mathematical [[356: Nerd Sniping|nerd sniping]] that Randall included for seemingly no better reason than trolling us. It consists of some of advanced trigonometry and other assorted college-level concepts that will in all likelihood just bore you if you don't care about them already. You can walk away right now thinking "Randall is just nerd sniping us" and still get the joke. If you REALLY want to know what all the math means, we'll try and work through it below...

"Octal expansion" refers to writing out the number in base-8. In base-8, only the numerals 0-7 are used to express numbers. This does not mean that values such as 18, 19, 28, 29, and so on do not exist; rather, said values are represented with a more limited range of numerals.

For the sake of simplicity in this next demonstration, we will only acknowledge whole numbers with positive values.

In base-8, the numbers 1 through 7 have the same values as in base-10. The next number, eight, is written out as 10. This is because the "ones" digit has run out of unique numerals to express this value, so it rolls over to the "eights" digit. Nine is 11. Ten is 12.  Numbering continues in this manner, up to fifteen (17). The "ones" digit must roll over to the "eights" digit again, so sixteen is 20. Seventeen is 21. After twenty-three (27), it rolls over again, giving us twenty-four (30). 

Counting by eights, the next numbers are thirty-two (40), forty (50), forty-eight (60), and fifty-six (70). At sixty-three (77), both the "ones" and "eights" digit has run out of unique numerals, so the excess value must roll over to the "sixty-fours" digit, giving us sixty-four (100). If we keep counting, we will eventually reach five-hundred-eleven (777). A new "five-hundred-twelves" digit is created. The next number is five-hundred-twelve (1000).

As you can see, numbers written in base-8 tend to be longer and less economical to write than in base-10, but it does serve its purpose. Trust us on this.

In this next demonstration, we will look at how to write non-integers in base-8. Again, we will acknowledge only positive values.

In base-8, all the numerals that follow the period are not known as the "decimal", but as the "octal". This is because "decimal" specifically refers to tenths, while "octal" refers to eighths.

In decimal, the first place after the periods depicts "tenths", the next place "hundredths", the next "thousandths", and so on. In octal, the first place represents "eighths", the next "sixty-fourths", the next "five-hundred-twelfths", etc.

One eighth is 0.1. Two eighths, or one fourth, is 0.2. Four eighths, or one half, is 0.4.
One sixty-fourth is 0.01. Five sixty-fourths is 0.05. Nine sixty-fourths, or one eighth plus one sixty-fourth, is 0.11.
One five-hundred-twelfth is 0.001. Five-hundred-eleven five-hundred-twelfths is 0.777.

Unfortunately, this entire lesson has a very disappointing end. As it turns out, the title text for the comic is incorrect. The first 200 digits of 'pau' in octal are:
<pre>
4.5545743763144164432362345144750501224254715730156503147633545270030431677126116550546747570313312523403514716576464333172731124310201076447270723624573721640220437652155065544220143116155742515634462
</pre>
The sequence '666' does not occur at all in it.

Possibly, [[Randall]] used [http://www.wolframalpha.com/ Wolfram|Alpha] to calculate the result (he uses it a lot, for example [http://what-if.xkcd.com/70/ What-if 70: The Constant Groundskeeper] or [http://what-if.xkcd.com/62/ What-if 62: Falling With Helium]).
However, as of November 18, 2013, there's a bug in Wolfram|Alpha so that, when getting 200 octal digits from "pau", it just calculates the decimal value rounded to 15 significant digits (this is 4.71238898038469) and expands that as octal digits as far as needed.

This gives a periodically repeating number. In the first 200 digits of the octal expansion, the sequences 666 and 6666 do occur, but each only once. There are 4 occurrences, however, in the first 300 digits:
<pre>
4.554574376314416445676661714336617116240444076666510533533077631151350452060436452476274022621206136310000177621674175071262255702044274154476005744176002676623042402346036604733130522524127534777714554305412763636566643022106616734723661726160312772574551366370203115523402704104015532221722772357666</pre>
Expansion that long indeed does contain 666 (the {{w|Number of the beast|number of the beast}}) four times (with one instance as 6666). It also contains 0000, 222, 444, and 7777, but they only appear once in a run.

{{w|Mathematical coincidence|Coincidentally}}, e+2 is also very similar to 1.5pi, although only to a few digits.
<pre>
1.5π = 4.71238898038...
e+2  = 4.71828182845...
</pre>

The "Devil's Ratio" may be an allusion to the "{{w|Tritone|Devil's Interval}}", aka the "Devil's Chord" or 'Diabolus in Musica' ('The Devil in music'), which is the name sometimes given to the harmony between a root note and its tritone/augmented fourth/diminished fifth.  This note is situated halfway between octaves, and is named for its dissonant quality.  It is possibly a cross-reference between this and the "{{w|golden ratio}}".

==Transcript==
:[On the left is a "forbidden"-style slashed circle with the π symbol, captioned "Pi". On the right is a "forbidden"-style slashed circle with 2π, captioned "Tau". In the middle it reads 1.5π, captioned "Pau".]
:A compromise solution to the Pi Tau dispute

==Trivia==
*For Pi the sequence '666' occurs for the first time at position 2440. Many more occurrences can be found here: [http://www.angio.net/pi/ The Pi-Search Page].
* Note that pau is Catalan for peace, which is a good solution for the pi/tau dispute.
* In the discussion it has been theorized that Randall used [[356: Nerd Sniping|Nerd Sniping]]. In which case he was aware of the mistake in Wolfram!
*For an entertaining introduction to the concept, see this [https://www.khanacademy.org/math/recreational-math/vi-hart/pi-tau/v/pi-is--still--wrong Vi Hart video].

{{comic discussion}}
[[Category:Comics with color]]
[[Category:Math]]
[[Category:Compromise]]