# 1184: Circumference Formula

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The {{w|circumference}} C of a {{w|circle}} is 2{{w|pi|π}}''r'', where ''r'' is the {{w|radius}} of the circle. Randall then makes a footnote about ''r'', using <sup>2</sup>. This creates a typographical ambiguity, since a superscript 2 can also be an exponent (as in ''x''<sup>2</sup>). Randall's formula now looks like a strange hybrid of the correct formula and the formula for the ''{{w|area}}'' of the circle: A = π''r'' <sup>2</sup>. | The {{w|circumference}} C of a {{w|circle}} is 2{{w|pi|π}}''r'', where ''r'' is the {{w|radius}} of the circle. Randall then makes a footnote about ''r'', using <sup>2</sup>. This creates a typographical ambiguity, since a superscript 2 can also be an exponent (as in ''x''<sup>2</sup>). Randall's formula now looks like a strange hybrid of the correct formula and the formula for the ''{{w|area}}'' of the circle: A = π''r'' <sup>2</sup>. | ||

− | The ′ and ″ that follows the r | + | The ′ and ″ that follows the r's in the title text are the {{w|prime symbol|prime}} and double prime symbols. {{w|Prime (symbol)#Use in mathematics, statistics, and science|Among other things}}, these are often used in mathematics to tell apart corresponding properties of different objects. Building up on the ambiguity theme, r′ can also be interpreted as "the first derivative of r with respect to the independent variable", r″ as the second derivative, and so on. |

In this case, ''r'' is already used to notate the radius of any circle. ''r'' ′ ("r-prime") could then be used to notate radii of things that are not circles (like the [http://www.mathsisfun.com/definitions/radius-polygon-.html radius of any regular polygon], or the radius of any {{w|sphere}}). [[Randall]] chooses the radius of {{w|Earth Prime}}. | In this case, ''r'' is already used to notate the radius of any circle. ''r'' ′ ("r-prime") could then be used to notate radii of things that are not circles (like the [http://www.mathsisfun.com/definitions/radius-polygon-.html radius of any regular polygon], or the radius of any {{w|sphere}}). [[Randall]] chooses the radius of {{w|Earth Prime}}. | ||

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"Prime" can mean first or original, and "Earth Prime" is a term sometimes used in works of fiction involving parallel universes to refer to ''our'' Earth (e.g. {{w|The Long Earth}}), or to a world with a minimum of divergence points from Earth as we know it. (The radius of Earth Prime would thus not differ much from the {{w|radius of the Earth}}; 6 371 km.) | "Prime" can mean first or original, and "Earth Prime" is a term sometimes used in works of fiction involving parallel universes to refer to ''our'' Earth (e.g. {{w|The Long Earth}}), or to a world with a minimum of divergence points from Earth as we know it. (The radius of Earth Prime would thus not differ much from the {{w|radius of the Earth}}; 6 371 km.) | ||

− | The double prime symbol (″) is also | + | The double prime symbol (″) is also ambiguous, as it can be used to designate {{w|inches}} and {{w|arcseconds}}. |

<small><sup>2</sup> These are not intended to indicate the presence of a footn... oh, nevermind.</small> | <small><sup>2</sup> These are not intended to indicate the presence of a footn... oh, nevermind.</small> |

## Revision as of 15:53, 13 March 2013

Circumference Formula |

Title text: Assume r' refers to the radius of Earth Prime, and r'' means radius in inches. |

## Explanation

The circumference C of a circle is 2π*r*, where *r* is the radius of the circle. Randall then makes a footnote about *r*, using ^{2}. This creates a typographical ambiguity, since a superscript 2 can also be an exponent (as in *x*^{2}). Randall's formula now looks like a strange hybrid of the correct formula and the formula for the *area* of the circle: A = π*r* ^{2}.

The ′ and ″ that follows the r's in the title text are the prime and double prime symbols. Among other things, these are often used in mathematics to tell apart corresponding properties of different objects. Building up on the ambiguity theme, r′ can also be interpreted as "the first derivative of r with respect to the independent variable", r″ as the second derivative, and so on.

In this case, *r* is already used to notate the radius of any circle. *r* ′ ("r-prime") could then be used to notate radii of things that are not circles (like the radius of any regular polygon, or the radius of any sphere). Randall chooses the radius of Earth Prime.

"Prime" can mean first or original, and "Earth Prime" is a term sometimes used in works of fiction involving parallel universes to refer to *our* Earth (e.g. The Long Earth), or to a world with a minimum of divergence points from Earth as we know it. (The radius of Earth Prime would thus not differ much from the radius of the Earth; 6 371 km.)

The double prime symbol (″) is also ambiguous, as it can be used to designate inches and arcseconds.

^{2} These are not intended to indicate the presence of a footn... oh, nevermind.

## Transcript

- Circumference of a circle:
- 2πr
^{2} ^{2}The circle's radius

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# Discussion

Tau x Radius, superscript 2

- Since tau is more commonly used for the Golden Ratio, that's a silly idea. 121.74.169.237 11:13, 23 September 2013 (UTC)
- You may be confusing tau with phi. I've never seen the golden ratio represented by anything other than phi. I've also never seen tau representing anything other than 2pi. 108.162.219.7 19:25, 10 May 2014 (UTC)

- Leaves one wondering what the superscript 1 refers. -- 74.215.40.250 (talk)
*(please sign your comments with ~~~~)*- It's 2
*πr*^{2},**not***τr*^{2}. —173.199.215.5 05:37, 11 March 2013 (UTC)- You're missing the point.
*τ*== 2*π*and is considered better than using*π*by some people -- 138.195.69.136 (talk)*(please sign your comments with ~~~~)*- Only for very loose definitions of "better." 71.201.53.130 14:59, 11 March 2013 (UTC)
- Whoa! Never heard about that before, but after 2 hrs or so, I think I'm getting convinced! Check this site out: http://tauday.com/ What do you think? –St.nerol (talk) 18:06, 11 March 2013 (UTC)

- You're missing the point.

- It's 2

- I think tau is pointless. Using tau what then happens to Euler's famous formula, the most beautiful equation of them all? Pi shows up in so many different ways and places in mathematics. Tau appears pretty much only in the formula for a circle's circumference. Why bother needlessly proliferating symbols? J Milstein (talk) 18:17, 11 March 2013 (UTC)
- Surface area of a sphere is 2τr^2, or if you want to get pi in there
*π*d^2 108.162.218.101 (talk)*(please sign your comments with ~~~~)*

- Surface area of a sphere is 2τr^2, or if you want to get pi in there

- I think tau is pointless. Using tau what then happens to Euler's famous formula, the most beautiful equation of them all? Pi shows up in so many different ways and places in mathematics. Tau appears pretty much only in the formula for a circle's circumference. Why bother needlessly proliferating symbols? J Milstein (talk) 18:17, 11 March 2013 (UTC)

- RE: Euler's Identity: e^(tau*i) - 1 = 0 --Max Nanasy (talk) 18:27, 11 March 2013 (UTC)
- Ok, that works J Milstein (talk) 17:05, 13 March 2013 (UTC)
- Why not just e^(tau*i) = 1. Do you routinely do 2 + 2 - 4 = 0?206.181.86.98 20:31, 13 March 2013 (UTC)
- Because:
- Symmetry wrt the original Euler's Identity (e^(pi*i) + 1 = 0)
- According to http://en.wikipedia.org/wiki/Euler's_identity#Mathematical_beauty, "in algebra and other areas of mathematics, equations are commonly written with zero on one side of the equals sign."

- --Max Nanasy (talk) 00:35, 14 March 2013 (UTC)
- I think Euler only did that because he disliked negative numbers. It really is less a deal than people make of it.206.181.86.98 03:02, 15 March 2013 (UTC)

- Because:
- The tau variant of Euler's identity above,
*e^(tau*i)=1*, appears to miss the point. Normally, a positivt number to the power of any real number is positive. Thus*i*could be any normal number. Well, not any number.*i*could be 0 and the equation will hold. With pi however,*e^(pi*i)=-1*,*i*must be magical. /David A 141.101.80.111 23:53, 9 November 2013 (UTC)

- The tau manifesto fairly well convinced me that all occurances of π in mathematics utimately trace back from the formula C = 2
*πr*. If so, π naturally*enter*calculations as 2π. Can anyone find a counterexample to this thesis? –St.nerol (talk) 00:29, 14 March 2013 (UTC)- How could there be a counter-example? I think it is true. In complex analysis, it really should be 2π, and thus Gaussian integrals. And then number theory applications. Even this neat result really stems from trig identities, so it really is a result for 2π. 206.181.86.98 02:59, 15 March 2013 (UTC)

- RE: Euler's Identity: e^(tau*i) - 1 = 0 --Max Nanasy (talk) 18:27, 11 March 2013 (UTC)

- From what I understand, the thesis from the tau-proponents is that 2*pi is the fundamental natural constant, and that virtually
*every time*that pi shows up without the factor 2, there originally was a factor two that was cancelled out. –St.nerol (talk) 01:53, 12 March 2013 (UTC) - For everyone who suddenly started a debate about 2pi and tau: http://xkcd.com/1292/ 108.162.222.61 06:58, 4 May 2014 (UTC)

Not completely sure Earth Prime is from Sliders, but it's true it's the only one named exactly that ... -- Hkmaly (talk) 09:54, 11 March 2013 (UTC)

There's also a Prime Earth now. Just so DC can screw with us. Hogtree Octovish (talk) 10:40, 11 March 2013 (UTC)

I still don't get it.49.176.102.213 12:41, 11 March 2013 (UTC)

- If you don't get it, you don't need to get it J Milstein (talk) 18:07, 11 March 2013 (UTC)

Well, that was lame. --87.122.60.227 17:19, 11 March 2013 (UTC)

This comic illustrates the strategy of "The Unconsummated Asterisk", from the essay "Mathmanship" by Nicholas Vanserg (available at [1]).

The other side of the asterisk gambit is to use a superscript as a key to a real footnote. The knowledge‐seeker reads that S is – 36.7^{14}calories and thinks "Gee what a whale of a lot of calories" until he reads to the bottom of the page, finds footnote 14 and says "oh."

For bonus points, Randall could have used also "Pi-Throwing":

For example every schoolboy knows what π stands for so you can hold him at bay by heaving some entirely different kind of π into the equation. The poor fellow will automatically multiply by 3.1416, then begin wondering how a π got into the act anyhow, and finally discover that all the while π was osmotic pressure. If you are careful not to warn him, this one is good for a delay of about an hour and a half.Chymicus (talk) 19:01, 11 March 2013 (UTC)

- Another good one is π as a symbol for profit in financial discussions. -DrGaellon (talk | contribs) 23:23, 25 March 2013 (UTC)

I believe the current description of prime as denoting derivatives is true but irrelevant. Since the area and circumference refers to geometry (not really calculus), it's more likely that the title text is referring to the common use of primes in geometry. For example, there might be two or more parallel lines that are denoted by x, x′, x′′, etc. Wikipedia also notes another geometric use of prime: "if a point is represented by the Cartesian coordinates (x, y), then that point rotated, translated or reflected might be represented as (x′, y′)." S (talk) 23:32, 11 March 2013 (UTC)

that is so wrong, i feel my mind corrupted now. -- Anarcat (talk) 23:57, 11 March 2013 (UTC)

This explanation was hillarious -- where is the up-vote button ?? Spongebog (talk)

So, where's todays comic? How many times has Randal been late?70.199.225.225 16:15, 13 March 2013 (UTC)

- Today's comic was posted just a few minutes ago. I'm anxiously awaiting its explanation as it picks on a programming language I'm not familiar with (possibly SQL). Smperron (talk) 16:33, 13 March 2013 (UTC)

It uses pseudocode. The new one is about sorting algorithms in general, not any particular language. 130.245.231.101 17:00, 13 March 2013 (UTC)

Perhaps it's just me, but did no one see the "square the circle" gag...? --128.232.142.37 09:24, 14 March 2013 (UTC) No one but you saw the square-the-circle gag, because it's not there. For it to be there, it would require this: (2πr)² J Milstein (talk) 15:31, 14 March 2013 (UTC)

This one threw me for a loop for the longest time because I learned to use πd to find circumference, not 2πr. Anyone else learn that way? (Knowing how my brain works, it is equally possible I taught myself to use πd as a shortcut, and was in fact taught 2πr by my teachers.) Boct1584 (talk) 22:20, 31 March 2015 (UTC)

I thought the joke in the title text was that primes can refer to successive derivatives. 108.162.221.150 05:05, 19 August 2015 (UTC)

Ok, but the 2nd sentence in the explanation has a grammar mistake. The sentence reads, "Randall then makes a footnote about r, using." In this case "²" is an indication for a footnote, isn't it ? 162.158.83.240 (talk) *(please sign your comments with ~~~~)*