1047: Approximations

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Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.
Title text: Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.


Per wikipedia: A twin prime is a prime number that differs from another prime number by two.

“Rent Method” refers to the song “Seasons of Love” from the musical “Rent.” The song asks, “How do you measure a year?” One line says “525,600 minutes” while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one.

Incidentally, 75^4 overstates the number of seconds in a year by 29 hours.

Jenny’s Number = (867)-5309, “please don’t change your number on me” But since this is *explain*xkcd, let’s add that it’s from a song by Tommy Tutone: [1]

The complicated formula for the White House switchboard yields 0.2024561415. (202) 456-1415 was (at least during the Bush administration) the phone number for the White House switchboard.

All these approximations actually work astonishingly well. There are re-occurring math jokes along the lines of, “3/5 + pi/(7-pi) – sqrt(2) = 0, but your calculator is probably not good enough to compute this correctly”, which are mainly used to troll geeks. Those interested in number theory may easily compute that sqrt(2) is not even algebraic in the quotient field of Z[pi], which disproves the equality.

Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”.

Randall makes fun of both of these, using rather strange approximations (honestly: you may handle 22/7, but who can calculate in a sensible way with 99^8, let alone 30^(pi^e)?) to calculate some constants that are easy enough to handle in the decimal system, and stating such “slightly wrong” trick equations, one of which *is* actually correct (which may astonish only those who are not familiar with cosines).

Jenny’s number and the White House switchboard have already been explained. The other constants are either self-explanatory or simple physical constants that have decimal values you may google in about 1 sec. I am not going to explain what these constants mean. Three things to note: near the bottom, there is a constant which you may easily confuse for a 9. Instead, it’s a g, the standard downward acceleration due to gravity on Earth.

Now to the accuracy values: The fine structure constant is 0.007297 something, which is approximately 1/137. Randall’s point here is that 1/137 is not a very useful approximation: Firstly, 0.007297 is input into a calculator as fast as 1/137, and it’s more accurate. Secondly, if you do not have a calculator, 1/137 earns you nothing, for 137 is a prime and therefore does not ease further computation. That’s why Randall stated he’s had enough of this crap.

The ruby laser wavelength varies because “ruby” is not clearly defined.

The mean earth radius varies because there is not one single way to make a sphere out of the earth. Theoretically, it should be possible to measure the distance from the center for any point on the Earth’s surface and compute the mean integral, but practically it’s not, so geodesy has defined some sets of radii to take the mean of, yielding different mean radii. If you are interested in details, ask Wikipedia. Randall’s value lies somewhere in between, thus actually being a possible definition for Earth’s mean radius.

The image text gives more or less useful information. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.

Pi is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it’s not supposed to occur anywhere in an exponent (unless you deal with complex numbers). Thus a sensible use of the pi-th root would be as if you found an English-speaking extraterrestrian community – not really impossible, but of such low probability that nobody would believe you.

Same goes for the e-th power: e only appears in the basis of a power, not in the exponent.


A table of slightly wrong equations and identities useful for approximations and/or trolling teachers. (Found using a mix of trial-and-error, Mathematica, and Robert Munafo's Ries tool.) All units are SI MKS unless otherwise noted.

  • Relation: One light year(m) ~= 99^8

Accurate to within: one part in 40

  • Relation: Earth Surface(m^2) ~= 69^8

Accurate to within: one part in 130

  • Relation: Ocean's volume(m^3) ~= 9^19

Accurate to within: one part in 70

  • Relation: Seconds in a year ~= 75^4

Accurate to within: one part in 400

  • Relation: Seconds in a year (rent method) ~= 525,600 x 60

Accurate to within: one part in 1400

  • Relation: Age of the universe (seconds) ~= 15^15

Accurate to within: one part in 70

  • Relation: Planck's constant ~= 1/(30^pi^e)

Accurate to within: one part in 110

  • Relation: Fine structure constant ~= 1/140

Accurate to within: [I've had enough of this 137 crap]

  • Relation: Fundamental charge ~= 3/(14 * pi^pi^pi)

Accurate to within: one part in 500

  • Relation: White House Switchboard ~= 1/(e^((1 + 8^(1/(e-1)))^(1/pi)))
  • Relation: Jenny's Constant ~= (7^(e/1 - 1/e) - 9) * pi^2

Intermission: World Population Estimate which should stay current for a decade or two:

Take the last two digits of the current year

Example: 20[14]

Subtract the number of leap years since hurricane Katrina

Example:14 (minus 2008 and 2012) is 12

Add a decimal point

Example: 1.2

Add 6

Example: 6 + 1.2

7.2 ~= World population in billions.

Version for US population:

Example: 20[14]

Subtract 10

Example: 4

Multiply by 3

Example: 12

Add 10

Example: 3[22] million

  • Relation: Electron rest energy ~= e/7^16 Joules

Accurate to within: one part in 1000

  • Relation: Light-year(miles) ~= 2^(42.42)

Accurate to within: one part in 1000

  • Relation: sin(60 degrees) = (3^(1/2))/2 ~= e/pi

Accurate to within: one part in 1000

  • Relation: 3^(1/2) ~= 2e/pi

Accurate to within: one part in 1000

  • Relation: gamma(Euler's gamma constant) ~= 1/(3^(1/2))

Accurate to within: One part in 4000

  • Relation: Feet in a meter ~= 5/(pi^(1/e))

Accurate to within: one part in 4000

  • Relation: 5^(1/2) ~= 2/e + 3/2

Accurate to within: one part in 7000

  • Relation: Avogadro's number ~= 69^pi^(5^(1/2))

Accurate to within: one part in 25,000

  • Relation: Gravitational constant G ~= 1/(e^((pi - 1)^(pi + 1)))

Accurate to within: one part in 25,000

  • Relation: R(gas constant) ~= (e+1) * (5^(1/2))

Accurate to within: one part in 50,000

  • Relation: Proton-electron mass ratio ~= 6*pi^5

Accurate to within: one part in 50,000

  • Relation: Liters in a gallon ~= 3 + pi/4

Accurate to within: one part in 500,000

  • Relation: g ~= 6 + ln(45)

Accurate to within: one part in 750,000

  • Relation: Proton-electron mass ratio ~= (e^8 - 10)/phi

Accurate to within: one part in 5,000,000

  • Relation: Ruby laser wavelength ~= 1/(1200^2)

Accurate to within: [within actual variation]

  • Relation: Mean Earth Radius ~= (5^8)*6e

Accurate to within: [within actual variation]

Protip - not all of these are wrong:

  • 2^(1/2) ~= 3/5 + pi/(7-pi)
  • cos(pi/7) + cos(3pi/7) + cos(5pi/7) ~= 1/2
  • gamma(Euler's gamma constant) ~= e/3^4 + e/5
  • 5^(1/2) ~= (13 + 4pi)/(24 - 4pi)
  • sigma(1/n^n) ~= ln(3)^e
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They're actually quite accurate. I've used these in calculations, and they seem to give close enough answers. Davidy22[talk] 14:03, 8 January 2013 (UTC)

I only see a use for the liters in a gallon one. The rest are for trolling or simple amusement. The cosine identity bit our math team in the butt at a competition. It was painful. --Quicksilver (talk) 05:27, 17 August 2013 (UTC)

Annoyingly this explanation does not cover 42 properly, it does not say that Douglas Adams got the number 42 from Lewis Carroll, who is more relevant to the page because he was a mathematician named Charles Lutwidge Dodgson. He was obsessed with the number forty-two. The original plate illustrations of Alice in Wonderland drawn by him numbered forty-two. Rule Forty-Two in Alice in Wonderland is "All persons more than a mile high to leave the court", There is also a Code of Honour in the preface of The Hunting of the Snark, an extremely long poem written by him when he was 42 years old, in which rule forty-two is "No one shall speak to the Man at the Helm". The queens in Alice Through the Looking Glass the White Queen announces her age as "one hundred and one, five months and a day", which - if the best possible date is assumed for the action of Through the Looking-Glass - gives a total of 37,044 days. With the further (textually unconfirmed) assumption that both Queens were born on the same day their combined age becomes 74,088 days, which is 42 x 42 x 42. -- 02:43, 29 August 2013 (UTC)

This explanation covers 42 adequately, and would probably be made slightly worse if such information were added. The very widely known cultural reference is to Adams's interpretation, not Dodgson's original obsession. Adding it would be akin to introducing the MPLM into the explanation for the hijacking of Renaissance artists' names by the TMNT. I definitely concede that it does not cover 42 exhaustively, but I think it can be considered complete and in working order without such an addition. If it really irks you, be bold and add it! --Quicksilver (talk) 00:37, 30 August 2013 (UTC)

"sqrt(2) is not even algebraic in the quotient field of Z[pi]" is not correct. Q is part of the quotient field of Z[pi] and sqrt(2) is algebraic of it. The needed facts are that pi is not algebraic, but the formula implies it is in Q(sqrt(2)). --DrMath 06:47, 7 September 2013 (UTC)

13/15 is a better approximation to sqrt(3)/2 than is e/pi. Continued fraction approximations are great! --DrMath 07:23, 7 September 2013 (UTC)

How could he forget 1 gallon ≈ 0.1337 ft³?! 00:51, 8 September 2013 (UTC)

Worth mentioning that Wolfram Alpha now officially recognizes the White House switchboard constant and the Jenny constant. 18:28, 8 October 2013 (UTC)

Maybe we should add the [Extension:LaTeXSVG LaTeX extension] to make it easier to transcribe these equations. -- 23:02, 16 December 2013 (UTC)

Protip - Does anyone see the correct equation?

Maybe this is just an other Wolfram Alpha error, like we recently have had here: 1292: Pi vs. Tau. All equations still look invalid to me.

  • √2 = 3/5 + π/(7-π): is impossible because √2 is an irrational number and no equation can match.
  • cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2: could only match if cos(x) + cos(3x) + cos(5x) = 1/2 would be valid, because π/7 is also an irrational number.
  • γ = e/34 + e/5 or γ = e/54 + e/5: would mean that a sum of two irrational numbers do fit to the Gamma Constant. Impossible.
  • √5 = 13 + 4π / 24 - 4π: √5 and π are irrational numbers, there is no way to match them in any equation like this.
  • Σ 1/nn = ln(3)e: doesn't make any sense either.

Maybe Miss Lenhart can help. --Dgbrt (talk) 21:41, 17 December 2013 (UTC)

cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 is exactly correct.

Let a=π/7, b=3π/7, and c=5π/7, then (cosa+cosb+cosc)⋅2sina=2cosasina+2cosbsina+2coscsina=sin2a+sin(b+a)−sin(b−a)+sin(c+a)−sin(c−a)=sin(2π/7)+sin(4π/7)−sin(2π/7)+sin(6π/7)−sin(4π/7)=sin(6π/7)=sin(π/7)=sina

Hence, cos(π/7) + cos(3π/7) + cos(5π/7) = sin(π/7) / 2sin(π/7) = 1/2 01:57, 16 January 2014 (UTC)

What is this: sin(6π/7)=sin(π/7) ? A new math is born... --Dgbrt (talk) 20:49, 16 January 2014 (UTC)
Actually it does. My proof is geometric: the sines of two supplementary angles (angle a + angle b = π (in radians)) are equivalent because they necessarily have the same x height in a Cartesian plane. Look on a unit circle, or even a sine function. Also, Calculus and most other mathematics use radians over degrees because they make the functions simpler and eliminate irrationality when a trig function shows up, but physics uses degrees because it's easier to understand and taught first. Anonymous 01:27, 13 February 2014 (UTC)
As an aside, just how far along in math are you? Radian measure is taught in high school (at least the good ones). Anonymous 13:24, 13 February 2014 (UTC)
Sure, I was wrong at my last statement. sin(6π/7)=sin(π/7) is correct by using the radian measure. But just change π/7 to π/77 would give a very different result on that formular here. I still can't figure out why PI divided by the number 7 should be that unique, PI divided by 77 should be the same. My fault is: I still can't find the Nerd Sniping here. And we all do know that Randall did use wrong WolframAlpha results here. According to the last question: I'm very well on Math, that's because I want to understand this. This is like 0.999=1. --Dgbrt (talk) 22:01, 13 February 2014 (UTC)
Ah, I see. I think it has to do with the way e^i*π breaks down, as one of the answers shown in the corresponding link explains, but other answers rely on various angle identities (including the supplementary sines one in the proof above). Anonymous 03:10, 14 February 2014 (UTC) (PS, have you checked 545 lately? I answered your question there, too)
As per the derivation from january 16 , you can use any a,b,c that satisfies this set of equations: 2 a = b - a, a + b = c- a, c + a = π - a. This is due to the fact that sin(x) = sin(π-x), and what was derived the 16th. 12:38, 21 February 2014 (UTC)
So, still incomplete?

Where's our (in)complete judge? 19:21, 18 December 2013 (UTC)

The protip is still a mystery. I'm calling for help a few lines above. --Dgbrt (talk) 21:16, 18 December 2013 (UTC)

The 'Seconds in a year' ones remind me of one of my favorite quotes: "How many seconds are there in a year? If I tell you there are 3.155 x 10^7, you won't even try to remember it. On the other hand, who could forget that, to within half a percent, pi seconds is a nanocentury" -- Tom Duff, Bell Labs. Beolach (talk) 19:14, 17 April 2014 (UTC)

Please do not change former discussions. --Dgbrt (talk) 23:57, 17 April 2014 (UTC)
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