epi – pi

By Phil Plait | January 31, 2007 1:26 pm

I didn’t know this! From a recent XKCD comic:

This is correct! The value of epi – pi is 19.999099979 (plus a trailing infinite list of numbers). I strongly suspect this is just a coincidence — after all, why not pie – pi (19.317565) or pie – e (19.740875)? If you pick enough numbers, one of them is bound to come out near an integer coincidentally. Hoagland bases his whole "Face on Mars" nonsense on this numerology fallacy.

It’s funny how coincidences seem like more than they are. Our brains are wired for that sort of thing.


Comments (55)

  1. monkey

    Um…I like the thoughts and I find interest in them…but….am I missing part of this….is ACM par tof the joke? Sorry….perhaps a rediculously unintelligent way to start a post link but, oh well. I gatz ta no

  2. monkey

    And to make the post even more simplistic, I made a spelling mistake. Sorry.

    should be ‘part of the joke’.


  3. Beren

    This is just the first frame of the joke (: There are two more frames. Click the link (:

  4. Randall

    Left-handed smilies making brain hurt!

  5. You may not have heard of the ‘Feynman point’ in PI- which has its own Wikipedia article:


  6. I just ran it thru Mathematica and here is the result to 100 figures:

    19.99909997918947576726644298466904449606893684322510617247010181721652594440424378488893717172543215 …

    Sometime in the 1970s, Martin Gardner presented a similar expression in an April issue of Scientific American – and the joke was that the result was an integer ( =x.99999999 ….). In those days it was difficult to find the solution to an arbitrary degree of precision. Actually the result of the expression diverged from a series of 9s after only a few digits.
    I have to go find that article!

  7. Hey, I’m 17, nearly 18, and I live near US 17, and there are approximately 17 donuts in a box of 17 donuts.


    I think not.

  8. Paul M.

    Yet another coincidence – I saw this just yesterday on Wikipedia under Gelfond’s constant. It’s even given the heading “Numerical coincidences”

  9. This is a wierd coincidence: The Riemann Hypothesis predicts a regular pattern primes might display in their distribution,which turns out to correspond with quantum properties of atoms. If you compare a strip of zeros from Riemann’s critical line, they seem to correspond to energy levels predicted by quantum physicists in the nucleus of heavy atoms.


  10. seaducer

    God, I am so stupid. This is why I am not a scientist. I have no idea what you guys are talking about. And I think life is better that way. lol

  11. David S-D

    Some coincidences are even weirder, and maybe even hint at something going on underneath: try e^(pi * sqrt(163)), for instance. I’m not a number theorist, but apparently they can tell you why it turns out the way it does. (Kuwaiti Demon — is this expression what you were referring to?)

    Also, there’s http://www.math.harvard.edu/~elkies/Misc/pi10.pdf.

  12. David S-D

    Looks like the above link included the period at the end of my sentence. It should be http://www.math.harvard.edu/~elkies/Misc/pi10.pdf

  13. Tim G

    Not purely mathematical, but astronomy related:

    light-year / astronomical unit = 63239.67
    mile / inch = 63360

    Now you can visualize vastly different astronomical distances

    Also, knowing this will save three keystrokes on your calculator and still allow greater precision:

    mile / km = 1.609344
    ln(5) = 1.609437912

  14. George

    I wondered where you were going with it and I like where you went.

    Is there a numerical version of pareidolia, say pireidolia?

  15. Mondoz

    OK Phil & Co. Please spell it out so a business major like myself can understand…

    What, exactly, is special about 19.999099979?

  16. df

    !([ExponentialE]^Ï€ – (1 + Ï€^2 + Ï€^4 +
    Ï€^7 + Ï€^8 – Ï€^10 + Ï€^17)/Ï€^16) = 20

    (to 9 decimal places). Hope it comes out ok –

  17. df

    it didn’t so here it is

    exp(pi) – (1 + pi^2+pi^2+pi^7+pi^8-pi^10+pi^17)/pi^16

  18. df

    and i made an error –

    exp(pi) – (1 + pi^2+pi^4+pi^7+pi^8-pi^10+pi^17)/pi^16

  19. Can you prove epi > pie? Is it a coincidence?

  20. I meant. e^pi > pi^e.

  21. TomPaine

    Proof: This is why mathematicians do not succeed in comedy.

  22. HvP

    “Proof: This is why mathematicians do not succeed in comedy.”

    TomPaine gets my vote for comment of the week 😛

  23. ACM = Association for Computing Machinery

    e^pi > pi ^e – Just do the math! 23.14069… > 22.45915…

    Now, that this near integer diverges as quickly as the 1/10,000s place is unimpressive 😉 I work in an industry where we measure availability of ourt systems at the 1/1000000 place (that’d be .99999x – or 99.999x % available) and worry when that 6th digit isn’t also close to 9.


  24. ourt? I don’t remember that T being there…. :0

    Obviously, ourt = our.

  25. MKR

    Cruel jokes are the best!

  26. ABR

    John B. Sandlin: Aw, I preferred ourt systems. More exotic. Random ‘t’s from the Ourt. Or is that the definition of comets? Sorry, long day.

    Being a mere BioBABlogee, I am impressed with the precision of your equipment. Here’s a bit of inanity this group may chuckle over.

    I once worked with a small group of fellow biologists in a federal chemistry lab. One day the Lab Quality Assurance officer paid my boss (also a biologist) a visit to discuss data management issues and brought up the term “significant figures”. Without blinking, my boss replied, “Oh, ALL of our figures are significant.”

  27. A fun almost integer involving pi, phi, and e is:
    ( 7 * pi ) / ( 5 * phi * e ) = 0.9999902974580796214743940244…
    where phi is, of course, “the golden ratio.”

  28. Troy

    I don’t see anything special in the number derived except the digits all are 9s or 0s. Pi itself is much more interesting in the first 10 digits you get all 10 digits. Of course this is in the base 10 number system which only appeals to us because we have 10 fingers. In base 3 or hexadecimal or octal you’d get a different pattern. e and pi are sort of opposite sides of the same coin so the fact you get a non random pattern isn’t all that stunning to me. As for Hoagland maybe we should take up a collection so he can get some meds, I mean he’s either a flake or insane or some combination of the two! I can’t believe anyone actually pays to buy stuff he wrote! I’m surprised my little joke of the ‘face’ being a ‘butte’ hasn’t caught on. Of course I always thought the Great Lakes Huron and Erie were funny because one sounds like ‘hear’ and the other like ‘ear’.

  29. Some numerical coincidences are useful!

    When I took geophysics, we would cancel out the pi that results from the earth’s orbit in various geophysical calculations by approximating a year as pi x 10^7 seconds. Obviously you don’t want the guys who do spacecraft trajectories using this trick, but it’s close enough for the order-of-magnitude type calculations that we did in undergrad planetary.

  30. Kullat Nunu

    My favorite is

    e^(i * pi) + 1 = 0

    Three basic arithmetic functions (+, *, ^), five fundamental constants (0, 1, pi, e, i [= imaginary unit, i^2 = -1]), one equation.

    And it is not a coincidence.

    Ah, Dr. Goulu already mentioned it.

  31. AntiQuest

    I (physics student) caught out a few of the Comp Sci majors with this today. Not really significant but funny to see them struggle with thier program.

    Oh and TomPaine, check out Tom Lehrer before you say mathematicians aren’t funny. Especially “That’s mathematics” (and “Werner von Braun” for the Phil’s interest…).

  32. DennyMo

    I was gonna make some pithy comment about those stupid emails that have you go through some silly math gymnastics and arrive at some result that sounds impressive or surprising. But then you all start throwing around imaginary numbers, natural logarithms, etc., makes my comment seem kinda sophomoric, so I’m gonna just keep my mouth shut…

  33. APP

    Pi itself is much more interesting in the first 10 digits you get all 10 digits. Of course this is in the base 10 number system which only appeals to us because we have 10 fingers.



    Two 1’s, two 3’s, two 5’s, no 0’s, 7’s, or 8’s.

  34. It’s a coincidence.

    Kind of like the coincidence that

    10! (=10*9*8*…*3*2*1) is exactly equal to the number of seconds in 6 weeks.

    Given that there are an infinite number of numbers, coincidences can always be found.

  35. Rob

    Sweet Lord Christ, could someone please explain what the $%*k this means to those of us who don’t get it?

    Maybe we should just accept that we’re out of our league and leave you Matharians to enjoy yourselves, but I feel I’m smart enough to play with you guys if you’ll help me understand.


  36. Quiet_Desperation

    I used to like math humor until my uncle was killed in a tragic long division accident.

  37. Bill Nettles

    Physics has interesting numerical coincidences which suggest that our standard measurements need tweaking:

    The number of seconds in a tropical year is about 3.1557 x 10^7 seconds, remarkably close to pi x 10^7. If the second was just a little “longer”, then they would match, as long as the earth doesn’t slow down.

    In SI units, the acceleration of gravity near the surface of the earth is very close to the SQRT(pi). Hmmmm…

    Finally, we know where Ol’ McDonalds Farm is: on the Argand Plain [sic] about 0.6517 units from the origin (wherever THAT is), 38.22 degress east of south. Check out e^(i^(e^(i^0))). Oh, and the imaginary component of the farm has a magnitude close to (slightly larger than) the mass-energy of the electron in MeV. Ooooo…

  38. Bill Nettles

    In Si units, the acceleration of gravity near the surface of the earth is very close to pi^2. Hmmmm. [I SQRT’d the wrong thing…that can be dangerous]

  39. Patrick

    Phil is commenting on the fact that the result is near 20, but that’s not the end of joke. Read the rest of the comic!

  40. Steve

    Phill you B*$*£”% Hadn’t been aware of that webcomic before, thanks for giving me another way to waste time at work (as if i didn’t have enough already) :)

  41. Brian

    Thanx all for some very interesting posts. In response to Samuel’s hint of a deeper reason that e^pi > pi^e, let f(x) = ln(x)/x. Then f'(x)=(1-ln(x))/(x^2).
    f'(x)=0 if and only if x=e, which yields a maximum at x=e for all positive values of x. So if A>0 and A not equal to e, f(e)>f(A)
    e^pi>pi^e is a special case.

  42. Brian


  43. Some other guy

    Does anyone know the official definition of “e”?

    I know one fact, that if you tetrate (see http://en.wikipedia.org/wiki/Tetration) e-rt(e) (The base-e root of e) to infinity, the result is e, but if you tetrate anything higher than that to infinity, be it just 10^-1000000000 higher, it goes to infinity. (Well, I actually found this out from experimentation.)

  44. Sephiroth

    e=the tangent line of f(x)=e^x where x=0

  45. Patrick

    The most explicit definition of e is e = 1 + 1/1! + 1/2! + 1/3! + …

    This is a rapidly converging sum (remember n! = n times (n-1) times … times 1).

  46. I agree with you, I think it’s just a coincidence.

    There are plenty of them if u look for them.

    Eg sqrt(10) approx = 22/7 approx = pi.

    Even though none of those are integers.

  47. adam

    Hey, no one’s mentioned the alt-text in that XKCD comic yet:

    “Also, I hear the 4th root of (9^2 + 19^2/22) is pi.”

    Someone with Mathematica try that out.

  48. For those who are pissed because they don’t get the whole fuss:

    1) Click on the image. It will take you to the actual comic. The image is only the first of three frames.
    2) The joke in a nutshell: e^pi – pi = 19.999099979, which is very very close to 20. So close that if you don’t know better, you’ll feel tempted to believe you’ve introduced some rounding error on the calculation. The math guy exploits this to drive the computer guys from ACM nuts trying to find where they were making the mistake.

  49. ElBosso

    1111,1111^2 = 1234567,87654 |*7,2
    = 8888888,71109 |sqrt (4 times)
    = 2,718335 -> nearly e^1 !
    cool, isn´t it?
    found out by myself!

  50. Bas Joosten

    In reply to Adam:
    “Also, I hear the 4th root of (9^2 + 19^2/22) is pi.”
    I have no mathematica, but I can do high precision calculations. Found out it is approximately pi up to the 9th digit. Aproximately equal is still unequal, of course.

  51. Julie

    “Hey, no one’s mentioned the alt-text in that XKCD comic yet:

    “Also, I hear the 4th root of (9^2 + 19^2/22) is pi.”

    Someone with Mathematica try that out.”

    Why the hell would you need mathematica to do that? Put it in a calculator or use Google calculator.

  52. Gerson W. Barbosa

    I’ve come up with a slightly more weird expression:

    (sqrt(2)*(pi^17-4*e^pi*(e^pi-pi)))/4 = 99999999.979

    I guess this would have worked in the XKCD comic as well :-)


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