217: e to the pi Minus pi

Explain xkcd: It's 'cause you're dumb.
Jump to: navigation, search
e to the pi Minus pi
Also, I hear the 4th root of (9^2 + 19^2/22) is pi.
Title text: Also, I hear the 4th root of (9^2 + 19^2/22) is pi.


e is a mathematical constant roughly equal to 2.71828182846. π is another, roughly equal to 3.14159265359. Both are transcendental numbers.

The first panel discusses eππ, which is around 19.999099979 — very close to 20. Black Hat explains how he tricked a programming team into believing that eππ really equals 20 - instead of just being weirdly close - thus that any noticeable deviation from 20 results from errors in the code. This made them waste a lot of time trying to find a nonexistent bug until they realized that Black Hat was lying (clearly they had not known him for very long).

Floating point numbers are how computers store non-integer real numbers as decimals — or rather, in most cases, approximate them: infinite amounts of data would be required to represent most numbers in decimal form (exceptions are whole numbers and terminating decimals). The "floating-point handlers" would be the code performing the eππ calculation.

ACM is the Association for Computing Machinery, sponsoring the International Collegiate Programming Contest.

Some random facts about the math here:

  • eππ is an irrational number, but this is not a trivial fact. It was proven by Yuri Valentinovich Nesterenko in the late 20th century.
  • The mysterious almost-equation is believed to be a mathematical coincidence, or a numerical relationship that "just happens" with no satisfactory explanation. It can be rearranged to (π + 20)i ≈ −1, so cos(ln(π + 20)) ≈ −1. Piling on a few more cosines gives cos(π cos(π cos(ln(π + 20)))) ≈ −1, which is off by less than 10−35!

The title text pokes fun at another coincidence: ∜(9² + 19²/22) ≈ 3.1415926525, close to π (deviating only in the 9th decimal place). The humor comes from the fact that π is transcendental. Transcendental numbers are numbers that cannot be expressed through basic arithmetic with integers; one cannot end up with the exact value for any transcendental number (including π) by adding, subtracting, multiplying, dividing, exponentiating, and/or taking the nth root of any rational number, meaning the title text cannot possibly be true.

A much later comic, 1047: Approximations, puts forth quite a few more mathematical coincidences.


Cueball: Hey, check it out: eπ − π is 19.999099979. That's weird.
Black Hat: Yeah. That's how I got kicked out of the ACM in college.
Cueball: ...what?
Black Hat: During a competition, I told the programmers on our team that eπ − π was a standard test of floating-point handlers -- it would come out to 20 unless they had rounding errors.
Cueball: That's awful.
Black Hat: Yeah, they dug through half their algorithms looking for the bug before they figured it out.

comment.png add a comment! ⋅ comment.png add a topic (use sparingly)! ⋅ Icons-mini-action refresh blue.gif refresh comments!


Asserting that the programmers' algorithms truncated to three decimal digits is an unsupported and unnecessary extrapolation. Most floating-point implementations use binary, not decimal, and 19.999099979 looks very much like a rounding error in binary floating-point that has accumulated over several operations. Daddy (talk) 12:39, 29 April 2013 (UTC)

Fixed. Xhfz (talk) 22:57, 16 August 2013 (UTC)

The third bullet-point above needs changing... (9^2+(19^2/22))=97.4090909091 which is close to pi to the fourth power, so it should be (as noted in the text) (9^2+(19^2/22))^1/4 Squirreltape (talk) 19:27, 25 February 2014 (UTC)

Actually, in-case you didn't notice, it says "∜(9² + 19²/22)", not just the sum on its own. I checked the sum on my calculator, and it is equal to what the page is saying. "∜(9² + 19²/22)" means "4th root of (9^2+19^2/22)" (What the title text is saying), or on Windows Calculator, "(9^2+19^2/22) yroot(4)" (Basically what the sum is saying). So, the 3rd bullet point is correct. --Katavschi (talk) 22:48, 23 April 2014 (UTC)

It says above that (π + 20)^i ≈ -i, but this should be (π + 20)^i ≈ -1. Proof: π + 20 ≈ e^π => (π + 20)^i ≈ (e^π)^i = e^(πi) = -1.

The ACM competitions are famous for being under tight time pressure. Making your own team waste time would absolutely get you kicked out (and make enemies) Mountain Hikes (talk) 04:40, 23 September 2015 (UTC)

"If they thought about the mathematics"

hm, are you saying it is obvious that e^ pi - pi is not 20? How would you know without approximating it? The sum of two irrationals is not necessarily irrational. 01:58, 26 October 2015 (UTC)

approximate e^pi using slightly bigger numbers than e and pi (say e: 2.7183 and pi: 3.1416) and subtract a value that is slightly smaller than pi (say 3.1415). The result is less than 20 and a upper limit for e^pi - pi 19:59, 22 August 2016 (UTC)