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(xkcd) Note: ’1 year = pi x 10^{7} seconds’ is popular with physicists. For this
list, I’ve tried to stick to approximations that I noticed on my own.
A Table of Slightly Wrong Equations and Identities Useful for Approximations and/or Trolling Teachers (Found Using a Mix of TrialandError, Mathematica, and Robert Munafo’s Ries tool.) All units are SI MKS unless otherwise noted.
Scene: Three Column Chart
Over first two columns is “Relation:” and over the the third is “Accurate to within:”
Relations:  Accurate to within:  
One light year (m)  99 to the power of 8  One part in 40  
Earth Surface (m to the power of 2)  69 to the power of 8  One part in 130  
Ocean’s volume (m to the power of 2)  9 to the power of 19  One part in 70  
Seconds in a year  75 to the power of 4  One part in 400  
Seconds in a year (Rent method)  525,600 times 60  One part in 1400  
Age of the Universe (seconds)  15 to the power of 15  One part in 70  
Planck’s Constant  1/(30 to the power of (pi to the power of e))  One part in 110  
Fine Structure Constant  1/140  [I’ve had enough of this 137 crap]  
Fundamental Charge  3/(14 times (pi to the power of (pi to the power of pi))  
White House Switch Board

1/(e to the power of(the pi root of (1 + (the e1 root of 8))))  
Jenny’s Constant

(7 to the power of ((e/1)(1/e)) – 9) times pi to the power of 2  
Intermission:World Population Estimate which should stay current for a decade of two:
Take the last two digits of the current year (example: “2014” with a square drawn around the “14”. An arrow runs from that arrow down to another square with “14” in it.). Subtract the number of leap years since Hurricane Katrina. (An arrow runs from the “14” square to a new square with “12” in it. Next to that square are “2008” and “2012” stacked and an arrow running from them to the arrow between the “14” and the “12” square.) Add a decimal point. (An arrow runs from the “12” square to a new square with “1.2” in it.) Add 6. (An arrow runs from the “1.2” square to a new “1.2” square with a “6 +” to the right. To the left is an equal sign and a square with “7.2” in it.) Next to that square is “World Population in Billions.” (Next to all of that is a gray version) Version for US Population: “2014” with a square drawn around the “14.” An arrow runs to a box with “subtract 10” in it. An arrow runs to another square with the resultant “4” in it. An arrow runs to the instruction “Multiply by 3” and an arrow runs from that to a square with the product, “12,” in it. An arrow runs from that to a box with “Add 10” in it and an arrow runs from that up to a square with the result, “22,” in it. To the right of that square is a “3” and to the left is the word “Million.” 

Electron Rest Energy  (e/7 to the power of 16) times J  One part in 1000  
Lightyear (miles)  2 to the power of 42.42  One part in 1000  
Sin(60 degrees) = (square root of 3)/2 =e/pi  One part in 1000  
Square root of 3 = (2 times e/pi)  One part in 1000  
Gamma (Euler’s Gamma Constant)  1/(square root of 3)  One part in 4000  
Feet in a meter  5/(the e root of pi)  One part in 4000  
Square root of 5 = (2/3 + 3/2)  One part in 7000  
Avogadro’s Number  69 to the power of (pi to the power of (the square root of 5))  One part in 25,000  
Gravitational Constant G  1/(e to the power of (pi1 to the power of (pi +1)))  One part in 50,000  
R (Gas Constant)  (e +1) (square root of 5)  One part in 50,000  
Liters in a Gallon  3 + (pi/4)  One part in 500,000  
g  6 + ln(45)  One part in 750,000  
ProtonElectron Mass Ratio  ((e to the power of 8 ) – 10)/ Phi  One part in 5,000,000  
Ruby laser wavelength  1/(1200 to the power of 2)  [Within actual variation]  
Mean Earth Radius  (5 to the power of 8 ) * 6 * e  [Within actual variation]  
Protip—Not all of these are Wrong:  
Square root of 2 = (3/5) – (pi/(7pi))  Cos(pi/7) + Cos3(pi/7) + Cos(5pi/7) = 1/2  
(Euler’s Gamma Constant) Gamma =(e/(3 to the power of 4) + (e/5))  Square root of 5 = (13 + 4pi)/(244pi)  Sigma (1/n to the power of n) = ln(3) to the power of 3 
Hover text: Two tips: 1) 8675309 is not just prime, it’s a twin prime, and 2) if you ever find yourself raising log(anything) to the power of e or taking the pith root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.
Warning: This comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberalarts majors). Comic by xkcd.com. Described by BlindGadget under the Creative Commons license.
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