Approximations

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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:
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
{{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.}}