April 25, 2012
#1047: Approximations explain
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.
{| class=“wikitable”
|-
|colspan=“2” align=“center” | Relation:
|align=“center” | Accurate to within:
|-
|align=“center” | One light-year(m)
|align=“center” | 998
|align=“center” | one part in 40
|-
|align=“center” | Earth Surface(m2)
|align=“center” | 698
|align=“center” | one part in 130
|-
|align=“center” | Oceans’ volume(m3)
|align=“center” | 919
|align=“center” | one part in 70
|-
|align=“center” | Seconds in a year
|align=“center” | 754
|align=“center” | one part in 400
|-
|align=“center” | Seconds in a year (Rent method)
|align=“center” | 525,600 x 60
|align=“center” | one part in 1400
|-
|align=“center” | Age of the universe (seconds)
|align=“center” | 1515
|align=“center” | one part in 70
|-
|align=“center” | Planck’s constant
|align=“center” | 1/(30πe)
|align=“center” | one part in 110
|-
|align=“center” | Fine structure constant
|align=“center” | 1/140
|align=“center” | [I’ve had enough of this 137 crap]
|-
|align=“center” | Fundamental charge
|align=“center” | 3/(14 * πππ)
|align=“center” | one part in 500
|-
|align=“center”|White House Switchboard
|colspan=“2” align=“center”|1 / (eπ√(1 + (e-1)√8))
|-
|align=“center”|Jenny’s Constant
|colspan=“2” align=“center”|(7(e/1 - 1/e) - 9) * π2
|-
|colspan=“3” align=“center”|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
|-
|align=“center”|Electron rest energy
|align=“center”|e/716 J
|align=“center”|one part in 1000
|-
|align=“center”|Light-year(miles)
|align=“center”|2(42.42)
|align=“center”|one part in 1000
|-
|colspan=“2” align=“center”|sin(60°) = √3/2 = e/π
|align=“center”|one part in 1000
|-
|colspan=“2” align=“center”|√3 = 2e/π
|align=“center”|one part in 1000
|-
|align=“center”|γ(Euler’s gamma constant)
|align=“center”|1/√3
|align=“center”|one part in 4000
|-
|align=“center”|Feet in a meter
|align=“center”|5/(e√π)
|align=“center”|one part in 4000
|-
|colspan=“2” align=“center”|√5 = 2/e + 3/2
|align=“center”|one part in 7000
|-
|align=“center”|Avogadro’s number
|align=“center”|69π√5
|align=“center”|one part in 25,000
|-
|align=“center”|Gravitational constant G
|align=“center”|1 / e(π - 1)(π + 1)
|align=“center”|one part in 25,000
|-
|align=“center”|R (gas constant)
|align=“center”|(e+1) √5
|align=“center”|one part in 50,000
|-
|align=“center”|Proton-electron mass ratio
|align=“center”|6*π5
|align=“center”|one part in 50,000
|-
|align=“center”|Liters in a gallon
|align=“center”|3 + π/4
|align=“center”|one part in 500,000
|-
|align=“center”|g
|align=“center”|6 + ln(45)
|align=“center”|one part in 750,000
|-
|align=“center”|Proton-electron mass ratio
|align=“center”|(e8 - 10) / ϕ
|align=“center”|one part in 5,000,000
|-
|align=“center”|Ruby laser wavelength
|align=“center”|1 / (12002)
|align=“center”|[within actual variation]
|-
|align=“center”|Mean Earth Radius
|align=“center”|(58)*6e
|align=“center”|[within actual variation]
|-
|colspan=“3” align=“center”|Protip - not all of these are wrong:
|-
|colspan=“2” align=“center”|√2 = 3/5 + π/(7-π)
|align=“center”|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
|align=“center”|γ(Euler’s gamma constant) = e/34 + e/5
|align=“center”|√5 = (13 + 4π) / (24 - 4π)
|align=“center”|Σ 1/nn = ln(3)e
|}