Approximations
Approximations
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: Accurate to within: One light-year(m) 99 8 one part in 40 Earth Surface(m 2 ) 69 8 one part in 130 Oceans' volume(m 3 ) 9 19 one part in 70 Seconds in a year 75 4 one part in 400 Seconds in a year ( Rent method) 525,600 x 60 one part in 1400 Age of the universe (seconds) 15 15 one part in 70 Planck's constant 1/(30 π e ) one part in 110 Fine structure constant 1/140 [I've had enough of this 137 crap] Fundamental charge 3/(14 * π π π ) one part in 500 White House Switchboard 1 / (e π √(1 + (e-1) √8) ) Jenny's Constant (7 (e/1 - 1/e) - 9) * π 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 Electron rest energy e/7 16 J one part in 1000 Light-year(miles) 2 (42.42) one part in 1000 sin(60°) = √3/2 = e/π one part in 1000 √3 = 2e/π one part in 1000 γ(Euler's gamma constant) 1/√3 one part in 4000 Feet in a meter 5/( e √π) one part in 4000 √5 = 2/e + 3/2 one part in 7000 Avogadro's number 69 π √5 one part in 25,000 Gravitational constant G 1 / e (π - 1) (π + 1) one part in 25,000 R (gas constant) (e+1) √5 one part in 50,000 Proton-electron mass ratio 6*π 5 one part in 50,000 Liters in a gallon 3 + π/4 one part in 500,000 g 6 + ln(45) one part in 750,000 Proton-electron mass ratio (e 8 - 10) / ϕ one part in 5,000,000 Ruby laser wavelength 1 / (1200 2 ) [within actual variation] Mean Earth Radius (5 8 )*6e [within actual variation] Protip - not all of these are wrong: √2 = 3/5 + π/(7-π) cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 γ(Euler's gamma constant) = e/3 4 + e/5 √5 = (13 + 4π) / (24 - 4π) Σ 1/n n = ln(3) e

This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.
Approximations like these are sometimes used as mnemonics by mathematicians and physicists, though most of Randall's approximations are too convoluted to be useful as mnemonics.  Perhaps the best known mnemonic approximation (though not used here by Randall) is that "π is approximately equal to 22/7".  Randall does mention (and mock) the common mnemonic among physicists that the fine structure constant is approximately 1/137.  Although Randall gives approximations for the number of seconds in a year, he does not mention the common physicists' mnemonic that it is "π × 10 7 ", though he later added a statement to the top of the comic page addressing this point.
At the bottom of the comic are expressions involving transcendental numbers (namely π and e) that are tantalizingly close to being exactly true but are not (indeed, they cannot be, due to the nature of transcendental numbers).  Such near-equations were previously discussed in 217: e to the pi Minus pi .  One of the entries, though, is a "red herring" that is exactly true.
Randall says he compiled this table through "a mix of trial-and-error, Mathematica , and Robert Munafo's Ries tool."  "Ries" is a " reverse calculator " that forms equations matching a given number.
The world population estimate for 2020 is still accurate. The estimate is 7.7 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: United States Census Bureau - U.S. and World Population Clock . Nevertheless there are other numbers listed by different sources.
The first part of the title text notes that "Jenny's constant," which is actually a telephone number referenced in Tommy Tutone's 1982 song 867-5309/Jenny , is not only prime but a twin prime because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.  Twin primes were also referenced in 1310: Goldbach Conjectures .
The second part of the title text makes fun of the unusual mathematical operations contained in the comic. π is a useful number in many contexts, but it doesn't usually occur anywhere in an exponent. Even when it does, such as with complex numbers, taking the πth root is rarely helpful.  A rare exception is an identity for the pi-th root of 4 discovered by Bill Gosper.  Similarly, e typically appears in the basis of a power (forming the exponential function ), not in the exponent. (This is later referenced in Lethal Neutrinos ).
One of the "approximations" actually is precisely correct: .  Here is a proof:

Multiplying by 1 (or by a nonzero number divided by itself) leaves the equation unchanged:

The on the top of the fraction is multiplied through the original equation:

Use the trigonometric identity on the second and third terms in the numerator:

Use the trigonometric identity on the first term in the numerator:

Noting that and that the sines of supplementary angles (angles that sum to π) are equal:

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: Accurate to within: One light-year(m) 99 8 one part in 40 Earth Surface(m 2 ) 69 8 one part in 130 Oceans' volume(m 3 ) 9 19 one part in 70 Seconds in a year 75 4 one part in 400 Seconds in a year ( Rent method) 525,600 x 60 one part in 1400 Age of the universe (seconds) 15 15 one part in 70 Planck's constant 1/(30 π e ) one part in 110 Fine structure constant 1/140 [I've had enough of this 137 crap] Fundamental charge 3/(14 * π π π ) one part in 500 White House Switchboard 1 / (e π √(1 + (e-1) √8) ) Jenny's Constant (7 (e/1 - 1/e) - 9) * π 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 Electron rest energy e/7 16 J one part in 1000 Light-year(miles) 2 (42.42) one part in 1000 sin(60°) = √3/2 = e/π one part in 1000 √3 = 2e/π one part in 1000 γ(Euler's gamma constant) 1/√3 one part in 4000 Feet in a meter 5/( e √π) one part in 4000 √5 = 2/e + 3/2 one part in 7000 Avogadro's number 69 π √5 one part in 25,000 Gravitational constant G 1 / e (π - 1) (π + 1) one part in 25,000 R (gas constant) (e+1) √5 one part in 50,000 Proton-electron mass ratio 6*π 5 one part in 50,000 Liters in a gallon 3 + π/4 one part in 500,000 g 6 + ln(45) one part in 750,000 Proton-electron mass ratio (e 8 - 10) / ϕ one part in 5,000,000 Ruby laser wavelength 1 / (1200 2 ) [within actual variation] Mean Earth Radius (5 8 )*6e [within actual variation] Protip - not all of these are wrong: √2 = 3/5 + π/(7-π) cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 γ(Euler's gamma constant) = e/3 4 + e/5 √5 = (13 + 4π) / (24 - 4π) Σ 1/n n = ln(3) e
