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This comic lists some approximations for numbers, most of them mathematical and physical constants. All of them work astonishingly well. There are reoccurring math jokes along the lines of, “3/5 + π/(7 – π) – √2 = 0, but your calculator is probably not good enough to compute this correctly”, which are mainly used to troll geeks.
Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”.
Randall makes fun of both of these, using rather strange approximations (honestly: you may handle 22/7, but who can calculate in a sensible way with 99^8, let alone 30^(pi^e)?) to calculate some constants that are easy enough to handle in the decimal system, and stating such “slightly wrong” trick equations, one of which is actually correct (which may astonish only those who are not familiar with cosines).
There are a few cultural references in this comic:
- 998 and 698 are sexual references.
- “Rent Method” refers to the song “Seasons of Love” from the musical “Rent.” The song asks, “How do you measure a year?” One line says “525,600 minutes” while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one.
- (202) 456-1414 is the phone number for the White House switchboard. Truncated, Randall's formula yields 0.2024561414.
- Jenny's constant comes from Tommy Tutone's tune 867-5309/Jenny. Randall's formula gives approximately 867.530901981685.
- 42 is, according to Douglas Adams' The Hitchhiker's Guide to the Galaxy, the Answer to the Ultimate Question of Life, the Universe, and Everything.
And here are some of the mathematical and physical ones, with Wikipedia links.
- Informally, the Planck constant is the smallest action possible in quantum mechanics.
- The fine structure constant indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. At one point it was believed to be exactly the reciprocal of 137, and many people have tried to find a simple formula explaining this (with a pinch of numerology thrown in at times), including the infamous Sir Arthur Adding-One.
- In mathematics, the Euler-Mascheroni constant (Euler gamma constant) is a mysterious number describing the relationship between the harmonic series and the natural logarithm.
- The gravitational constant relates to, uh, gravity.
- The gas constant relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and molar amount (hence the name).
- ϕ is the golden ratio, or (1 + √5)/2. It has many interesting geometrical properties.
- The ruby laser wavelength varies because “ruby” is not clearly defined.
- The mean earth radius varies because there is not one single way to make a sphere out of the earth. Randall's value lies within the actual variation of Earth's radius.
The correct equation in the "Pro tip - Not all of these are wrong" section is cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2 as shown here. If you're still confused, the functions use radians, not degrees.
The number 8675309 at the title text refers to the song 867-5309/Jenny as mentioned above, causing a fad of people dialing this number and asking for "Jenny". The number is in fact 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.
Pi is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it doesn't usually occur anywhere in an exponent. When it does, such as with complex numbers, taking the pi-th root is rarely helpful. For example, if we try to derive:
eπi + 1 = 0
eπi = -1
(ei)π = -1
ei = π√(-1)
We get nowhere.
Same goes for the e-th power: e typically appears in the basis of a power (forming the exponential function), not in the exponent. (This is later referenced in [Lethal Neutrinos http://what-if.xkcd.com/73/].
The software referred to in the comic is ries, a 'reverse calculator' which forms equations matching a given number.
Actual Approximation One light year(m) 9.46x1015 998 9.23x1015 Earth Surface(m2) 698 Ocean's volume(m3) 919 Seconds in a year 31557600 754 34640625 Seconds in a year (rent method) 31557600 525,600 x 60 31536000 Age of the universe (seconds) 1515 4.379x1017 Planck's constant 6.626x10-34 1/(30πe) 6.685x10-34 Fine structure constant 7.297x10-3 1/140 7.143x10-3 Fundamental charge 1.602x10-19 3/(14 * πππ) 1.599x10-19
|This transcript is incomplete. Please help editing it! Thanks.|
- 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) 998 one part in 40 Earth Surface(m2) 698 one part in 130 Ocean's volume(m3) 919 one part in 70 Seconds in a year 754 one part in 400 Seconds in a year (rent method) 525,600 x 60 one part in 1400 Age of the universe (seconds) 1515 one part in 70 Planck's constant 1/(30πe) one part in 110 Fine structure constant 1/140 [I've had enough of this 137 crap] Fundamental charge 3/(14 * πππ) one part in 500 White House Switchboard 1 /
π√(e(1 + (e-1)√8)
Jenny's Constant (7(e/1 - 1/e) - 9) * π2 Intermission:
World Population Estimate
which should stay current
for a decade or two:
Take the last two digits of the current year
Subtract the number of leap years since hurricane Katrina
Example:14 (minus 2008 and 2012) is 12
Add a decimal point
Example: 6 + 1.2
7.2 = World population in billions.
Version for US population:
Multiply by 3
Example: 3 million
Electron rest energy e/716 Joules 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 gamma(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 69π√5 one part in 25,000 Gravitational constant G 1 / e(pi - 1)(pi + 1) one part in 25,000 R(gas constant) (e+1) √5 one part in 50,000 Proton-electron mass ratio 6*π5 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 e8 - 10 / ϕ one part in 5,000,000 Ruby laser wavelength 1 / (12002) [within actual variation] Mean Earth Radius (58)*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/34 + e/5 √5 = 13 + 4π / 24 - 4π Σ 1/nn = ln(3)e