| This explanation may be incomplete or incorrect: The layout is still bad. And the cos(pi/7) + cos(3pi/7) + cos(5pi/7) issue is still not explained. See discussion.|
If you can address this issue, please edit the page! Thanks.
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).
Name and formula proposed: Formula proposed: Resulting approximate value: Correct value: Discussion: One light year(m) 998 9,227,446,944,279,201 9.46x1015 998 and 698 are sexual references. Earth Surface(m2) 698 513,798,374,428,641 998 and 698 are sexual references. Ocean's volume(m3) 919 1,350,851,717,672,992,089 Seconds in a year 754 31,640,625 31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar) After this comic was released Randall got many responses by viewers. So he did add this statement to the top of the comic page:
"Lots of emails mention the physicist favorite, 1 year = pi x 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." Pi x 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number pi, but it should be multiplied by the factor of ten.
Using the traditional definitions that a second is 1/60th of a minute, a minute is 1/60th of an hour, and an hour is 1/24th of a day, a 365-day year is exactly 31,536,000 seconds (the "rent method approximation). Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.
Seconds in a year (rent method) 525,600 x 60 31,536,000 31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar) “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. Age of the universe (seconds) 1515 437,893,890,380,859,375 Planck's constant 1/(30πe) 6.685x10-34 6.626x10-34 Informally, the Planck constant is the smallest action possible in quantum mechanics. Fine structure constant 1/140 0.00714285717142857171428571, etc. (repeating 71428571) Approximately 0.007297, close to 1/137 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. Fundamental charge 3/(14 * πππ) 1.599x10-19 1.602x10-19 Telephone number for the White House Switchboard 1/
π√(e(1 + (e-1)√8)
.2024561414 (truncated) 2024561414 Jenny's Constant (7(e/1 - 1/e) - 9) * π2 approximately 867.530901981685. 8675309 Jenny's constant comes from Tommy Tutone's tune 867-5309/Jenny. 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. World Population Estimate Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion U.S. Population Estimate Equivalent to 310+3*(Year - 2010) million Electron rest energy e/716 Joules Light-year(miles) 2(42.42) 5884267614436.97 (rounded) 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. sin(60°) = 3√/2 e/π 0.8652559794 (rounded) 0.8660254038 (rounded) √3 2e/π 1.7305119589 (rounded) 1.7320508076 (rounded) gamma(Euler's gamma constant) 1/√3 0.5773502692 (rounded) Feet in a meter 5/(e√π) 3.2815481951 1/.3048 (exact) = 3.280839895 (rounded) √5 2/e + 3/2 2.2357588823 (rounded) 2.2360679775 (rounded) Avogadro's number 69π√5 6.02191201246329*1023 (rounded) Gravitational constant G 1 / e(pi - 1)(pi + 1) The gravitational constant relates to, uh, gravity. R(gas constant) (e+1) √5 The gas constant relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and molar amount (hence the name). Proton-electron mass ratio 6*π5 1836.1181087117 (rounded) Liters in a gallon 3 + π/4 3.7853981634 (rounded) g 6 + ln(45) 9.8066624898 (rounded) The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary. Proton-electron mass ratio e8 - 10 / ϕ ϕ is the golden ratio, or (1 + √5)/2. It has many interesting geometrical properties. Ruby laser wavelength 1 / (12002) The ruby laser wavelength varies because “ruby” is not clearly defined. Mean Earth Radius (58)*6e 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. √2 3/5 + π/(7-π) cos(π/7) + cos(3π/7) + cos(5π/7) 1/2 0.5 0.5 (exact) 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. γ(Euler's gamma constant) e/34 + e/5 In mathematics, the Euler-Mascheroni constant (Euler gamma constant) is a mysterious number describing the relationship between the harmonic series and the natural logarithm. √5 13 + 4π / 24 - 4π Σ 1/nn ln(3)e
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.
The software referred to in the comic is ries, a 'reverse calculator' which forms equations matching a given number.
- 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
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