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 × 107", 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.
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. 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).
|Thing to be approximated:||Formula proposed||Resulting approximate value||Correct value||Discussion|
|One light year (meters)||998||9,227,446,944,279,201||9,460,730,472,580,800 (exact)||Based on 365.25 days per year (see below). 998 and 698 are sexual references.|
|Earth's surface (m2)||698||513,798,374,428,641||5.10072 × 1014||998 and 698 are sexual references.|
|Oceans' volume (m3)||919||1,350,851,717,672,992,089||1.332 × 1018|
|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 × 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.
Using the traditional definitions that a second is 1/60 of a minute, a minute is 1/60 of an hour, and an hour is 1/24 of a day, a 365-day common year is exactly 31,536,000 seconds (the "Rent method" approximation) and the 366-day leap year is 31,622,400 seconds. 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 seconds. 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 × 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||4.354 ± 0.012 × 1017 (best estimate; exact value unknown)||This one will slowly get more accurate as the universe ages.|
|Planck's constant||6.6849901410 × 10−34||6.62606957 × 10−34||Informally, the Planck constant is the smallest action possible in quantum mechanics.|
|Fine structure constant||0.00714285||0.0072973525664 (accepted value as of 2014), close to 1/137||The fine structure constant indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. The joke here is that Randall chose to write 140 as the denominator, when 137 is much closer to reality and just as many digits (although 137 is a less "round" number than 140, and Randall writes in the table that he's "had enough" of it). At one point the fine structure constant 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" Eddington who argued very strenuously that the fine structure constant "should" be 1/136 when that was what the best measurements suggested, and then argued just as strenuously for 1/137 a few years later as measurements improved.|
|Fundamental charge||1.59895121062716 × 10−19||1.602176565 × 10−19||This is the charge of the proton, symbolized e for electron (whose charge is actually −e. You can blame Benjamin Franklin for that.)|
|Telephone number for the White House switchboard||0.2024561414||202-456-1414|
|Jenny's constant||867.5309019||867-5309||A telephone number referenced in Tommy Tutone's 1982 song 867-5309/Jenny. As mentioned in the title text, the number not only prime but a twin prime because 8675311 is also a prime.|
|World population estimate (billions)||Equivalent to||2005 — 6.5
2006 — 6.6
|Grows by 75 million every year on average. As of 2021, roughly correct.|
|U.S. population estimate (millions)||Equivalent to||2000 — 280
2001 — 283
|Grows by 3 million each year. As of 2021 the actual number is ~13 million smaller.|
|Electron rest energy (joules)||8.17948276564429 × 10−14||8.18710438 × 10−14|
|Light year (miles)||242.42||5,884,267,614,436.97||5,878,625,373,183.61 = 9,460,730,472,580,800 (meters in a light-year, by definition) / 1609.344 (meters in a mile)||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.|
|1.7305119589||1.7320508076||Same as the above|
|γ (Euler's gamma constant)||0.5773502692||0.5772156649||The Euler–Mascheroni constant (denoted γ) is a mysterious number describing the relationship between the harmonic series and the natural logarithm.|
|Feet in a meter||3.2815481951||3.280839895||Exactly 1/0.3048, as the international foot is defined as 0.3048 meters.|
|Avogadro's number||6.02191201246329 × 1023||6.02214129 × 1023||Also called a mole for shorthand, Avogadro's number is (roughly) the number of individual atoms in 12 grams of pure carbon. Used in basically every application of chemistry. In 2019 the constant was redefined to 6.02214076 × 1023, making the Approximation slightly more correct.|
|Gravitational constant G||6.6736110685 × 10−11||6.67385 × 10−11||The universal gravitational constant G is equal to Fr2/Mm, where F is the gravitational force between two objects, r is the distance between them, and M and m are their masses.|
|R (gas constant)||8.3143309279||8.3144622||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||1836.1181087117||1836.15267246||The proton-to-electron mass ratio is the ratio between the rest mass of the proton divided by the rest mass of the electron.|
|Liters in a gallon||3.7853981634||3.785411784 (exact)||A U.S. liquid gallon is defined by law as 231 cubic inches|
|g0 or gn||6 + ln(45)||9.8066624898||9.80665||Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 km/h/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.
Randall used a letter g without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.
|Proton–electron mass ratio||1836.1530151398||1836.15267246||φ is the golden ratio, or . It has many interesting geometrical properties.|
|Ruby laser wavelength (meters)||6.9444 × 10−7||~6.943 × 10−7||The ruby laser wavelength varies because "ruby" is not clearly defined.|
|Mean Earth radius (meters)||6,370,973.035||6,371,008.7 (IUGG definition)||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 International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).|
|1.4142200581||1.4142135624||There are recurring math jokes along the lines of, ", but your calculator is probably not good enough to compute this correctly". See also 217: e to the pi Minus pi.|
|0.5||0.5 (exact)||This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also here. If you're still confused, the functions use radians, not degrees.|
|γ (Euler's gamma constant)||0.5772154006||0.5772156649||The Euler–Mascheroni constant (denoted γ) is a mysterious number describing the relationship between the harmonic series and the natural logarithm.|
One of the "approximations" actually is precisely correct: . Here is a proof:
Multiplying by 1 (or by a 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) 998 one part in 40 Earth Surface(m2) 698 one part in 130 Oceans' 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 / (eπ√(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 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 69π√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*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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