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| | | date = April 25, 2012 | | | date = April 25, 2012 |
| | | title = Approximations | | | title = Approximations |
| − | | before = [[#Explanation|↓ Skip to explanation ↓]]
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| | | image = approximations.png | | | image = approximations.png |
| | | titletext = Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong. | | | titletext = Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong. |
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| | ==Explanation== | | ==Explanation== |
| | + | {{incomplete}} |
| | + | 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. Those interested in number theory may easily compute that sqrt(2) is not even algebraic in the quotient field of Z[pi], which disproves the equality. |
| | | | |
| − | This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.
| + | Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”. |
| | | | |
| − | Approximations like these are sometimes used as {{w|mnemonic}}s 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 {{w|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<sup>7</sup>", though he later added a statement to the top of the comic page addressing this point.
| + | [[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). |
| | | | |
| − | At the bottom of the comic are expressions involving {{w|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.
| + | There are a few cultural references in this comic: |
| | | | |
| − | Randall says he compiled this table through "a mix of trial-and-error, ''{{w|Mathematica}}'', and Robert Munafo's [http://mrob.com/pub/ries/ Ries] tool." "Ries" is a "{{w|Closed-form expression#Conversion from numerical forms|reverse calculator}}" that forms equations matching a given number.
| + | * 99<sup>8</sup> and 69<sup>8</sup> are sexual references. |
| | + | * “Rent Method” refers to the song “Seasons of Love” from the musical “{{w|Rent (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) 456-1415 was (at least during the Bush administration) the phone number for the White House switchboard. Randall's formula gives approximately 0.2024561415. |
| | + | * Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. Randall's formula gives approximately 867.530901981685. |
| | + | * {{w|42 (number)|42}} is, according to Douglas Adams' ''The Hitchhiker's Guide to the Galaxy'', the Answer to Life, the Universe, and Everything. |
| | | | |
| − | The {{w|world population}} estimate for 2023 is still accurate. The estimate is 7.9 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [https://www.census.gov/popclock/ United States Census Bureau - U.S. and World Population Clock]. Nevertheless there are other numbers listed by different sources.
| + | And here are some of the mathematical and physical ones, with Wikipedia links. |
| | | | |
| − | 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 {{w|867-5309/Jenny}}, is not only prime but a {{w|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]]. | + | * Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics. |
| | + | * The {{w|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 {{w|numerology}} thrown in at times). |
| | + | * In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}. |
| | + | * The {{w|gravitational constant}} relates to, uh, gravity. |
| | + | * The {{w|gas constant}} relates energy to temperature in physics. |
| | + | * ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties. |
| | + | * The ruby laser wavelength varies because “ruby” is not clearly defined. |
| | + | * The {{w|Earth radios#mean radii|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 second part of the title text makes fun of the unusual mathematical operations contained in the comic. {{w|Pi|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 [http://gosper.org/4%5E1%C3%B7%CF%80.png identity] for the pi-th root of 4 discovered by Bill Gosper. Similarly, {{w|e (mathematical constant)|e}} typically appears in the base of a power (forming the {{w|exponential function}}), not in the exponent. (This is later referenced in [http://what-if.xkcd.com/73/ Lethal Neutrinos]). | + | 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 [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 shown here]. |
| | | | |
| − | ===Equations===
| + | 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 {{w|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. |
| | | | |
| − | {| class="wikitable"
| + | {{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it really shouldn't occur anywhere in an exponent (unless you deal with complex numbers). |
| − | |-
| |
| − | !align="center"|Thing to be approximated:
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| − | !align="center"|Formula proposed
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| − | !align="center"|Resulting approximate value
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| − | !align="center"|Correct value
| |
| − | !align="center"|Discussion
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| − | |-
| |
| − | |align="center"|One {{w|light year}} (meters)
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| − | |align="center"|99<sup>8</sup>
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| − | |align="center"|9,227,446,944,279,201
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| − | |align="center"|9,460,730,472,580,800 (exact)
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| − | |align="left"|Based on 365.25 days per year (see below). 99<sup>8</sup> and 69<sup>8</sup> are [[487: Numerical Sex Positions|sexual references]].
| |
| − | |-
| |
| − | |align="center"|Earth's surface (m<sup>2</sup>)
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| − | |align="center"|69<sup>8</sup>
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| − | |align="center"|513,798,374,428,641
| |
| − | |align="center"|5.10072 × 10<sup>14</sup>
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| − | |align="left"|99<sup>8</sup> and 69<sup>8</sup> are [[487: Numerical Sex Positions|sexual references]].
| |
| − | |-
| |
| − | |align="center"|Oceans' volume (m<sup>3</sup>)
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| − | |align="center"|9<sup>19</sup>
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| − | |align="center"|1,350,851,717,672,992,089
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| − | |align="center"|1.332 × 10<sup>18</sup>
| |
| − | |align="left"|
| |
| − | |-
| |
| − | |align="center"|Seconds in a year
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| − | |align="center"|75<sup>4</sup>
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| − | |align="center"|31,640,625
| |
| − | |align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
| |
| − | |align="left"|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 × 10<sup>7</sup> seconds. 75<sup>4</sup> is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 10<sup>7</sup> is nearly equal to 31,415,926.536, and 75<sup>4</sup> 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.
| + | Same goes for the e-th power: e typically appears in the basis of a power (forming the {{w|exponential function}}), not in the exponent. |
| − | |-
| |
| − | |align="center"|Seconds in a year (''Rent'' method)
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| − | |align="center"|525,600 × 60
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| − | |align="center"|31,536,000
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| − | |align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
| |
| − | |align="left"|"''Rent'' Method" refers to the song "{{w|Seasons of Love}}" from the musical ''{{w|Rent (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. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
| |
| − | |-
| |
| − | |align="center"|Age of the universe (seconds)
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| − | |align="center"|15<sup>15</sup>
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| − | |align="center"|437,893,890,380,859,375
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| − | |align="center"|(4.354 ± 0.012) × 10<sup>17</sup> (best estimate; exact value unknown)
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| − | |align="left"|This one will slowly get more accurate as the universe ages.
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| − | |-
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| − | |align="center"|Planck's constant
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| − | |align="center"|<math>\frac {1} {30^{\pi^e}}</math>
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| − | |align="center"|6.6849901410 × 10<sup>−34</sup>
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| − | |align="center"|6.62606957 × 10<sup>−34</sup>
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| − | |align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
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| − | |-
| |
| − | |align="center"|Fine structure constant
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| − | |align="center"|<math>\frac{1}{140}</math>
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| − | |align="center"|0.00<span style="text-decoration: overline;">714285</span>
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| − | |align="center"|0.0072973525664 (accepted value as of 2014), close to 1/137
| |
| − | |align="left"|The {{w|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 {{w|numerology}} thrown in at times), including the infamous {{w|Arthur Eddington|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.
| |
| − | |-
| |
| − | |align="center"|Fundamental charge
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| − | |align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
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| − | |align="center"|1.59895121062716 × 10<sup>−19</sup>
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| − | |align="center"|1.602176565 × 10<sup>−19</sup>
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| − | |align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
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| − | |-
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| − | |align="center"|Telephone number for the {{w|White House}} switchboard
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| − | |align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math>
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| − | |align="center"|0.2024561414932
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| − | |align="center"|202-456-1414
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| − | |align="left"|
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| − | |-
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| − | |align="center"|Jenny's constant
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| − | |align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
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| − | |align="center"|867.5309019
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| − | |align="center"|867-5309
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| − | |align="left"|A telephone number referenced in {{w|Tommy Tutone}}'s 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number is not only prime but a {{w|twin prime}} because 8675311 is also a prime.
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| − | |-
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| − | |align="center"|World population estimate (billions)
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| − | |align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
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| − | |align="center"|2005 — 6.5<br>
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| − | 2006 — 6.6<br>
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| − | 2007 — 6.7<br>
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| − | 2008 — 6.7<br>
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| − | 2009 — 6.8<br>
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| − | 2010 — 6.9<br>
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| − | 2011 — 7.0<br>
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| − | 2012 — 7.0<br>
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| − | 2013 — 7.1<br>
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| − | 2014 — 7.2<br>
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| − | 2015 — 7.3<br>
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| − | 2016 — 7.3<br>
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| − | 2017 — 7.4<br>
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| − | 2018 — 7.5<br>
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| − | 2019 — 7.6<br>
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| − | 2020 — 7.6<br>
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| − | 2021 — 7.7<br>
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| − | 2022 — 7.8<br>
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| − | 2023 — 7.9<br>
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| − | 2024 — 7.9<br>
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| − | 2025 — 8.0<br>
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| − | 2026 — 8.1<br>
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| − | 2027 — 8.2<br>
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| − | 2028 — 8.2<br>
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| − | 2029 — 8.3<br>
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| − | 2030 — 8.4<br>
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| − | 2031 — 8.5<br>
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| − | 2032 — 8.5<br>
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| − | 2033 — 8.6<br>
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| − | 2034 — 8.7<br>
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| − | 2035 — 8.8<br>
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| − | |align="center"|
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| − | |align="left"|Grows by 75 million every year on average. As of 2023, a bit too small.
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| − | |-
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| − | |align="center"|U.S. population estimate (millions)
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| − | |align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
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| − | |align="center"|2000 — 280<br>
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| − | 2001 — 283<br>
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| − | 2002 — 286<br>
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| − | 2003 — 289<br>
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| − | 2004 — 292<br>
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| − | 2005 — 295<br>
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| − | 2006 — 298<br>
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| − | 2007 — 301<br>
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| − | 2008 — 304<br>
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| − | 2009 — 307<br>
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| − | 2010 — 310<br>
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| − | 2011 — 313<br>
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| − | 2012 — 316<br>
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| − | 2013 — 319<br>
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| − | 2014 — 322<br>
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| − | 2015 — 325<br>
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| − | 2016 — 328<br>
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| − | 2017 — 331<br>
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| − | 2018 — 334<br>
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| − | 2019 — 337<br>
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| − | 2020 — 340<br>
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| − | 2021 — 343<br>
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| − | 2022 — 346<br>
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| − | 2023 — 349<br>
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| − | 2024 — 352<br>
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| − | 2025 — 355<br>
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| − | 2026 — 358<br>
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| − | 2027 — 361<br>
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| − | 2028 — 364<br>
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| − | 2029 — 367<br>
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| − | 2030 — 370<br>
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| − | 2031 — 373<br>
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| − | 2032 — 376<br>
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| − | 2033 — 379<br>
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| − | 2034 — 382<br>
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| − | 2035 — 385<br>
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| − | |align="center"|
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| − | |align="left"|Grows by 3 million each year. As of 2021 the actual number is ~13 million smaller.
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| − | |-
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| − | |align="center"|Electron rest energy (joules)
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| − | |align="center"|<math>\frac {e} {7^{16}}</math>
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| − | |align="center"|8.17948276564429 × 10<sup>−14</sup>
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| − | |align="center"|8.18710438 × 10<sup>−14</sup>
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| − | |align="left"|
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| − | |-
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| − | |align="center"|Light year (miles)
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| − | |align="center"|2<sup>42.42</sup>
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| − | |align="center"|5,884,267,614,436.97
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| − | |align="center"|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)
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| − | |align="left"|{{w|42 (number)|42}} is, according to {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
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| − | |-
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| − | |align="center"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
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| − | |align="center"|<math>\frac{e}{\pi}</math>
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| − | |align="center"|0.8652559794
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| − | |align="center"|0.8660254038
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| − | |align="left"|
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| − | |-
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| − | |align="center"|<math>\sqrt 3</math>
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| − | |align="center"|<math>\frac{2e}{\pi}</math>
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| − | |align="center"|1.7305119589
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| − | |align="center"|1.7320508076
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| − | |align="left"|Same as the above
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| − | |-
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| − | |align="center"|γ (Euler's gamma constant)
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| − | |align="center"|<math>\frac {1} {\sqrt 3}</math>
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| − | |align="center"|0.5773502692
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| − | |align="center"|0.5772156649
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| − | |align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
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| − | |-
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| − | |align="center"|Feet in a meter
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| − | |align="center"|<math>\frac {5} {\sqrt[e]\pi}</math>
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| − | |align="center"|3.2815481951
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| − | |align="center"|3.280839895
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| − | |align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
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| − | |-
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| − | |align="center"|<math>\sqrt 5</math>
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| − | |align="center"|<math>\frac{2}{e} + \frac32</math>
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| − | |align="center"|2.2357588823
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| − | |align="center"|2.2360679775
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| − | |align="left"|
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| − | |-
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| − | |align="center"|Avogadro's number
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| − | |align="center"|<math>69^{\pi^\sqrt{5}}</math>
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| − | |align="center"|6.02191201246329 × 10<sup>23</sup>
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| − | |align="center"|6.02214129 × 10<sup>23</sup>
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| − | |align="left"|Also called a mole for shorthand, {{w|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 × 10<sup>23</sup>, making the Approximation slightly more correct.
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| − | |-
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| − | |align="center"|Gravitational constant ''G''
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| − | |align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
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| − | |align="center"|6.6736110685 × 10<sup>−11</sup>
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| − | |align="center"|6.67385 × 10<sup>−11</sup>
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| − | |align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''<sup>2</sup>/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
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| − | |-
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| − | |align="center"|''R'' (gas constant)
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| − | |align="center"|<math>(e + 1) \sqrt 5</math>
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| − | |align="center"|8.3143309279
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| − | |align="center"|8.3144622
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| − | |align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).
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| − | |-
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| − | |align="center"|Proton–electron mass ratio
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| − | |align="center"|<math>6 \pi^5</math>
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| − | |align="center"|1836.1181087117
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| − | |align="center"|1836.15267246
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| − | |align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
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| − | |-
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| − | |align="center"|Liters in a {{w|gallon}}
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| − | |align="center"|<math>3 + \frac{\pi}{4}</math>
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| − | |align="center"|3.7853981634
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| − | |align="center"|3.785411784 (exact)
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| − | |align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches. The British imperial gallon would be about 20% larger (but the litre is the same thing as the US liter).
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| − | |-
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| − | |align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub>
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| − | |align="center"|6 + ln(45)
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| − | |align="center"|9.8066624898
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| − | |align="center"|9.80665
| |
| − | |align="left"|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/s<sup>2</sup>, which is exactly 35.30394 km/h/s (about 32.174 ft/s<sup>2</sup>, 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.
| + | The software referred to in the comic is [http://mrob.com/pub/ries/ ries], a 'reverse calculator' which forms equations matching a given number. |
| − | |-
| |
| − | |align="center"|Proton–electron mass ratio
| |
| − | |align="center"|<math>\frac {e^8 - 10} {\phi}</math>
| |
| − | |align="center"|1836.1530151398
| |
| − | |align="center"|1836.15267246
| |
| − | |align="left"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
| |
| − | |-
| |
| − | |align="center"|Ruby laser wavelength (meters)
| |
| − | |align="center"|<math>\frac{1}{1200^2}</math>
| |
| − | |align="center"|6.9<span style="text-decoration: overline;">444</span> × 10<sup>−7</sup>
| |
| − | |align="center"|~6.943 × 10<sup>−7</sup>
| |
| − | |align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
| |
| − | |-
| |
| − | |align="center"|Mean Earth radius (meters)
| |
| − | |align="center"|<math>5^8 6e</math>
| |
| − | |align="center"|6,370,973.035
| |
| − | |align="center"|6,371,008.7 (IUGG definition)
| |
| − | |align="left"|The {{w|Earth radius#mean radii|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).
| |
| − | |-
| |
| − | |align="center"|<math>\sqrt 2</math>
| |
| − | |align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
| |
| − | |align="center"|1.4142200581
| |
| − | |align="center"|1.4142135624
| |
| − | |align="left"|There are recurring math jokes along the lines of, "<math>\textstyle{ \frac35 + \frac{\pi}{7-\pi} - \sqrt{2} = 0}</math>, but your calculator is probably not good enough to compute this correctly". See also [[217: e to the pi Minus pi]].
| |
| − | |-
| |
| − | |align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
| |
| − | |align="center"|<math>\frac12</math>
| |
| − | |align="center"|0.5
| |
| − | |align="center"|0.5 (exact)
| |
| − | |align="left"|This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}.
| |
| − | |-
| |
| − | |align="center"|γ (Euler's gamma constant)
| |
| − | |align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
| |
| − | |align="center"|0.5772154006
| |
| − | |align="center"|0.5772156649
| |
| − | |align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
| |
| − | |-
| |
| − | |align="center"|<math>\sqrt 5</math>
| |
| − | |align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
| |
| − | |align="center"|2.2360678094
| |
| − | |align="center"|2.2360679775
| |
| − | |align="left"|
| |
| − | |-
| |
| − | |align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
| |
| − | |align="center"|<math>\ln(3)^e</math>
| |
| − | |align="center"|1.2912987577
| |
| − | |align="center"|1.2912859971
| |
| − | |align="left"|
| |
| − | |}
| |
| | | | |
| − | ===Proof=== | + | ==Transcript== |
| − | | + | 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. |
| − | One of the "approximations" actually is precisely correct: <math>\textstyle{ \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = \frac12 }</math>. Here is a proof:
| |
| − | | |
| − | :<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
| |
| − | | |
| − | Multiplying by 1 (or by a nonzero number divided by itself) leaves the equation unchanged:
| |
| − | | |
| − | :<math>= \left( \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} \right) \frac{2 \sin\frac{\pi}{7}}{2 \sin\frac{\pi}{7}}</math>
| |
| − | | |
| − | The <math>\textstyle{ 2 \sin\frac{\pi}{7} }</math> on the top of the fraction is multiplied through the original equation:
| |
| − | | |
| − | :<math>= \frac {2 \cos \frac{\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{3\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{5\pi}{7} \sin\frac{\pi}{7}} {2 \sin\frac{\pi}{7}}</math>
| |
| − | | |
| − | Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin B = \sin (A+B) - \sin(A-B)}</math> on the second and third terms in the numerator:
| |
| − | | |
| − | :<math>\begin{align}
| |
| − | &= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \left(\frac{3\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{3\pi}{7} - \frac{\pi}{7}\right) \right] + \left[\sin \left(\frac{5\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{5\pi}{7} - \frac{\pi}{7}\right) \right]} {2 \sin\frac{\pi}{7}} \\
| |
| − | &= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}}
| |
| − | \end{align}</math>
| |
| − | | |
| − | Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin A = \sin 2A }</math> on the first term in the numerator:
| |
| | | | |
| − | :<math>\begin{align} | + | *Relation: One light year(m) ~= 99^8 |
| − | &= \frac {\sin \frac{2\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
| + | Accurate to within: one part in 40 |
| − | &= \frac {\sin \frac{6\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{4\pi}{7} \right] + \left[\sin \frac{2\pi}{7} - \sin \frac{2\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
| |
| − | &= \frac {\sin \frac{6\pi}{7} } {2 \sin\frac{\pi}{7}}
| |
| − | \end{align}</math>
| |
| | | | |
| − | Noting that <math>\textstyle{\frac{6\pi}{7} + \frac{\pi}{7} = \pi}</math> and that the sines of supplementary angles (angles that sum to π) are equal:
| + | *Relation: Earth Surface(m^2) ~= 69^8 |
| | + | Accurate to within: one part in 130 |
| | | | |
| − | :<math>\begin{align} | + | *Relation: Ocean's volume(m^3) ~= 9^19 |
| − | &= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
| + | Accurate to within: one part in 70 |
| − | &= \frac12 \quad \quad \quad \text{Q.E.D.}
| |
| − | \end{align}</math>
| |
| | | | |
| − | To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7) <!--<math>\textstyle{ \cos \frac{2k\pi}{7} }</math>--> is the real part of the k-th 7-th root of unity, exp(2 k i pi/7)<!--<math>\textstyle{ \exp \frac{2 k i\pi}{7} }</math>-->. The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:
| + | *Relation: Seconds in a year ~= 75^4 |
| | + | Accurate to within: one part in 400 |
| | | | |
| − | <!--:<math>0 = \cos \frac{0\pi}{7} + \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} + \cos \frac{8\pi}{7} + \cos \frac{10\pi}{7} + \cos \frac{12\pi}{7} </math>-->
| + | *Relation: Seconds in a year (rent method) ~= 525,600 x 60 |
| − | :0 = cos(0 pi/7) + cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) + cos(8 pi/7) + cos(10 pi/7) + cos(12 pi/7) | + | Accurate to within: one part in 1400 |
| | | | |
| − | But one of these roots is just 1, and all other root go by pairs of conjugate roots, which have the same real part (alternatively, consider that cos(x) = cos(2 pi - x)):
| + | *Relation: Age of the universe (seconds) ~= 15^15 |
| | + | Accurate to within: one part in 70 |
| | | | |
| − | <!--:<math>0 = 1 + 2 ( \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} ) </math>-->
| + | *Relation: Planck's constant ~= 1/(30^pi^e) |
| − | :0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7)) | + | Accurate to within: one part in 110 |
| | | | |
| − | Hence
| + | *Relation: Fine structure constant ~= 1/140 |
| | + | Accurate to within: [I've had enough of this 137 crap] |
| | | | |
| − | <!--:<math>\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = - 1/2 </math>-->
| + | *Relation: Fundamental charge ~= 3/(14 * pi^pi^pi) |
| − | :cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2
| + | Accurate to within: one part in 500 |
| | | | |
| − | which, because cos(x) = cos(pi - x),<!--<math>\cos (x) = - \cos(\pi - x)</math>,--> can be rewritten as
| + | *Relation: White House Switchboard ~= 1/(e^((1 + 8^(1/(e-1)))^(1/pi))) |
| | | | |
| − | <!--:<math>\cos \frac{5\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{pi}{7} = 1/2 </math>-->
| + | *Relation: Jenny's Constant ~= (7^(e/1 - 1/e) - 9) * pi^2 |
| − | :cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2
| |
| | | | |
| − | Q.E.D.
| + | ---- |
| | + | Intermission: World Population Estimate which should stay current for a decade or two: |
| | | | |
| − | ==Transcript==
| |
| − | :'''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" | 99<sup>8</sup>
| |
| − | |align="center" | one part in 40
| |
| − | |-
| |
| − | |align="center" | Earth Surface(m<sup>2</sup>)
| |
| − | |align="center" | 69<sup>8</sup>
| |
| − | |align="center" | one part in 130
| |
| − | |-
| |
| − | |align="center" | Oceans' volume(m<sup>3</sup>)
| |
| − | |align="center" | 9<sup>19</sup>
| |
| − | |align="center" | one part in 70
| |
| − | |-
| |
| − | |align="center" | Seconds in a year
| |
| − | |align="center" | 75<sup>4</sup>
| |
| − | |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" | 15<sup>15</sup>
| |
| − | |align="center" | one part in 70
| |
| − | |-
| |
| − | |align="center" | Planck's constant
| |
| − | |align="center" | 1/(30<sup>π<sup>e</sup></sup>)
| |
| − | |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 * π<sup>π<sup>π</sup></sup>)
| |
| − | |align="center" | one part in 500
| |
| − | |-
| |
| − | |align="center"|White House Switchboard
| |
| − | |colspan="2" align="center"|1 / (e<sup><sup>π</sup>√(1 + <sup>(e-1)</sup>√8)</sup>)
| |
| − | |-
| |
| − | |align="center"|Jenny's Constant
| |
| − | |colspan="2" align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup>
| |
| − | |-
| |
| − | |colspan="3" align="center"|Intermission:<br /> World Population Estimate<br /> which should stay current<br /> for a decade or two:<br />
| |
| | Take the last two digits of the current year | | Take the last two digits of the current year |
| | | | |
| Line 426: |
Line 87: |
| | Subtract the number of leap years since hurricane Katrina | | Subtract the number of leap years since hurricane Katrina |
| | | | |
| − | Example: 14 (minus 2008 and 2012) is 12 | + | Example:14 (minus 2008 and 2012) is 12 |
| | | | |
| | Add a decimal point | | Add a decimal point |
| Line 436: |
Line 97: |
| | Example: 6 + 1.2 | | Example: 6 + 1.2 |
| | | | |
| − | 7.2 = World population in billions. | + | 7.2 ~= World population in billions. |
| − | | |
| | | | |
| | Version for US population: | | Version for US population: |
| Line 454: |
Line 114: |
| | | | |
| | Example: 3[22] million | | Example: 3[22] million |
| − | |-
| + | ---- |
| − | |align="center"|Electron rest energy
| + | |
| − | |align="center"|e/7<sup>16</sup> J
| + | *Relation: Electron rest energy ~= e/7^16 Joules |
| − | |align="center"|one part in 1000
| + | Accurate to within: one part in 1000 |
| − | |-
| + | |
| − | |align="center"|Light-year(miles)
| + | *Relation: Light-year(miles) ~= 2^(42.42) |
| − | |align="center"|2<sup>(42.42)</sup>
| + | Accurate to within: one part in 1000 |
| − | |align="center"|one part in 1000
| + | |
| − | |-
| + | *Relation: sin(60 degrees) = (3^(1/2))/2 ~= e/pi |
| − | |colspan="2" align="center"|sin(60°) = √3/2 = e/π
| + | Accurate to within: one part in 1000 |
| − | |align="center"|one part in 1000
| + | |
| − | |-
| + | *Relation: 3^(1/2) ~= 2e/pi |
| − | |colspan="2" align="center"|√3 = 2e/π
| + | Accurate to within: one part in 1000 |
| − | |align="center"|one part in 1000
| + | |
| − | |-
| + | *Relation: gamma(Euler's gamma constant) ~= 1/(3^(1/2)) |
| − | |align="center"|γ(Euler's gamma constant)
| + | Accurate to within: One part in 4000 |
| − | |align="center"|1/√3
| + | |
| − | |align="center"|one part in 4000
| + | *Relation: Feet in a meter ~= 5/(pi^(1/e)) |
| − | |-
| + | Accurate to within: one part in 4000 |
| − | |align="center"|Feet in a meter
| + | |
| − | |align="center"|5/(<sup>e</sup>√π)
| + | *Relation: 5^(1/2) ~= 2/e + 3/2 |
| − | |align="center"|one part in 4000
| + | Accurate to within: one part in 7000 |
| − | |-
| + | |
| − | |colspan="2" align="center"|√5 = 2/e + 3/2
| + | *Relation: Avogadro's number ~= 69^pi^(5^(1/2)) |
| − | |align="center"|one part in 7000
| + | Accurate to within: one part in 25,000 |
| − | |-
| + | |
| − | |align="center"|Avogadro's number
| + | *Relation: Gravitational constant G ~= 1/(e^((pi - 1)^(pi + 1))) |
| − | |align="center"|69<sup>π<sup>√5</sup></sup>
| + | Accurate to within: one part in 25,000 |
| − | |align="center"|one part in 25,000
| + | |
| − | |-
| + | *Relation: R(gas constant) ~= (e+1) * (5^(1/2)) |
| − | |align="center"|Gravitational constant G
| + | Accurate to within: one part in 50,000 |
| − | |align="center"|1 / e<sup>(π - 1)<sup>(π + 1)</sup></sup>
| + | |
| − | |align="center"|one part in 25,000
| + | *Relation: Proton-electron mass ratio ~= 6*pi^5 |
| − | |-
| + | Accurate to within: one part in 50,000 |
| − | |align="center"|R (gas constant)
| + | |
| − | |align="center"|(e+1) √5
| + | *Relation: Liters in a gallon ~= 3 + pi/4 |
| − | |align="center"|one part in 50,000
| + | Accurate to within: one part in 500,000 |
| − | |-
| + | |
| − | |align="center"|Proton-electron mass ratio
| + | *Relation: g ~= 6 + ln(45) |
| − | |align="center"|6*π<sup>5</sup>
| + | Accurate to within: one part in 750,000 |
| − | |align="center"|one part in 50,000
| + | |
| − | |-
| + | *Relation: Proton-electron mass ratio ~= (e^8 - 10)/phi |
| − | |align="center"|Liters in a gallon
| + | Accurate to within: one part in 5,000,000 |
| − | |align="center"|3 + π/4
| + | |
| − | |align="center"|one part in 500,000
| + | *Relation: Ruby laser wavelength ~= 1/(1200^2) |
| − | |-
| + | Accurate to within: [within actual variation] |
| − | |align="center"|g
| + | |
| − | |align="center"|6 + ln(45)
| + | *Relation: Mean Earth Radius ~= (5^8)*6e |
| − | |align="center"|one part in 750,000
| + | Accurate to within: [within actual variation] |
| − | |-
| + | |
| − | |align="center"|Proton-electron mass ratio
| + | ---- |
| − | |align="center"|(e<sup>8</sup> - 10) / ϕ
| + | Protip - not all of these are wrong: |
| − | |align="center"|one part in 5,000,000
| + | |
| − | |-
| + | *2^(1/2) ~= 3/5 + pi/(7-pi) |
| − | |align="center"|Ruby laser wavelength
| + | |
| − | |align="center"|1 / (1200<sup>2</sup>)
| + | *cos(pi/7) + cos(3pi/7) + cos(5pi/7) ~= 1/2 |
| − | |align="center"|[within actual variation]
| + | |
| − | |-
| + | *gamma(Euler's gamma constant) ~= e/3^4 + e/5 |
| − | |align="center"|Mean Earth Radius
| + | |
| − | |align="center"|(5<sup>8</sup>)*6e
| + | *5^(1/2) ~= (13 + 4pi)/(24 - 4pi) |
| − | |align="center"|[within actual variation]
| + | |
| − | |-
| + | *sigma(1/n^n) ~= ln(3)^e |
| − | |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/3<sup>4</sup> + e/5
| |
| − | |align="center"|√5 = (13 + 4π) / (24 - 4π)
| |
| − | |align="center"|Σ 1/n<sup>n</sup> = ln(3)<sup>e</sup>
| |
| − | |}
| |
| | | | |
| | {{comic discussion}} | | {{comic discussion}} |
