Editing 1047: Approximations
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| date = April 25, 2012 | | date = April 25, 2012 | ||
| title = Approximations | | title = Approximations | ||
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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. | ||
}} | }} | ||
− | + | {{incomplete|The layout is still bad. And the cos(pi/7) + cos(3pi/7) + cos(5pi/7) issue is still not explained. See discussion.}} | |
==Explanation== | ==Explanation== | ||
+ | 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). | |
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− | + | :{| class="wikitable" | |
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− | {| class="wikitable" | ||
|- | |- | ||
− | + | |align="center"|Thing to be approximated: | |
− | + | |align="center"|Formula proposed: | |
− | + | |align="center"|Resulting approximate value: | |
− | + | |align="center"|Correct value: | |
− | + | |align="center"|Discussion: | |
|- | |- | ||
− | |align="center"|One | + | |align="center"|One light year(m) |
|align="center"|99<sup>8</sup> | |align="center"|99<sup>8</sup> | ||
|align="center"|9,227,446,944,279,201 | |align="center"|9,227,446,944,279,201 | ||
|align="center"|9,460,730,472,580,800 (exact) | |align="center"|9,460,730,472,580,800 (exact) | ||
− | |align="left"| | + | |align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references. |
|- | |- | ||
− | |align="center"|Earth | + | |align="center"|Earth Surface(m<sup>2</sup>) |
|align="center"|69<sup>8</sup> | |align="center"|69<sup>8</sup> | ||
|align="center"|513,798,374,428,641 | |align="center"|513,798,374,428,641 | ||
− | |align="center"|5.10072 | + | |align="center"|5.10072*10<sup>14</sup> |
− | |align="left"|99<sup>8</sup> and 69<sup>8</sup> are | + | |align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references. |
|- | |- | ||
− | |align="center"| | + | |align="center"|Ocean's volume(m<sup>3</sup>) |
|align="center"|9<sup>19</sup> | |align="center"|9<sup>19</sup> | ||
|align="center"|1,350,851,717,672,992,089 | |align="center"|1,350,851,717,672,992,089 | ||
− | |align="center"|1 | + | |align="center"|1,332*10<sup>18</sup> |
|align="left"| | |align="left"| | ||
|- | |- | ||
Line 55: | Line 44: | ||
|align="center"|75<sup>4</sup> | |align="center"|75<sup>4</sup> | ||
|align="center"|31,640,625 | |align="center"|31,640,625 | ||
− | |align="center"|31,557,600 (Julian calendar) | + | |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: | |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 | + | "Lots of emails mention the physicist favorite, 1 year = pi x 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." Pi x 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 pi, but it should be multiplied by the factor of ten.<br><br> |
− | + | 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. | |
− | Using the traditional definitions that a second is 1/ | ||
|- | |- | ||
− | |align="center"|Seconds in a year ( | + | |align="center"|Seconds in a year (rent method) |
− | |align="center"|525,600 | + | |align="center"|525,600 x 60 |
|align="center"|31,536,000 | |align="center"|31,536,000 | ||
− | |align="center"|31,557,600 (Julian calendar) | + | |align="center"|31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar) |
− | |align="left"| | + | |align="left"|“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. |
|- | |- | ||
|align="center"|Age of the universe (seconds) | |align="center"|Age of the universe (seconds) | ||
|align="center"|15<sup>15</sup> | |align="center"|15<sup>15</sup> | ||
|align="center"|437,893,890,380,859,375 | |align="center"|437,893,890,380,859,375 | ||
− | |align="center"| | + | |align="center"|4.354±0.012*10<sup>17</sup> (best estimate; exact value unknown) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
|align="center"|Planck's constant | |align="center"|Planck's constant | ||
− | |align="center"|< | + | |align="center"|1/(30<sup>π<sup>e</sup></sup>) |
− | |align="center"|6. | + | |align="center"|6.68499014108082*10<sup>-34</sup> (rounded) |
− | |align="center"|6.62606957 | + | |align="center"|6.62606957*10<sup>-34</sup> |
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics. | |align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics. | ||
|- | |- | ||
|align="center"|Fine structure constant | |align="center"|Fine structure constant | ||
− | |align="center"| | + | |align="center"|1/140 |
− | |align="center"|0. | + | |align="center"|0.00714285717142857171428571, etc. (repeating 71428571) |
− | |align="center"|0. | + | |align="center"|0.00729735257 (accepted value as of 2011), 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. | + | |align="left"|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), including the infamous {{w|Arthur Eddington|Sir Arthur Adding-One}}. |
|- | |- | ||
|align="center"|Fundamental charge | |align="center"|Fundamental charge | ||
− | |align="center"|< | + | |align="center"|3/(14 * π<sup>π<sup>π</sup></sup>) |
− | |align="center"|1.59895121062716 | + | |align="center"|1.59895121062716*10<sup>-19</sup> (rounded) |
− | |align="center"|1.602176565 | + | |align="center"|1.602176565*10<sup>-19</sup> (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"|Telephone number for the | + | |align="center"|Telephone number for the White House Switchboard |
− | |align="center"|< | + | |align="center"|1/<br /> |
− | |align="center"| | + | <sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>) |
− | |align="center"| | + | |align="center"|.2024561414 (truncated) |
+ | |align="center"|2024561414 | ||
|align="left"| | |align="left"| | ||
|- | |- | ||
− | |align="center"|Jenny's | + | |align="center"|Jenny's Constant |
− | |align="center"|< | + | |align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup> |
− | |align="center"|867. | + | |align="center"|867.530901981685 (approximately) |
− | |align="center"| | + | |align="center"|8675309 |
− | |align="left"| | + | |align="left"|Jenny's constant comes from Tommy Tutone's tune {{w|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 {{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. |
|- | |- | ||
− | |align="center"|World | + | |align="center"|World Population Estimate (billions) |
− | |align="center"|Equivalent to | + | |align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion |
− | |align="center"|2005 | + | |align="center"|2005 6.5 |
− | 2006 | + | 2006 6.6 |
− | 2007 | + | 2007 6.7 |
− | 2008 | + | 2008 6.7 |
− | 2009 | + | 2009 6.8 |
− | 2010 | + | 2010 6.9 |
− | 2011 | + | 2011 7 |
− | 2012 | + | 2012 7 |
− | 2013 | + | 2013 7.1 |
− | 2014 | + | 2014 7.2 |
− | 2015 | + | 2015 7.3 |
− | 2016 | + | 2016 7.3 |
− | 2017 | + | 2017 7.4 |
− | 2018 | + | 2018 7.5 |
− | 2019 | + | 2019 7.6 |
− | 2020 | + | 2020 7.6 |
− | 2021 | + | 2021 7.7 |
− | 2022 | + | 2022 7.8 |
− | 2023 | + | 2023 7.9 |
− | 2024 | + | 2024 7.9 |
− | 2025 | + | 2025 8 |
− | 2026 | + | 2026 8.1 |
− | 2027 | + | 2027 8.2 |
− | 2028 | + | 2028 8.2 |
− | 2029 | + | 2029 8.3 |
− | 2030 | + | 2030 8.4 |
− | 2031 | + | 2031 8.5 |
− | 2032 | + | 2032 8.5 |
− | |||
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|align="center"| | |align="center"| | ||
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"|U.S. | + | |align="center"|U.S. Population Estimate (millions) |
− | |align="center"|Equivalent to | + | |align="center"|Equivalent to 310+3*(Year - 2010) million |
− | |align="center"|2000 | + | |align="center"|2000 280 |
− | 2001 | + | 2001 283 |
− | 2002 | + | 2002 286 |
− | 2003 | + | 2003 289 |
− | 2004 | + | 2004 292 |
− | 2005 | + | 2005 295 |
− | 2006 | + | 2006 298 |
− | 2007 | + | 2007 301 |
− | 2008 | + | 2008 304 |
− | 2009 | + | 2009 307 |
− | 2010 | + | 2010 310 |
− | 2011 | + | 2011 313 |
− | 2012 | + | 2012 316 |
− | 2013 | + | 2013 319 |
− | 2014 | + | 2014 322 |
− | 2015 | + | 2015 325 |
− | 2016 | + | 2016 328 |
− | 2017 | + | 2017 331 |
− | 2018 | + | 2018 334 |
− | 2019 | + | 2019 337 |
− | 2020 | + | 2020 340 |
− | 2021 | + | 2021 343 |
− | 2022 | + | 2022 346 |
− | 2023 | + | 2023 349 |
− | 2024 | + | 2024 352 |
− | 2025 | + | 2025 355 |
− | 2026 | + | 2026 358 |
− | 2027 | + | 2027 361 |
− | 2028 | + | 2028 364 |
− | 2029 | + | 2029 367 |
− | 2030 | + | 2030 370 |
− | 2031 | + | 2031 373 |
− | 2032 | + | 2032 376 |
− | |||
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− | |||
|align="center"| | |align="center"| | ||
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"|Electron rest energy | + | |align="center"|Electron rest energy |
− | |align="center"|< | + | |align="center"|e/7<sup>16</sup> Joules |
− | |align="center"|8.17948276564429 | + | |align="center"|8.17948276564429*10<sup>-14</sup> |
− | |align="center"|8.18710438 | + | |align="center"|8.18710438*10<sup>-14</sup> (rounded) |
|align="left"| | |align="left"| | ||
|- | |- | ||
− | |align="center"|Light year (miles) | + | |align="center"|Light-year(miles) |
− | |align="center"|2<sup>42.42</sup> | + | |align="center"|2<sup>(42.42)</sup> |
− | |align="center"| | + | |align="center"|5884267614436.97 (rounded) |
− | |align="center"| | + | |align="center"|9460730472580800 (meters in a light-year, by definition) / 1609.344 (meters in a mile) = 8212439646337500/1397 (exact) = 5878625373183.61 (rounded) |
− | |align="left"|{{w|42 (number)|42}} is, according to | + | |align="left"|{{w|42 (number)|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. |
|- | |- | ||
− | |align="center"| | + | |align="center"|sin(60°) = <sup>3</sup>√/2 |
− | |align="center"| | + | |align="center"|e/π |
− | |align="center"|0.8652559794 | + | |align="center"|0.8652559794 (rounded) |
− | |align="center"|0.8660254038 | + | |align="center"|0.8660254038 (rounded) |
|align="left"| | |align="left"| | ||
|- | |- | ||
− | |align="center"| | + | |align="center"|√3 |
− | |align="center"| | + | |align="center"|2e/π |
− | |align="center"|1.7305119589 | + | |align="center"|1.7305119589 (rounded) |
− | |align="center"|1.7320508076 | + | |align="center"|1.7320508076 (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"| | + | |align="center"|gamma(Euler's gamma constant) |
− | |align="center"| | + | |align="center"|1/√3 |
− | |align="center"|0.5773502692 | + | |align="center"|0.5773502692 (rounded) |
− | |align="center"|0. | + | |align="center"|0.5772156649015328606065120900824024310421... |
− | |align="left"| | + | |align="left"|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}}. |
|- | |- | ||
|align="center"|Feet in a meter | |align="center"|Feet in a meter | ||
− | |align="center"|< | + | |align="center"|5/(<sup>e</sup>√π) |
|align="center"|3.2815481951 | |align="center"|3.2815481951 | ||
− | |align="center"|3.280839895 | + | |align="center"|1/.3048 (exact) = 3.280839895 (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"| | + | |align="center"|√5 |
− | |align="center"| | + | |align="center"|2/e + 3/2 |
− | |align="center"|2.2357588823 | + | |align="center"|2.2357588823 (rounded) |
− | |align="center"|2.2360679775 | + | |align="center"|2.2360679775 (rounded) |
|align="left"| | |align="left"| | ||
|- | |- | ||
|align="center"|Avogadro's number | |align="center"|Avogadro's number | ||
− | |align="center"|< | + | |align="center"|69<sup>π<sup>√5</sup></sup> |
− | |align="center"|6.02191201246329 | + | |align="center"|6.02191201246329*10<sup>23</sup> (rounded) |
− | |align="center"|6.02214129 | + | |align="center"|6.02214129*10<sup>23</sup> (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"|Gravitational constant | + | |align="center"|Gravitational constant G |
− | |align="center"|< | + | |align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup> |
− | |align="center"|6. | + | |align="center"|6.67361106850561*10<sup>-11</sup> (rounded) |
− | |align="center"|6.67385 | + | |align="center"|6.67385*10<sup>-11</sup> (rounded) |
− | |align="left"|The universal {{w|gravitational constant}} G is equal to | + | |align="left"|The universal {{w|gravitational constant}} G is equal to F*r<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. |
|- | |- | ||
− | |align="center"| | + | |align="center"|R(gas constant) |
− | |align="center"| | + | |align="center"|(e+1) √5 |
− | |align="center"|8.3143309279 | + | |align="center"|8.3143309279 (rounded) |
− | |align="center"|8.3144622 | + | |align="center"|8.3144622 (rounded) |
|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). | |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). | ||
|- | |- | ||
− | |align="center"| | + | |align="center"|Proton-electron mass ratio |
− | |align="center"|< | + | |align="center"|6*π<sup>5</sup> |
− | |align="center"|1836.1181087117 | + | |align="center"|1836.1181087117 (rounded) |
− | |align="center"|1836.15267246 | + | |align="center"|1836.15267246 (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"|Liters in a | + | |align="center"|Liters in a gallon (U.S. liquid gallon, defined by law as 231 cubic inches) |
− | |align="center"| | + | |align="center"|3 + π/4 |
− | |align="center"|3.7853981634 | + | |align="center"|3.7853981634 (rounded) |
|align="center"|3.785411784 (exact) | |align="center"|3.785411784 (exact) | ||
− | |align="left"| | + | |align="left"| |
|- | |- | ||
|align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub> | |align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub> | ||
|align="center"|6 + ln(45) | |align="center"|6 + ln(45) | ||
− | |align="center"|9.8066624898 | + | |align="center"|9.8066624898 (rounded) |
− | |align="center"|9.80665 | + | |align="center"|9.80665 (standard) |
− | |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 | + | |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/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.<br><br>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. |
− | |||
− | Randall used a letter | ||
|- | |- | ||
− | |align="center"| | + | |align="center"|Proton-electron mass ratio |
− | |align="center"|< | + | |align="center"|(e<sup>8</sup> - 10) / ϕ |
− | |align="center"|1836.1530151398 | + | |align="center"|1836.1530151398 (rounded) |
− | |align="center"|1836.15267246 | + | |align="center"|1836.15267246 (rounded) |
− | |align="left"| | + | |align="left"|ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties. |
|- | |- | ||
− | |align="center"|Ruby laser wavelength | + | |align="center"|Ruby laser wavelength |
− | |align="center"|< | + | |align="center"|1 / (1200<sup>2</sup>) |
− | |align="center"| | + | |align="center"|.00000069444444444444... (repeating decimal) |
− | |align="center"| | + | |align="center"|694.3 nm |
− | |align="left"|The | + | |align="left"|The ruby laser wavelength varies because “ruby” is not clearly defined. |
|- | |- | ||
− | |align="center"|Mean Earth | + | |align="center"|Mean Earth Radius |
− | |align="center"|< | + | |align="center"|(5<sup>8</sup>)*6e |
− | |align="center"|6,370,973. | + | |align="center"|2343750e (exact), 6,370,973.035450887 (6370 km, 973 m, 3 cm, 5 mm, 450,887 nm) (rounded) |
− | |align="center"|6,371,008.7 ( | + | |align="center"|6,371,008.7 (International Union of Geodesy and Geophysics 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 | + | |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"| | + | |align="center"|√2 |
− | |align="center"| | + | |align="center"|3/5 + π/(7-π) |
− | |align="center"|1.4142200581 | + | |align="center"|1.4142200581 (rounded) |
− | |align="center"|1.4142135624 | + | |align="center"|1.4142135624 (rounded) |
− | |align="left"| | + | |align="left"| |
|- | |- | ||
− | |align="center"| | + | |align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) |
− | |align="center"| | + | |align="center"|1/2 |
|align="center"|0.5 | |align="center"|0.5 | ||
|align="center"|0.5 (exact) | |align="center"|0.5 (exact) | ||
− | |align="left"| | + | |align="left"|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]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}. |
|- | |- | ||
− | |align="center"|γ (Euler's gamma constant) | + | |align="center"|γ(Euler's gamma constant) |
− | |align="center"|< | + | |align="center"|e/3<sup>4</sup> + e/5 |
− | |align="center"|0.5772154006 | + | |align="center"|0.5772154006 (rounded) |
− | |align="center"|0. | + | |align="center"|0.5772156649015328606065120900824024310421... |
− | |align="left"| | + | |align="left"|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}}. |
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− | |align="center"| | + | |align="center"|√5 |
− | |align="center"| | + | |align="center"|(13 + 4π) / (24 - 4π) |
− | |align="center"|2.2360678094 | + | |align="center"|2.2360678094 (rounded) |
− | |align="center"|2.2360679775 | + | |align="center"|2.2360679775 (rounded) |
|align="left"| | |align="left"| | ||
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− | |align="center"|< | + | |align="center"|Σ 1/n<sup>n</sup> |
− | |align="center"| | + | |align="center"|ln(3)<sup>e</sup> |
− | |align="center"|1.2912987577 | + | |align="center"|1.2912987577 (rounded) |
− | |align="center"|1.2912859971 | + | |align="center"|1.2912859971 (rounded) |
|align="left"| | |align="left"| | ||
|} | |} | ||
− | + | {{w|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: | |
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− | + | ''e''<sup>π''i''</sup> + 1 = 0 | |
− | + | ''e''<sup>π''i''</sup> = -1 | |
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− | + | (''e''<sup>''i''</sup>)<sup>π</sup> = -1 | |
− | < | + | ''e''<sup>''i''</sup> = <sup>π</sup>√(-1) |
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− | + | We get nowhere. | |
− | + | 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. (This is later referenced in [http://what-if.xkcd.com/73/ Lethal Neutrinos]). | |
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− | + | The software referred to in the comic is [http://mrob.com/pub/ries/ ries], a 'reverse calculator' which forms equations matching a given number. | |
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==Transcript== | ==Transcript== | ||
Line 377: | Line 316: | ||
|align="center" | Accurate to within: | |align="center" | Accurate to within: | ||
|- | |- | ||
− | |align="center" | One light | + | |align="center" | One light year(m) |
|align="center" | 99<sup>8</sup> | |align="center" | 99<sup>8</sup> | ||
|align="center" | one part in 40 | |align="center" | one part in 40 | ||
Line 385: | Line 324: | ||
|align="center" | one part in 130 | |align="center" | one part in 130 | ||
|- | |- | ||
− | |align="center" | | + | |align="center" | Ocean's volume(m<sup>3</sup>) |
|align="center" | 9<sup>19</sup> | |align="center" | 9<sup>19</sup> | ||
|align="center" | one part in 70 | |align="center" | one part in 70 | ||
Line 393: | Line 332: | ||
|align="center" | one part in 400 | |align="center" | one part in 400 | ||
|- | |- | ||
− | |align="center" | Seconds in a year ( | + | |align="center" | Seconds in a year (rent method) |
|align="center" | 525,600 x 60 | |align="center" | 525,600 x 60 | ||
|align="center" | one part in 1400 | |align="center" | one part in 1400 | ||
Line 414: | Line 353: | ||
|- | |- | ||
|align="center"|White House Switchboard | |align="center"|White House Switchboard | ||
− | |colspan="2" align="center"|1 / | + | |colspan="2" align="center"|1/<br /> |
+ | <sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>) | ||
|- | |- | ||
|align="center"|Jenny's Constant | |align="center"|Jenny's Constant | ||
Line 426: | Line 366: | ||
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 437: | Line 377: | ||
7.2 = World population in billions. | 7.2 = World population in billions. | ||
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Version for US population: | Version for US population: | ||
Line 456: | Line 395: | ||
|- | |- | ||
|align="center"|Electron rest energy | |align="center"|Electron rest energy | ||
− | |align="center"|e/7<sup>16</sup> | + | |align="center"|e/7<sup>16</sup> Joules |
|align="center"|one part in 1000 | |align="center"|one part in 1000 | ||
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Line 463: | Line 402: | ||
|align="center"|one part in 1000 | |align="center"|one part in 1000 | ||
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− | |colspan="2" align="center"|sin(60°) = | + | |colspan="2" align="center"|sin(60°) = <sup>3</sup>√/2 = e/π |
|align="center"|one part in 1000 | |align="center"|one part in 1000 | ||
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Line 469: | Line 408: | ||
|align="center"|one part in 1000 | |align="center"|one part in 1000 | ||
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− | |align="center"| | + | |align="center"|gamma(Euler's gamma constant) |
|align="center"|1/√3 | |align="center"|1/√3 | ||
− | |align="center"| | + | |align="center"|One part in 4000 |
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|align="center"|Feet in a meter | |align="center"|Feet in a meter | ||
Line 485: | Line 424: | ||
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|align="center"|Gravitational constant G | |align="center"|Gravitational constant G | ||
− | |align="center"|1 / e<sup>( | + | |align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup> |
|align="center"|one part in 25,000 | |align="center"|one part in 25,000 | ||
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− | |align="center"|R (gas constant) | + | |align="center"|R(gas constant) |
|align="center"|(e+1) √5 | |align="center"|(e+1) √5 | ||
|align="center"|one part in 50,000 | |align="center"|one part in 50,000 |