Editing 1047: Approximations

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| date      = April 25, 2012
 
| date      = April 25, 2012
 
| title    = Approximations
 
| title    = Approximations
−
| before    = [[#Explanation|↓ Skip to explanation ↓]]
 
 
| 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==
−
 
 
This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.
 
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 {{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.  
+
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 physicist's mnemonic that it is "π times 10<sup>7</sup>," 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 {{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.   
 
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.   
  
−
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.
+
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.
  
−
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.
+
The world population estimate for 2017 is still accurate. The estimate is 7.4 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.
  
 
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]].  
 
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]].  
  
−
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 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 π-th root is rarely helpful.  Similarly, {{w|e (mathematical constant)|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]).
−
 
 
−
===Equations===
 
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
−
!align="center"|Thing to be approximated:
+
|align="center"|Thing to be approximated:
−
!align="center"|Formula proposed
+
|align="center"|Formula proposed:
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!align="center"|Resulting approximate value
+
|align="center"|Resulting approximate value:
−
!align="center"|Correct value
+
|align="center"|Correct value:
−
!align="center"|Discussion
+
|align="center"|Discussion:
 
|-
 
|-
−
|align="center"|One {{w|light year}} (meters)
+
|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"|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="left"|Based on 365.25 days per year (see below). 99<sup>8</sup> and 69<sup>8</sup> are sexual references.  
 
|-
 
|-
−
|align="center"|Earth's surface (m<sup>2</sup>)
+
|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 × 10<sup>14</sup>
+
|align="center"|5.10072*10<sup>14</sup>
−
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are [[487: Numerical Sex Positions|sexual references]].
+
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references.
 
|-
 
|-
−
|align="center"|Oceans' volume (m<sup>3</sup>)
+
|align="center"|Oceans' 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"|1.332 × 10<sup>18</sup>
+
|align="center"|1,332*10<sup>18</sup>
 
|align="left"|
 
|align="left"|
 
|-
 
|-
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|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian 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 × 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.
+
"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." π 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 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.
+
 
 +
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 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.
 
|-
 
|-
 
|align="center"|Seconds in a year (''Rent'' method)
 
|align="center"|Seconds in a year (''Rent'' method)
−
|align="center"|525,600 × 60
+
|align="center"|525,600 x 60
 
|align="center"|31,536,000
 
|align="center"|31,536,000
 
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
 
|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="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"|(4.354 ± 0.012) × 10<sup>17</sup> (best estimate; exact value unknown)
+
|align="center"|4.354±0.012*10<sup>17</sup> (best estimate; exact value unknown)
 
|align="left"|This one will slowly get more accurate as the universe ages.
 
|align="left"|This one will slowly get more accurate as the universe ages.
 
|-
 
|-
 
|align="center"|Planck's constant
 
|align="center"|Planck's constant
−
|align="center"|<math>\frac {1} {30^{\pi^e}}</math>
+
|align="center"|1/(30<sup>π<sup>e</sup></sup>)
−
|align="center"|6.6849901410 × 10<sup>−34</sup>
+
|align="center"|6.68499014108082*10<sup>−34</sup> (rounded)
−
|align="center"|6.62606957 × 10<sup>−34</sup>
+
|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"|<math>\frac{1}{140}</math>
+
|align="center"|1/140
 
|align="center"|0.00<span style="text-decoration: overline;">714285</span>
 
|align="center"|0.00<span style="text-decoration: overline;">714285</span>
 
|align="center"|0.0072973525664 (accepted value as of 2014), close to 1/137
 
|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="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"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
+
|align="center"|3/(14 * π<sup>π<sup>π</sup></sup>)
−
|align="center"|1.59895121062716 × 10<sup>−19</sup>
+
|align="center"|1.59895121062716*10<sup>−19</sup> (rounded)
−
|align="center"|1.602176565 × 10<sup>−19</sup>
+
|align="center"|1.602176565*10<sup>−19</sup> (rounded)
−
|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]].)
+
|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]].)
 
|-
 
|-
−
|align="center"|Telephone number for the {{w|White House}} switchboard
+
|align="center"|Telephone number for the White House Switchboard
−
|align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math>
+
|align="center"|1/<br />
−
|align="center"|0.2024561414932
+
<sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8)</sup>)
−
|align="center"|202-456-1414
+
|align="center"|.2024561414 (truncated)
 +
|align="center"|2024561414
 
|align="left"|
 
|align="left"|
 
|-
 
|-
−
|align="center"|Jenny's constant
+
|align="center"|Jenny's Constant
−
|align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
+
|align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup>
−
|align="center"|867.5309019
+
|align="center"|867.530901981685 (approximately)
−
|align="center"|867-5309
+
|align="center"|8675309
−
|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 not only prime but a {{w|twin prime}} because 8675311 is also a prime.  
+
|align="left"|"Jenny's constant" is actually a telephone number referenced in Tommy Tutone's 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number not only prime but a {{w|twin prime}} because 8675311 is also a prime.  
 
|-
 
|-
−
|align="center"|World population estimate (billions)
+
|align="center"|World Population Estimate (billions)
−
|align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
+
|align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion
−
|align="center"|2005 — 6.5<br>
+
|align="center"|2005 6.5
−
2006 — 6.6<br>
+
2006 6.6
−
2007 — 6.7<br>
+
2007 6.7
−
2008 — 6.7<br>
+
2008 6.7
−
2009 — 6.8<br>
+
2009 6.8
−
2010 — 6.9<br>
+
2010 6.9
−
2011 — 7.0<br>
+
2011 7
−
2012 — 7.0<br>
+
2012 7
−
2013 — 7.1<br>
+
2013 7.1
−
2014 — 7.2<br>
+
2014 7.2
−
2015 — 7.3<br>
+
2015 7.3
−
2016 — 7.3<br>
+
2016 7.3
−
2017 — 7.4<br>
+
2017 7.4
−
2018 — 7.5<br>
+
2018 7.5
−
2019 — 7.6<br>
+
2019 7.6
−
2020 — 7.6<br>
+
2020 7.6
−
2021 — 7.7<br>
+
2021 7.7
−
2022 — 7.8<br>
+
2022 7.8
−
2023 — 7.9<br>
+
2023 7.9
−
2024 — 7.9<br>
+
2024 7.9
−
2025 — 8.0<br>
+
2025 8
−
2026 — 8.1<br>
+
2026 8.1
−
2027 — 8.2<br>
+
2027 8.2
−
2028 — 8.2<br>
+
2028 8.2
−
2029 — 8.3<br>
+
2029 8.3
−
2030 — 8.4<br>
+
2030 8.4
−
2031 — 8.5<br>
+
2031 8.5
−
2032 — 8.5<br>
+
2032 8.5
−
2033 — 8.6<br>
 
−
2034 — 8.7<br>
 
−
2035 — 8.8<br>
 
 
|align="center"|
 
|align="center"|
−
|align="left"|Grows by 75 million every year on average. As of 2023, a bit too small.
+
|align="left"|
 
|-
 
|-
−
|align="center"|U.S. population estimate (millions)
+
|align="center"|U.S. Population Estimate (millions)
−
|align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
+
|align="center"|Equivalent to 310+3*(Year - 2010) million
−
|align="center"|2000 — 280<br>
+
|align="center"|2000 280
−
2001 — 283<br>
+
2001 283
−
2002 — 286<br>
+
2002 286
−
2003 — 289<br>
+
2003 289
−
2004 — 292<br>
+
2004 292
−
2005 — 295<br>
+
2005 295
−
2006 — 298<br>
+
2006 298
−
2007 — 301<br>
+
2007 301
−
2008 — 304<br>
+
2008 304
−
2009 — 307<br>
+
2009 307
−
2010 — 310<br>
+
2010 310
−
2011 — 313<br>
+
2011 313
−
2012 — 316<br>
+
2012 316
−
2013 — 319<br>
+
2013 319
−
2014 — 322<br>
+
2014 322
−
2015 — 325<br>
+
2015 325
−
2016 — 328<br>
+
2016 328
−
2017 — 331<br>
+
2017 331
−
2018 — 334<br>
+
2018 334
−
2019 — 337<br>
+
2019 337
−
2020 — 340<br>
+
2020 340
−
2021 — 343<br>
+
2021 343
−
2022 — 346<br>
+
2022 346
−
2023 — 349<br>
+
2023 349
−
2024 — 352<br>
+
2024 352
−
2025 — 355<br>
+
2025 355
−
2026 — 358<br>
+
2026 358
−
2027 — 361<br>
+
2027 361
−
2028 — 364<br>
+
2028 364
−
2029 — 367<br>
+
2029 367
−
2030 — 370<br>
+
2030 370
−
2031 — 373<br>
+
2031 373
−
2032 — 376<br>
+
2032 376
−
2033 — 379<br>
 
−
2034 — 382<br>
 
−
2035 — 385<br>
 
 
|align="center"|
 
|align="center"|
−
|align="left"|Grows by 3 million each year. As of 2021 the actual number is ~13 million smaller.
+
|align="left"|
 
|-
 
|-
−
|align="center"|Electron rest energy (joules)
+
|align="center"|Electron rest energy
−
|align="center"|<math>\frac {e} {7^{16}}</math>
+
|align="center"|e/7<sup>16</sup> J
−
|align="center"|8.17948276564429 × 10<sup>−14</sup>
+
|align="center"|8.17948276564429*10<sup>−14</sup>
−
|align="center"|8.18710438 × 10<sup>−14</sup>
+
|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"|5,884,267,614,436.97
+
|align="center"|5884267614436.97 (rounded)
−
|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)
+
|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 {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
+
|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"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
+
|align="center"|sin(60°) = √3/2
−
|align="center"|<math>\frac{e}{\pi}</math>
+
|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"|<math>\sqrt 3</math>
+
|align="center"|√3
−
|align="center"|<math>\frac{2e}{\pi}</math>
+
|align="center"|2e/π
−
|align="center"|1.7305119589
+
|align="center"|1.7305119589 (rounded)
−
|align="center"|1.7320508076
+
|align="center"|1.7320508076 (rounded)
−
|align="left"|Same as the above
+
|align="left"|
 
|-
 
|-
−
|align="center"|γ (Euler's gamma constant)
+
|align="center"|γ(Euler's gamma constant)
−
|align="center"|<math>\frac {1} {\sqrt 3}</math>
+
|align="center"|1/√3
−
|align="center"|0.5773502692
+
|align="center"|0.5773502692 (rounded)
−
|align="center"|0.5772156649
+
|align="center"|0.5772156649015328606065120900824024310421...
−
|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="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"|<math>\frac {5} {\sqrt[e]\pi}</math>
+
|align="center"|5/(<sup>e</sup>√π)
 
|align="center"|3.2815481951
 
|align="center"|3.2815481951
−
|align="center"|3.280839895
+
|align="center"|1/0.3048 (exact) = 3.280839895 (rounded)
−
|align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
+
|align="left"|
 
|-
 
|-
−
|align="center"|<math>\sqrt 5</math>
+
|align="center"|√5
−
|align="center"|<math>\frac{2}{e} + \frac32</math>
+
|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"|<math>69^{\pi^\sqrt{5}}</math>
+
|align="center"|69<sup>π<sup>√5</sup></sup>
−
|align="center"|6.02191201246329 × 10<sup>23</sup>
+
|align="center"|6.02191201246329*10<sup>23</sup> (rounded)
−
|align="center"|6.02214129 × 10<sup>23</sup>
+
|align="center"|6.02214129*10<sup>23</sup> (rounded)
−
|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.
+
|align="left"|Also called a Mole for shorthand, this is (roughly) the number of individual atoms in twelve grams of pure Carbon. Used in basically every application of chemistry.
 
|-
 
|-
−
|align="center"|Gravitational constant ''G''
+
|align="center"|Gravitational constant G
−
|align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
+
|align="center"|1 / e<sup>(π - 1)<sup>(π + 1)</sup></sup>
−
|align="center"|6.6736110685 × 10<sup>−11</sup>
+
|align="center"|6.67361106850561*10<sup>−11</sup> (rounded)
−
|align="center"|6.67385 × 10<sup>−11</sup>
+
|align="center"|6.67385*10<sup>−11</sup> (rounded)
−
|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.
+
|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"|''R'' (gas constant)
+
|align="center"|R (gas constant)
−
|align="center"|<math>(e + 1) \sqrt 5</math>
+
|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"|Proton–electron mass ratio
+
|align="center"|Proton-electron mass ratio
−
|align="center"|<math>6 \pi^5</math>
+
|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"| 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.
+
|align="left"|
 
|-
 
|-
−
|align="center"|Liters in a {{w|gallon}}
+
|align="center"|Liters in a gallon (U.S. liquid gallon, defined by law as 231 cubic inches)
−
|align="center"|<math>3 + \frac{\pi}{4}</math>
+
|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"|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).
+
|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&nbsp;m/s<sup>2</sup>, which is exactly 35.30394&nbsp;km/h/s (about 32.174&nbsp;ft/s<sup>2</sup>, or 21.937&nbsp;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.
+
|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&nbsp;m/s<sup>2</sup>, which is exactly 35.30394&nbsp;(km/h)/s (about 32.174&nbsp;ft/s<sup>2</sup>, or 21.937&nbsp;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 ''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.
 
 
|-
 
|-
−
|align="center"|Proton–electron mass ratio
+
|align="center"|Proton-electron mass ratio
−
|align="center"|<math>\frac {e^8 - 10} {\phi}</math>
+
|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"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
+
|align="left"|ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties.
 
|-
 
|-
−
|align="center"|Ruby laser wavelength (meters)
+
|align="center"|Ruby laser wavelength
−
|align="center"|<math>\frac{1}{1200^2}</math>
+
|align="center"|1 / (1200<sup>2</sup>)
−
|align="center"|6.9<span style="text-decoration: overline;">444</span> × 10<sup>−7</sup>
+
|align="center"|0.00000069<span style="text-decoration: overline;">444</span>
−
|align="center"|~6.943 × 10<sup>−7</sup>
+
|align="center"|694.3&nbsp;nm
−
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
+
|align="left"|The ruby laser wavelength varies because "ruby" is not clearly defined.
 
|-
 
|-
−
|align="center"|Mean Earth radius (meters)
+
|align="center"|Mean Earth Radius
−
|align="center"|<math>5^8 6e</math>
+
|align="center"|(5<sup>8</sup>)*6e
−
|align="center"|6,370,973.035
+
|align="center"|2343750e (exact), 6,370,973.035450887270375673760982 (6370&nbsp;km, 973&nbsp;m, 35&nbsp;mm, 450&nbsp;&mu;m, 887&nbsp;nm, 270&nbsp;pm, 375&nbsp;fm, 673&nbsp;am, 760&nbsp;zm, 982&nbsp;ym) (rounded)
−
|align="center"|6,371,008.7 (IUGG definition)
+
|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&nbsp;m) plus 1/3 of the polar radius (6,356,752.3&nbsp;m).
 
|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&nbsp;m) plus 1/3 of the polar radius (6,356,752.3&nbsp;m).
 
|-
 
|-
−
|align="center"|<math>\sqrt 2</math>
+
|align="center"|√2
−
|align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
+
|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"|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="left"|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".
 
|-
 
|-
−
|align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
+
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7)
−
|align="center"|<math>\frac12</math>
+
|align="center"|1/2
 
|align="center"|0.5
 
|align="center"|0.5
 
|align="center"|0.5 (exact)
 
|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="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"|γ(Euler's gamma constant)
−
|align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
+
|align="center"|e/3<sup>4</sup> + e/5
−
|align="center"|0.5772154006
+
|align="center"|0.5772154006 (rounded)
−
|align="center"|0.5772156649
+
|align="center"|0.5772156649015328606065120900824024310421...
−
|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="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"|<math>\sqrt 5</math>
+
|align="center"|√5
−
|align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
+
|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"|
 
|-
 
|-
−
|align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
+
|align="center"|Σ 1/n<sup>n</sup>
−
|align="center"|<math>\ln(3)^e</math>
+
|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"|
 
|}
 
|}
Line 311: Line 301:
 
===Proof===
 
===Proof===
  
−
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:
+
One of the "approximations" actually is precisely correct: cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2.  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:
+
cos(π/7) + cos(3π/7) + cos(5π/7)
  
−
:<math>\begin{align}
+
* Multiplying by 1 (or by a number divided by itself) leaves the equation unchanged.
−
&= \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}} \\
 
−
&= \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:
+
= (cos(π/7) + cos(3π/7) + cos(5π/7)) (2sin(π/7)/(2sin(π/7)))
  
−
:<math>\begin{align}
+
* The 2sin(π/7) on the top of the fraction is multiplied through the original equation.
−
&= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
 
−
&= \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:
+
= (2cos(π/7)sin(π/7) + 2cos(3π/7)sin(π/7) + 2cos(5π/7)sin(π/7))/(2sin(π/7))
  
−
<!--:<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>-->
+
* Use the trigonometric identity 2cos(A)sin(B)=sin(A+B)-sin(A-B) on the 2nd two terms ([2cos(3π/7)sin(π/7)] + {2cos(5π/7)sin(π/7)}) /(2sin(π/7))
−
: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)
 
  
−
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)):
+
= (2cos(π/7)sin(π/7) + [sin(3π/7+π/7) - sin(3π/7-π/7)] + {sin(5π/7+π/7) - sin(5π/7-π/7)}) (1/2sin(π/7))<br>
 +
= (2cos(π/7)sin(π/7) + [sin(4π/7) - sin(2π/7)] + {sin(6π/7) - sin(4π/7)})/(2sin(π/7))
  
−
<!--:<math>0 = 1 + 2 ( \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} ) </math>-->
+
* Use the trigonometric identity 2cos(A)sin(A) = sin(2A) on the first term (2cos(π/7)sin(π/7))
−
:0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7))
 
  
−
Hence
+
= (sin(2π/7) + [sin(4π/7) - sin(2π/7)] + {sin(6π/7) - sin(4π/7)}) (1/2sin(π/7))<br>
 +
= (sin(6π/7) + [sin(2π/7) - sin(2π/7)] + {sin(4π/7) - sin(4π/7)}) (1/2sin(π/7))<br>
 +
= (sin(6π/7))/(2sin(π/7))
  
−
<!--:<math>\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = - 1/2 </math>-->
+
* Note that 6π/7 = (7π - π)/7 = 7π/7 - π/7 = π - π/7.
−
:cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2
 
  
−
which, because cos(x) = cos(pi - x),<!--<math>\cos (x) = - \cos(\pi - x)</math>,--> can be rewritten as
+
= (sin(π - π/7))/(2sin(π/7))
  
−
<!--:<math>\cos \frac{5\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{pi}{7} = 1/2 </math>-->
+
*Since sines of supplementary angles are equal.
−
:cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2
 
  
−
Q.E.D.
+
= (sin(π/7))/(2sin(π/7))<br>
 +
= (1/2) (sin(π/7)/sin(π/7))<br>
 +
= 1/2
  
 
==Transcript==
 
==Transcript==
Line 414: Line 381:
 
|-
 
|-
 
|align="center"|White House Switchboard
 
|align="center"|White House Switchboard
−
|colspan="2" align="center"|1 / (e<sup><sup>π</sup>√(1 + <sup>(e-1)</sup>√8)</sup>)
+
|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

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