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. | ||
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==Explanation== | ==Explanation== | ||
− | + | {{incomplete|Both tables are bad}} | |
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. | ||
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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 | + | The {{w|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 | + | 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. 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]). |
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{| class="wikitable" | {| 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 {{w|light year}} (meters) | |align="center"|One {{w|light year}} (meters) | ||
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|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 | + | |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's surface (m<sup>2</sup>) | ||
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|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 | + | |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>) | ||
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|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 | + | |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. |
|- | |- | ||
|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"|This one will slowly get more accurate as the universe ages. | |align="left"|This one will slowly get more accurate as the universe ages. | ||
|- | |- | ||
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|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. | + | |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 | ||
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|align="center"|Telephone number for the {{w|White House}} switchboard | |align="center"|Telephone number for the {{w|White House}} switchboard | ||
|align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math> | |align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math> | ||
− | |align="center"|0. | + | |align="center"|0.2024561414 |
|align="center"|202-456-1414 | |align="center"|202-456-1414 | ||
|align="left"| | |align="left"| | ||
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2030 — 8.4<br> | 2030 — 8.4<br> | ||
2031 — 8.5<br> | 2031 — 8.5<br> | ||
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|align="center"| | |align="center"| | ||
− | |align="left"|Grows by 75 million every year on average | + | |align="left"|Grows by 75 million every year on average. |
|- | |- | ||
|align="center"|U.S. population estimate (millions) | |align="center"|U.S. population estimate (millions) | ||
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2030 — 370<br> | 2030 — 370<br> | ||
2031 — 373<br> | 2031 — 373<br> | ||
− | 2032 — 376 | + | 2032 — 376 |
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|align="center"| | |align="center"| | ||
− | |align="left"|Grows by 3 million each year | + | |align="left"|Grows by 3 million each year. |
|- | |- | ||
|align="center"|Electron rest energy (joules) | |align="center"|Electron rest energy (joules) | ||
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|align="center"|6.02191201246329 × 10<sup>23</sup> | |align="center"|6.02191201246329 × 10<sup>23</sup> | ||
|align="center"|6.02214129 × 10<sup>23</sup> | |align="center"|6.02214129 × 10<sup>23</sup> | ||
− | |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 | + | |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. |
|- | |- | ||
|align="center"|Gravitational constant ''G'' | |align="center"|Gravitational constant ''G'' | ||
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|align="center"|1836.1181087117 | |align="center"|1836.1181087117 | ||
|align="center"|1836.15267246 | |align="center"|1836.15267246 | ||
− | |align="left"| | + | |align="left"| |
|- | |- | ||
|align="center"|Liters in a {{w|gallon}} | |align="center"|Liters in a {{w|gallon}} | ||
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|align="center"|3.7853981634 | |align="center"|3.7853981634 | ||
|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 | + | |align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches |
|- | |- | ||
|align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub> | |align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub> | ||
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|align="center"|<math>\frac{1}{1200^2}</math> | |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"|6.9<span style="text-decoration: overline;">444</span> × 10<sup>−7</sup> | ||
− | |align="center"|~6.943 | + | |align="center"|~6.943 |
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined. | |align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined. | ||
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:<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math> | :<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math> | ||
− | Multiplying by 1 (or by a | + | Multiplying by 1 (or by a 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> | :<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> | ||
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&= \frac12 \quad \quad \quad \text{Q.E.D.} | &= \frac12 \quad \quad \quad \text{Q.E.D.} | ||
\end{align}</math> | \end{align}</math> | ||
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==Transcript== | ==Transcript== | ||
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|- | |- | ||
|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 |