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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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 2020 is still accurate. The estimate is 7.7 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [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"|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"|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. As of | + | |align="left"|Grows by 75 million every year on average. As of 2019, roughly correct. |
|- | |- | ||
|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. As of | + | |align="left"|Grows by 3 million each year. As of 2019 the actual number is ~10 million smaller. |
|- | |- | ||
|align="center"|Electron rest energy (joules) | |align="center"|Electron rest energy (joules) | ||
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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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:<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== | ||
+ | {{incomplete transcript|bad table}} | ||
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.''' | :'''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.) | :(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.) |