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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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 | + | 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 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=== | ===Equations=== | ||
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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"|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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:<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> |