1047: Approximations
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==Explanation== | ==Explanation== | ||
| − | + | This comic lists some approximations for numbers, most of them mathematical and physical constants. All 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. Those interested in number theory may easily compute that sqrt(2) is not even algebraic in the quotient field of Z[pi], which disproves the equality. | |
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Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”. | 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 | + | [[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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| − | + | There are a few cultural references in this comic: | |
| − | The | + | * 99<sup>8</sup> and 69<sup>8</sup> are sexual references. |
| + | * “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. | ||
| + | * (202) 456-1415 was (at least during the Bush administration) the phone number for the White House switchboard. Randall's formula gives approximately 0.2024561415. | ||
| + | * Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. Randall's formula gives approximately 867.530901981685. | ||
| − | + | And here are Wikipedia links to the mathematical concepts: | |
| + | * {{w|Planck constant}} | ||
| + | * {{w|Fine structure constant}} is 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 for it (with a pinch of {{w|numerology}} thrown in at times). | ||
| + | * {{w|Euler-Mascheroni constant}} (Euler gamma constant) | ||
| + | * {{w|Gravitational constant}} | ||
| + | * {{w|Gas constant}} | ||
| + | * ϕ is the {{w|golden ratio}} | ||
| + | * The ruby laser wavelength varies because “ruby” is not clearly defined. | ||
| + | * The {{w|Earth radios#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 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]. | 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]. | ||
| − | The image text gives more or less useful information. Twin | + | The image text gives more or less useful information. {{w|Twin prime}}s 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. |
| − | {{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, | + | {{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it really shouldn't occur anywhere in an exponent (unless you deal with complex numbers). |
| − | Same goes for the e-th power: e | + | 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. |
==Transcript== | ==Transcript== | ||
Revision as of 19:58, 24 February 2013
| Approximations |
 Title text: 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. |
Explanation
This comic lists some approximations for numbers, most of them mathematical and physical constants. All 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. Those interested in number theory may easily compute that sqrt(2) is not even algebraic in the quotient field of Z[pi], which disproves the equality.
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).
There are a few cultural references in this comic:
- 998 and 698 are sexual references.
- “Rent Method” refers to the song “Seasons of Love” from the 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.
- (202) 456-1415 was (at least during the Bush administration) the phone number for the White House switchboard. Randall's formula gives approximately 0.2024561415.
- Jenny's constant comes from Tommy Tutone's tune 867-5309/Jenny. Randall's formula gives approximately 867.530901981685.
And here are Wikipedia links to the mathematical concepts:
- Planck constant
- Fine structure constant is 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 for it (with a pinch of numerology thrown in at times).
- Euler-Mascheroni constant (Euler gamma constant)
- Gravitational constant
- Gas constant
- ϕ is the golden ratio
- The ruby laser wavelength varies because “ruby” is not clearly defined.
- The 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 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 shown here.
The image text gives more or less useful information. 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.
Pi is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it really shouldn't occur anywhere in an exponent (unless you deal with complex numbers).
Same goes for the e-th power: e typically appears in the basis of a power (forming the exponential function), not in the exponent.
Transcript
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.) All units are SI MKS unless otherwise noted.
- Relation: One light year(m) ~= 99^8
Accurate to within: one part in 40
- Relation: Earth Surface(m^2) ~= 69^8
Accurate to within: one part in 130
- Relation: Ocean's volume(m^3) ~= 9^19
Accurate to within: one part in 70
- Relation: Seconds in a year ~= 75^4
Accurate to within: one part in 400
- Relation: Seconds in a year (rent method) ~= 525,600 x 60
Accurate to within: one part in 1400
- Relation: Age of the universe (seconds) ~= 15^15
Accurate to within: one part in 70
- Relation: Planck's constant ~= 1/(30^pi^e)
Accurate to within: one part in 110
- Relation: Fine structure constant ~= 1/140
Accurate to within: [I've had enough of this 137 crap]
- Relation: Fundamental charge ~= 3/(14 * pi^pi^pi)
Accurate to within: one part in 500
- Relation: White House Switchboard ~= 1/(e^((1 + 8^(1/(e-1)))^(1/pi)))
- Relation: Jenny's Constant ~= (7^(e/1 - 1/e) - 9) * pi^2
Intermission: World Population Estimate which should stay current for a decade or two:
Take the last two digits of the current year
Example: 20[14]
Subtract the number of leap years since hurricane Katrina
Example:14 (minus 2008 and 2012) is 12
Add a decimal point
Example: 1.2
Add 6
Example: 6 + 1.2
7.2 ~= World population in billions.
Version for US population:
Example: 20[14]
Subtract 10
Example: 4
Multiply by 3
Example: 12
Add 10
Example: 3[22] million
- Relation: Electron rest energy ~= e/7^16 Joules
Accurate to within: one part in 1000
- Relation: Light-year(miles) ~= 2^(42.42)
Accurate to within: one part in 1000
- Relation: sin(60 degrees) = (3^(1/2))/2 ~= e/pi
Accurate to within: one part in 1000
- Relation: 3^(1/2) ~= 2e/pi
Accurate to within: one part in 1000
- Relation: gamma(Euler's gamma constant) ~= 1/(3^(1/2))
Accurate to within: One part in 4000
- Relation: Feet in a meter ~= 5/(pi^(1/e))
Accurate to within: one part in 4000
- Relation: 5^(1/2) ~= 2/e + 3/2
Accurate to within: one part in 7000
- Relation: Avogadro's number ~= 69^pi^(5^(1/2))
Accurate to within: one part in 25,000
- Relation: Gravitational constant G ~= 1/(e^((pi - 1)^(pi + 1)))
Accurate to within: one part in 25,000
- Relation: R(gas constant) ~= (e+1) * (5^(1/2))
Accurate to within: one part in 50,000
- Relation: Proton-electron mass ratio ~= 6*pi^5
Accurate to within: one part in 50,000
- Relation: Liters in a gallon ~= 3 + pi/4
Accurate to within: one part in 500,000
- Relation: g ~= 6 + ln(45)
Accurate to within: one part in 750,000
- Relation: Proton-electron mass ratio ~= (e^8 - 10)/phi
Accurate to within: one part in 5,000,000
- Relation: Ruby laser wavelength ~= 1/(1200^2)
Accurate to within: [within actual variation]
- Relation: Mean Earth Radius ~= (5^8)*6e
Accurate to within: [within actual variation]
Protip - not all of these are wrong:
- 2^(1/2) ~= 3/5 + pi/(7-pi)
- cos(pi/7) + cos(3pi/7) + cos(5pi/7) ~= 1/2
- gamma(Euler's gamma constant) ~= e/3^4 + e/5
- 5^(1/2) ~= (13 + 4pi)/(24 - 4pi)
- sigma(1/n^n) ~= ln(3)^e
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