1047: Approximations
[edit] Explanation
This explanation may be incomplete or incorrect: see discussion If you can address this issue, please edit the page! Thanks. |
This comic lists some approximations for numbers, most of them mathematical and physical constants. All of 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.
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:
- 99^{8} and 69^{8} 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) 456-1414 is the phone number for the White House switchboard. Truncated, Randall's formula yields 0.2024561414.
- Jenny's constant comes from Tommy Tutone's tune 867-5309/Jenny. Randall's formula gives approximately 867.530901981685.
- 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.
And here are some of the mathematical and physical ones, with Wikipedia links.
- Informally, the Planck constant is the smallest action possible in quantum mechanics.
- The 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 numerology thrown in at times), including the infamous Sir Arthur Adding-One.
- In mathematics, the Euler-Mascheroni constant (Euler gamma constant) is a mysterious number describing the relationship between the harmonic series and the natural logarithm.
- The gravitational constant relates to, uh, gravity.
- The gas constant relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and molar amount (hence the name).
- ϕ is the golden ratio, or (1 + √5)/2. It has many interesting geometrical properties.
- 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. If you're still confused, the functions use radians, not degrees.
The number 8675309 at the title text refers to the song 867-5309/Jenny as mentioned above, causing a fad of people dialing this number and asking for "Jenny". The number is in fact a 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.
Pi is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it doesn't usually occur anywhere in an exponent. When it does, such as with complex numbers, taking the pi-th root is rarely helpful. For example, if we try to derive:
e^{πi} + 1 = 0
e^{πi} = -1
(e^{i})^{π} = -1
e^{i} = ^{π}√(-1)
We get nowhere.
Same goes for the e-th power: e typically appears in the basis of a power (forming the exponential function), not in the exponent. (This is later referenced in Lethal Neutrinos).
The software referred to in the comic is ries, a 'reverse calculator' which forms equations matching a given number.
Actual Approximation One light year(m) 9.46x10^{15} 99^{8} 9.23x10^{15} Earth Surface(m^{2}) 69^{8} Ocean's volume(m^{3}) 9^{19} Seconds in a year 31557600 75^{4} 34640625 Seconds in a year (rent method) 31557600 525,600 x 60 31536000 Age of the universe (seconds) 15^{15} 4.379x10^{17} Planck's constant 6.626x10^{-34} 1/(30^{πe}) 6.685x10^{-34} Fine structure constant 7.297x10^{-3} 1/140 7.143x10^{-3} Fundamental charge 1.602x10^{-19} 3/(14π^{ππ}) 1.599x10^{-19}
[edit] Transcript
This transcript is incomplete. Please help editing it! Thanks. |
- 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: Accurate to within: One light year(m) 99^{8} one part in 40 Earth Surface(m^{2}) 69^{8} one part in 130 Ocean's volume(m^{3}) 9^{19} one part in 70 Seconds in a year 75^{4} one part in 400 Seconds in a year (rent method) 525,600 x 60 one part in 1400 Age of the universe (seconds) 15^{15} one part in 70 Planck's constant 1/(30^{πe}) one part in 110 Fine structure constant 1/140 [I've had enough of this 137 crap] Fundamental charge 3/(14 * π^{ππ}) one part in 500 White House Switchboard 1/
^{π}√(e^{(1 + (e-1)√8})
Jenny's Constant (7^{(e/1 - 1/e)} - 9) * π^{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
Electron rest energy e/7^{16} Joules one part in 1000 Light-year(miles) 2^{(42.42)} one part in 1000 sin(60°) = ^{3}√/2 = e/π one part in 1000 √3 = 2e/π one part in 1000 gamma(Euler's gamma constant) 1/√3 One part in 4000 Feet in a meter 5/(^{e}√π) one part in 4000 √5 = 2/e + 3/2 one part in 7000 Avogadro's number 69^{π√5} one part in 25,000 Gravitational constant G 1 / e^{(pi - 1)(pi + 1)} one part in 25,000 R(gas constant) (e+1) √5 one part in 50,000 Proton-electron mass ratio 6*π^{5} one part in 50,000 Liters in a gallon 3 + π/4 one part in 500,000 g 6 + ln(45) one part in 750,000 Proton-electron mass ratio e^{8} - 10 / ϕ one part in 5,000,000 Ruby laser wavelength 1 / (1200^{2}) [within actual variation] Mean Earth Radius (5^{8})*6e [within actual variation] Protip - not all of these are wrong: √2 = 3/5 + π/(7-π) cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 γ(Euler's gamma constant) = e/3^{4} + e/5 √5 = 13 + 4π / 24 - 4π Σ 1/n^{n} = ln(3)^{e}
Discussion
They're actually quite accurate. I've used these in calculations, and they seem to give close enough answers. Davidy^{22}[talk] 14:03, 8 January 2013 (UTC)
I only see a use for the liters in a gallon one. The rest are for trolling or simple amusement. The cosine identity bit our math team in the butt at a competition. It was painful. --Quicksilver (talk) 05:27, 17 August 2013 (UTC)
Annoyingly this explanation does not cover 42 properly, it does not say that Douglas Adams got the number 42 from Lewis Carroll, who is more relevant to the page because he was a mathematician named Charles Lutwidge Dodgson. He was obsessed with the number forty-two. The original plate illustrations of Alice in Wonderland drawn by him numbered forty-two. Rule Forty-Two in Alice in Wonderland is "All persons more than a mile high to leave the court", There is also a Code of Honour in the preface of The Hunting of the Snark, an extremely long poem written by him when he was 42 years old, in which rule forty-two is "No one shall speak to the Man at the Helm". The queens in Alice Through the Looking Glass the White Queen announces her age as "one hundred and one, five months and a day", which - if the best possible date is assumed for the action of Through the Looking-Glass - gives a total of 37,044 days. With the further (textually unconfirmed) assumption that both Queens were born on the same day their combined age becomes 74,088 days, which is 42 x 42 x 42. --139.216.242.254 02:43, 29 August 2013 (UTC)
- This explanation covers 42 adequately, and would probably be made slightly worse if such information were added. The very widely known cultural reference is to Adams's interpretation, not Dodgson's original obsession. Adding it would be akin to introducing the MPLM into the explanation for the hijacking of Renaissance artists' names by the TMNT. I definitely concede that it does not cover 42 exhaustively, but I think it can be considered complete and in working order without such an addition. If it really irks you, be bold and add it! --Quicksilver (talk) 00:37, 30 August 2013 (UTC)
"sqrt(2) is not even algebraic in the quotient field of Z[pi]" is not correct. Q is part of the quotient field of Z[pi] and sqrt(2) is algebraic of it. The needed facts are that pi is not algebraic, but the formula implies it is in Q(sqrt(2)). --DrMath 06:47, 7 September 2013 (UTC)
13/15 is a better approximation to sqrt(3)/2 than is e/pi. Continued fraction approximations are great! --DrMath 07:23, 7 September 2013 (UTC)
How could he forget 1 gallon ≈ 0.1337 ft³?! 67.188.195.182 00:51, 8 September 2013 (UTC)
Worth mentioning that Wolfram Alpha now officially recognizes the White House switchboard constant and the Jenny constant. 86.164.243.91 18:28, 8 October 2013 (UTC)
Maybe we should add the [Extension:LaTeXSVG LaTeX extension] to make it easier to transcribe these equations. -- 108.162.219.220 23:02, 16 December 2013 (UTC)
- Protip - Does anyone see the correct equation?
Maybe this is just an other Wolfram Alpha error, like we recently have had here: 1292: Pi vs. Tau. All equations still look invalid to me.
- √2 = 3/5 + π/(7-π): is impossible because √2 is an irrational number and no equation can match.
- cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2: could only match if cos(x) + cos(3x) + cos(5x) = 1/2 would be valid, because π/7 is also an irrational number.
- γ = e/3^{4} + e/5 or γ = e/54 + e/5: would mean that a sum of two irrational numbers do fit to the Gamma Constant. Impossible.
- √5 = 13 + 4π / 24 - 4π: √5 and π are irrational numbers, there is no way to match them in any equation like this.
- Σ 1/n^{n} = ln(3)^{e}: doesn't make any sense either.
Maybe Miss Lenhart can help. --Dgbrt (talk) 21:41, 17 December 2013 (UTC)
cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 is exactly correct.
Let a=π/7, b=3π/7, and c=5π/7, then (cosa+cosb+cosc)⋅2sina=2cosasina+2cosbsina+2coscsina=sin2a+sin(b+a)−sin(b−a)+sin(c+a)−sin(c−a)=sin(2π/7)+sin(4π/7)−sin(2π/7)+sin(6π/7)−sin(4π/7)=sin(6π/7)=sin(π/7)=sina
Hence, cos(π/7) + cos(3π/7) + cos(5π/7) = sin(π/7) / 2sin(π/7) = 1/2 108.162.216.74 01:57, 16 January 2014 (UTC)
- What is this: sin(6π/7)=sin(π/7) ? A new math is born... --Dgbrt (talk) 20:49, 16 January 2014 (UTC)
- Actually it does. My proof is geometric: the sines of two supplementary angles (angle a + angle b = π (in radians)) are equivalent because they necessarily have the same x height in a Cartesian plane. Look on a unit circle, or even a sine function. Also, Calculus and most other mathematics use radians over degrees because they make the functions simpler and eliminate irrationality when a trig function shows up, but physics uses degrees because it's easier to understand and taught first. Anonymous 01:27, 13 February 2014 (UTC)
- As an aside, just how far along in math are you? Radian measure is taught in high school (at least the good ones). Anonymous 13:24, 13 February 2014 (UTC)
- Sure, I was wrong at my last statement. sin(6π/7)=sin(π/7) is correct by using the radian measure. But just change π/7 to π/77 would give a very different result on that formular here. I still can't figure out why PI divided by the number 7 should be that unique, PI divided by 77 should be the same. My fault is: I still can't find the Nerd Sniping here. And we all do know that Randall did use wrong WolframAlpha results here. According to the last question: I'm very well on Math, that's because I want to understand this. This is like 0.999=1. --Dgbrt (talk) 22:01, 13 February 2014 (UTC)
- Ah, I see. I think it has to do with the way e^i*π breaks down, as one of the answers shown in the corresponding link explains, but other answers rely on various angle identities (including the supplementary sines one in the proof above). Anonymous 03:10, 14 February 2014 (UTC) (PS, have you checked 545 lately? I answered your question there, too)
- As per the derivation from january 16 , you can use any a,b,c that satisfies this set of equations: 2 a = b - a, a + b = c- a, c + a = π - a. This is due to the fact that sin(x) = sin(π-x), and what was derived the 16th. 173.245.53.199 12:38, 21 February 2014 (UTC)
- Ah, I see. I think it has to do with the way e^i*π breaks down, as one of the answers shown in the corresponding link explains, but other answers rely on various angle identities (including the supplementary sines one in the proof above). Anonymous 03:10, 14 February 2014 (UTC) (PS, have you checked 545 lately? I answered your question there, too)
- Sure, I was wrong at my last statement. sin(6π/7)=sin(π/7) is correct by using the radian measure. But just change π/7 to π/77 would give a very different result on that formular here. I still can't figure out why PI divided by the number 7 should be that unique, PI divided by 77 should be the same. My fault is: I still can't find the Nerd Sniping here. And we all do know that Randall did use wrong WolframAlpha results here. According to the last question: I'm very well on Math, that's because I want to understand this. This is like 0.999=1. --Dgbrt (talk) 22:01, 13 February 2014 (UTC)
- So, still incomplete?
Where's our (in)complete judge? 199.27.128.186 19:21, 18 December 2013 (UTC)
- The protip is still a mystery. I'm calling for help a few lines above. --Dgbrt (talk) 21:16, 18 December 2013 (UTC)
The 'Seconds in a year' ones remind me of one of my favorite quotes: "How many seconds are there in a year? If I tell you there are 3.155 x 10^7, you won't even try to remember it. On the other hand, who could forget that, to within half a percent, pi seconds is a nanocentury" -- Tom Duff, Bell Labs. Beolach (talk) 19:14, 17 April 2014 (UTC)