explain xkcd - User contributions [en]

216: Romantic Drama Equation

2025-01-09T19:29:17Z

TheGeometrist: /* Formulas */

{{comic
| number = 216
| date = January 29, 2007
| title = Romantic Drama Equation
| image = romantic drama equation.png
| titletext = Real-life prospective-pairing curves over things like age can get depressing.
}}

==Explanation==
In a group of n people, such as the cast of a TV romantic drama like {{w|Queer as Folk (U.S. TV series)|Queer as Folk}}, the number of possible different pairs of people is n(n-1)/2. A romantic drama will often consider, over time, many possible romantic pairs of its cast members, even seeming to test the limit of how many pairs are possible. Through an austerely binaristic lens, this comic explores the implication of sex and sexual orientation in characterizing the possible pairs.

If everyone in the group is male or female, then each pair could be classified as gay or straight. The formulas in the comic give how many of the possible pairs are gay and how many are straight, as a function of the total number of people and how many are male (or, equivalently, how many are female.) For example, of the 9 principal cast of Firefly, 5 are men and 4 are women. With n=9 and x=5, we have 16 possible gay pairs and 20 possible straight pairs.

A graph shows how the relative number of males and females affects the number of gay pairs and straight pairs. When the group is all male (or all female), all of the possible pairs are gay, but as the minority sex's number is increased, more of the pairs are straight. When the group is half male and half female, the number of straight pairs is maximized, and straight pairs slightly outnumber gay pairs. The curves are labeled "gay cast" and "straight cast", perhaps implying that a "gay cast" would consider only gay pairs, and a "straight cast" would consider only straight pairs.

There is a note that the graph describes large casts. Because all the quantities involved are discrete, for a small [n] there are only a few points to plot on the graph, and the smooth, continuous curves seen in the comic are less recognizable.

The title-text mentions that Randall made a graph of his prospective dating pool over time and was depressed by the results. As he gets older, his dating pool gets smaller: fewer people his age are single. But as Randall later shows in [[314: Dating Pools]], age is not the problem--he is!

===Formulas===
The formulas may be derived as follows:

Each straight pair needs to include one of the x males and one of the (n-x) females, so there are x(n-x) possible ways of combining one of each. E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and 3 possible pairings involving the second, yielding 2(5-2)=6 possible pairs.

Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=x*%28n-x%29 x*(n-x)]

Each gay pair needs to include either two males or two females. To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males. However, that counts each possible pair twice. E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second. To avoid double-counting the pairs, we therefore need to divide the product by 2. So there are x(x-1)/2 possible pairs of two males. Similarly, there are (n-x)(n-x-1)/2 possible pairings of two females. Summing these, we get the total number of possible gay pairs as [x^2 - x + n^2 - nx - n - xn + x^2 + x]/2. That simplifies to [n^2 - n + 2 x^2 - 2 xn]/2. The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0]. Since the sum of these terms was divided by 2, we get that the total number of possible gay pairs is n(n-1)/2 - x(n-x), or n(n-1)/2 + x(x-n), which is what the cartoon says.

Wolfram|Alpha: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]

Desmos: https://www.desmos.com/3d/j0sgfhbcci

==Transcript==
:TV Romantic Drama Equation (Derived during a series of 'Queer as Folk' episodes)
:[A table shows equations for possible romantic pairings in a TV show. The equation under "gay" is n(n-1) 2+x(x-n); the equation under "straight" is x(n-x).]
:x: Number of male (or female) cast members.
:n: total number of cast members.

:[A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50/50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]

{{comic discussion}}
[[Category:Line graphs]]
[[Category:Homosexuality]]
[[Category:Math]]
[[Category:Romance]]
[[Category:Comics with color]]

1047: Approximations

2025-01-05T19:04:41Z

TheGeometrist: /* Equations */

{{comic
| number = 1047
| date = April 25, 2012
| title = Approximations
| before = [[#Explanation|↓ Skip to explanation ↓]]
| 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.
}}

==Explanation==

This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.

Approximations like these are sometimes used as {{w|mnemonic}}s by mathematicians and physicists, though most of Randall's approximations are too convoluted to be useful as mnemonics. Perhaps the best known mnemonic approximation (though not used here by Randall) is that "π is approximately equal to 22/7". Randall does mention (and mock) the common mnemonic among physicists that the {{w|fine structure constant}} is approximately 1/137. Although Randall gives approximations for the number of seconds in a year, he does not mention the common physicists' mnemonic that it is "π × 107", though he later added a statement to the top of the comic page addressing this point.

At the bottom of the comic are expressions involving {{w|transcendental numbers}} (namely π and e) that are tantalizingly close to being exactly true but are not (indeed, they cannot be, due to the nature of transcendental numbers). Such near-equations were previously discussed in [[217: e to the pi Minus pi]]. One of the entries, though, is a "red herring" that is exactly true.

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 2024 is still somewhat accurate. The estimate is 7.9 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 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 base 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]).

In [[217: e to the pi Minus pi]] and [[3023: The Maritime Approximation]] Randall gives other approximations based on numerical coincidences.

===Equations===

{| 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"|Error
|-
|align="center"|One {{w|light year}} (meters)
|align="center"|998
|align="center"|9,227,446,944,279,201
|align="center"|9,460,730,472,580,800 (exact)
|align="left"|Based on 365.25 days per year (see below). 998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|align="center"|2.3328353 × 1014
|-
|align="center"|Earth's surface (m2)
|align="center"|698
|align="center"|513,798,374,428,641
|align="center"|5.10072 × 1014
|align="left"|998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|align="center"|3.7263744 × 1012
|-
|align="center"|Oceans' volume (m3)
|align="center"|919
|align="center"|1,350,851,717,672,992,089
|align="center"|1.332 × 1018
|align="left"|
|-
|align="center"|Seconds in a year
|align="center"|754
|align="center"|31,640,625
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:
"Lots of emails mention the physicist favorite, 1 year = pi × 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.

Using the traditional definitions that a second is 1/60 of a minute, a minute is 1/60 of an hour, and an hour is 1/24 of a day, a 365-day common year is exactly 31,536,000 seconds (the "''Rent'' method" approximation) and the 366-day leap year is 31,622,400 seconds. Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600 seconds. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.
|-
|align="center"|Seconds in a year (''Rent'' method)
|align="center"|525,600 × 60
|align="center"|31,536,000
|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. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
|-
|align="center"|Age of the universe (seconds)
|align="center"|1515
|align="center"|437,893,890,380,859,375
|align="center"|(4.354 ± 0.012) × 1017 (best estimate; exact value unknown)
|align="left"|This one will slowly get more accurate as the universe ages.
|-
|align="center"|Planck's constant
|align="center"|<math>\frac {1} {30^{\pi^e}}</math>
|align="center"|6.6849901410 × 10−34
|align="center"|6.62606957 × 10−34
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
|-
|align="center"|Fine structure constant
|align="center"|<math>\frac{1}{140}</math>
|align="center"|0.00714285
|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. The joke here is that Randall chose to write 140 as the denominator, when 137 is much closer to reality and just as many digits (although 137 is a less "round" number than 140, and Randall writes in the table that he's "had enough" of it). At one point the fine structure constant 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" Eddington}} who argued very strenuously that the fine structure constant "should" be 1/136 when that was what the best measurements suggested, and then argued just as strenuously for 1/137 a few years later as measurements improved.
|-
|align="center"|Fundamental charge
|align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
|align="center"|1.59895121062716 × 10−19
|align="center"|1.602176565 × 10−19
|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
|-
|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"|0.2024561414932
|align="center"|202-456-1414
|align="left"|
|-
|align="center"|Jenny's constant
|align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
|align="center"|867.5309019
|align="center"|867-5309
|align="left"|A telephone number referenced in {{w|Tommy Tutone}}'s 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number is not only prime but a {{w|twin prime}} because 8675311 is also a prime.
|-
|align="center"|World population estimate (billions)
|align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
|align="center"|2005 — 6.5 
2006 — 6.6 
2007 — 6.7 
2008 — 6.7 
2009 — 6.8 
2010 — 6.9 
2011 — 7.0 
2012 — 7.0 
2013 — 7.1 
2014 — 7.2 
2015 — 7.3 
2016 — 7.3 
2017 — 7.4 
2018 — 7.5 
2019 — 7.6 
2020 — 7.6 
2021 — 7.7 
2022 — 7.8 
2023 — 7.9 
2024 — 7.9 
2025 — 8.0 
2026 — 8.1 
2027 — 8.2 
2028 — 8.2 
2029 — 8.3 
2030 — 8.4 
2031 — 8.5 
2032 — 8.5 
2033 — 8.6 
2034 — 8.7 
2035 — 8.8 
|align="center"|
|align="left"|Grows by 75 million every year on average (100 million every year, except for a pause every leap-year). As of 2024, still a bit too small. (7.8B)
|-
|align="center"|U.S. population estimate (millions)
|align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
|align="center"|2000 — 280 
2001 — 283 
2002 — 286 
2003 — 289 
2004 — 292 
2005 — 295 
2006 — 298 
2007 — 301 
2008 — 304 
2009 — 307 
2010 — 310 
2011 — 313 
2012 — 316 
2013 — 319 
2014 — 322 
2015 — 325 
2016 — 328 
2017 — 331 
2018 — 334 
2019 — 337 
2020 — 340 
2021 — 343 
2022 — 346 
2023 — 349 
2024 — 352 
2025 — 355 
2026 — 358 
2027 — 361 
2028 — 364 
2029 — 367 
2030 — 370 
2031 — 373 
2032 — 376 
2033 — 379 
2034 — 382 
2035 — 385 
|align="center"|
|align="left"|Grows by 3 million each year. As of 2024 the actual number is ~10 million smaller.
|-
|align="center"|Electron rest energy (joules)
|align="center"|<math>\frac {e} {7^{16}}</math>
|align="center"|8.17948276564429 × 10−14
|align="center"|8.18710438 × 10−14
|align="left"|
|-
|align="center"|Light year (miles)
|align="center"|242.42
|align="center"|5,884,267,614,436.97
|align="center"|5,878,625,373,183.61 = 9,460,730,472,580,800 (meters in a light-year, by definition) / 1609.344 (meters in a mile)
|align="left"|{{w|42 (number)|42}} is, according to {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
|-
|align="center"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
|align="center"|<math>\frac{e}{\pi}</math>
|align="center"|0.8652559794
|align="center"|0.8660254038
|align="left"|
|-
|align="center"|<math>\sqrt 3</math>
|align="center"|<math>\frac{2e}{\pi}</math>
|align="center"|1.7305119589
|align="center"|1.7320508076
|align="left"|Same as the above
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac {1} {\sqrt 3}</math>
|align="center"|0.5773502692
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|Feet in a meter
|align="center"|<math>\frac {5} {\sqrt[e]\pi}</math>
|align="center"|3.2815481951
|align="center"|3.280839895
|align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac{2}{e} + \frac32</math>
|align="center"|2.2357588823
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|Avogadro's number
|align="center"|<math>69^{\pi^\sqrt{5}}</math>
|align="center"|6.02191201246329 × 1023
|align="center"|6.02214129 × 1023
|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. In 2019 the constant was redefined to 6.02214076 × 1023, making the Approximation slightly more correct.
|-
|align="center"|Gravitational constant ''G''
|align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
|align="center"|6.6736110685 × 10−11
|align="center"|6.67385 × 10−11
|align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''2/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
|-
|align="center"|''R'' (gas constant)
|align="center"|<math>(e + 1) \sqrt 5</math>
|align="center"|8.3143309279
|align="center"|8.3144622
|align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>6 \pi^5</math>
|align="center"|1836.1181087117
|align="center"|1836.15267246
|align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
|-
|align="center"|Liters in a {{w|gallon}}
|align="center"|<math>3 + \frac{\pi}{4}</math>
|align="center"|3.7853981634
|align="center"|3.785411784 (exact)
|align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches. The British imperial gallon would be about 20% larger (but the litre is the same thing as the US liter).
|-
|align="center"|''g''0 or ''g''n
|align="center"|6 + ln(45)
|align="center"|9.8066624898
|align="center"|9.80665
|align="left"|Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 km/h/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.

Randall used a letter ''g'' without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>\frac {e^8 - 10} {\phi}</math>
|align="center"|1836.1530151398
|align="center"|1836.15267246
|align="left"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
|-
|align="center"|Ruby laser wavelength (meters)
|align="center"|<math>\frac{1}{1200^2}</math>
|align="center"|6.9444 × 10−7
|align="center"|~6.943 × 10−7
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
|-
|align="center"|Mean Earth radius (meters)
|align="center"|<math>5^8 6e</math>
|align="center"|6,370,973.035
|align="center"|6,371,008.7 (IUGG definition)
|align="left"|The {{w|Earth radius#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 International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).
|-
|align="center"|<math>\sqrt 2</math>
|align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
|align="center"|1.4142200581
|align="center"|1.4142135624
|align="left"|There are recurring math jokes along the lines of, "<math>\textstyle{ \frac35 + \frac{\pi}{7-\pi} - \sqrt{2} = 0}</math>, but your calculator is probably not good enough to compute this correctly". See also [[217: e to the pi Minus pi]].
|-
|align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
|align="center"|<math>\frac12</math>
|align="center"|0.5
|align="center"|0.5 (exact)
|align="left"|This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}: when an angular measure does not specify units, radians are the assumed default.
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
|align="center"|0.5772154006
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
|align="center"|2.2360678094
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
|align="center"|<math>\ln(3)^e</math>
|align="center"|1.2912987577
|align="center"|1.2912859971
|align="left"|
|}

===Proof===

One of the "approximations" actually is precisely correct: <math>\textstyle{ \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = \frac12 }</math>. Here is a proof:

:<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>

Multiplying by 1 (or by a nonzero 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>

The <math>\textstyle{ 2 \sin\frac{\pi}{7} }</math> on the top of the fraction is multiplied through the original equation:

:<math>= \frac {2 \cos \frac{\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{3\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{5\pi}{7} \sin\frac{\pi}{7}} {2 \sin\frac{\pi}{7}}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin B = \sin (A+B) - \sin(A-B)}</math> on the second and third terms in the numerator:

:<math>\begin{align}
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \left(\frac{3\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{3\pi}{7} - \frac{\pi}{7}\right) \right] + \left[\sin \left(\frac{5\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{5\pi}{7} - \frac{\pi}{7}\right) \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}}
\end{align}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin A = \sin 2A }</math> on the first term in the numerator:

:<math>\begin{align}
&= \frac {\sin \frac{2\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{4\pi}{7} \right] + \left[\sin \frac{2\pi}{7} - \sin \frac{2\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} } {2 \sin\frac{\pi}{7}}
\end{align}</math>

Noting that <math>\textstyle{\frac{6\pi}{7} + \frac{\pi}{7} = \pi}</math> and that the sines of supplementary angles (angles that sum to π) are equal:

:<math>\begin{align}
&= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
&= \frac12 \quad \quad \quad \text{Q.E.D.}
\end{align}</math>

To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7)  is the real part of the k-th 7-th root of unity, exp(2 k i pi/7). The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:


:0 = cos(0 pi/7) + cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) + cos(8 pi/7) + cos(10 pi/7) + cos(12 pi/7)

But one of these roots is just 1, and all other root go by pairs of conjugate roots, which have the same real part (alternatively, consider that cos(x) = cos(2 pi - x)):


:0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7))

Hence


:cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2

which, because cos(x) = cos(pi - x), can be rewritten as


:cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2

Q.E.D.

==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.

:{| class="wikitable"
|-
|colspan="2" align="center" | Relation:
|align="center" | Accurate to within:
|-
|align="center" | One light-year(m)
|align="center" | 998
|align="center" | one part in 40
|-
|align="center" | Earth Surface(m2)
|align="center" | 698
|align="center" | one part in 130
|-
|align="center" | Oceans' volume(m3)
|align="center" | 919
|align="center" | one part in 70
|-
|align="center" | Seconds in a year
|align="center" | 754
|align="center" | one part in 400
|-
|align="center" | Seconds in a year (''Rent'' method)
|align="center" | 525,600 x 60
|align="center" | one part in 1400
|-
|align="center" | Age of the universe (seconds)
|align="center" | 1515
|align="center" | one part in 70
|-
|align="center" | Planck's constant
|align="center" | 1/(30πe)
|align="center" | one part in 110
|-
|align="center" | Fine structure constant
|align="center" | 1/140
|align="center" | [I've had enough of this 137 crap]
|-
|align="center" | Fundamental charge
|align="center" | 3/(14 * πππ)
|align="center" | one part in 500
|-
|align="center"|White House Switchboard
|colspan="2" align="center"|1 / (eπ√(1 + (e-1)√8))
|-
|align="center"|Jenny's Constant
|colspan="2" align="center"|(7(e/1 - 1/e) - 9) * π2
|-
|colspan="3" align="center"|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
|-
|align="center"|Electron rest energy
|align="center"|e/716 J
|align="center"|one part in 1000
|-
|align="center"|Light-year(miles)
|align="center"|2(42.42)
|align="center"|one part in 1000
|-
|colspan="2" align="center"|sin(60°) = √3/2 = e/π
|align="center"|one part in 1000
|-
|colspan="2" align="center"|√3 = 2e/π
|align="center"|one part in 1000
|-
|align="center"|γ(Euler's gamma constant)
|align="center"|1/√3
|align="center"|one part in 4000
|-
|align="center"|Feet in a meter
|align="center"|5/(e√π)
|align="center"|one part in 4000
|-
|colspan="2" align="center"|√5 = 2/e + 3/2
|align="center"|one part in 7000
|-
|align="center"|Avogadro's number
|align="center"|69π√5
|align="center"|one part in 25,000
|-
|align="center"|Gravitational constant G
|align="center"|1 / e(π - 1)(π + 1)
|align="center"|one part in 25,000
|-
|align="center"|R (gas constant)
|align="center"|(e+1) √5
|align="center"|one part in 50,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|6*π5
|align="center"|one part in 50,000
|-
|align="center"|Liters in a gallon
|align="center"|3 + π/4
|align="center"|one part in 500,000
|-
|align="center"|g
|align="center"|6 + ln(45)
|align="center"|one part in 750,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|(e8 - 10) / ϕ
|align="center"|one part in 5,000,000
|-
|align="center"|Ruby laser wavelength
|align="center"|1 / (12002)
|align="center"|[within actual variation]
|-
|align="center"|Mean Earth Radius
|align="center"|(58)*6e
|align="center"|[within actual variation]
|-
|colspan="3" align="center"|Protip - not all of these are wrong:
|-
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
|align="center"|γ(Euler's gamma constant) = e/34 + e/5
|align="center"|√5 = (13 + 4π) / (24 - 4π)
|align="center"|Σ 1/nn = ln(3)e
|}

{{comic discussion}}
[[Category:Charts]]
[[Category:Math]]
[[Category:Physics]]
[[Category:Protip]]
[[Category:Large drawings]]

1047: Approximations

2025-01-05T18:57:54Z

TheGeometrist: /* Explanation */

{{comic
| number = 1047
| date = April 25, 2012
| title = Approximations
| before = [[#Explanation|↓ Skip to explanation ↓]]
| 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.
}}

==Explanation==

This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.

Approximations like these are sometimes used as {{w|mnemonic}}s by mathematicians and physicists, though most of Randall's approximations are too convoluted to be useful as mnemonics. Perhaps the best known mnemonic approximation (though not used here by Randall) is that "π is approximately equal to 22/7". Randall does mention (and mock) the common mnemonic among physicists that the {{w|fine structure constant}} is approximately 1/137. Although Randall gives approximations for the number of seconds in a year, he does not mention the common physicists' mnemonic that it is "π × 107", though he later added a statement to the top of the comic page addressing this point.

At the bottom of the comic are expressions involving {{w|transcendental numbers}} (namely π and e) that are tantalizingly close to being exactly true but are not (indeed, they cannot be, due to the nature of transcendental numbers). Such near-equations were previously discussed in [[217: e to the pi Minus pi]]. One of the entries, though, is a "red herring" that is exactly true.

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 2024 is still somewhat accurate. The estimate is 7.9 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 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 base 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]).

In [[217: e to the pi Minus pi]] and [[3023: The Maritime Approximation]] Randall gives other approximations based on numerical coincidences.

===Equations===

{| 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"|998
|align="center"|9,227,446,944,279,201
|align="center"|9,460,730,472,580,800 (exact)
|align="left"|Based on 365.25 days per year (see below). 998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|-
|align="center"|Earth's surface (m2)
|align="center"|698
|align="center"|513,798,374,428,641
|align="center"|5.10072 × 1014
|align="left"|998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|-
|align="center"|Oceans' volume (m3)
|align="center"|919
|align="center"|1,350,851,717,672,992,089
|align="center"|1.332 × 1018
|align="left"|
|-
|align="center"|Seconds in a year
|align="center"|754
|align="center"|31,640,625
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:
"Lots of emails mention the physicist favorite, 1 year = pi × 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.

Using the traditional definitions that a second is 1/60 of a minute, a minute is 1/60 of an hour, and an hour is 1/24 of a day, a 365-day common year is exactly 31,536,000 seconds (the "''Rent'' method" approximation) and the 366-day leap year is 31,622,400 seconds. Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600 seconds. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.
|-
|align="center"|Seconds in a year (''Rent'' method)
|align="center"|525,600 × 60
|align="center"|31,536,000
|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. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
|-
|align="center"|Age of the universe (seconds)
|align="center"|1515
|align="center"|437,893,890,380,859,375
|align="center"|(4.354 ± 0.012) × 1017 (best estimate; exact value unknown)
|align="left"|This one will slowly get more accurate as the universe ages.
|-
|align="center"|Planck's constant
|align="center"|<math>\frac {1} {30^{\pi^e}}</math>
|align="center"|6.6849901410 × 10−34
|align="center"|6.62606957 × 10−34
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
|-
|align="center"|Fine structure constant
|align="center"|<math>\frac{1}{140}</math>
|align="center"|0.00714285
|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. The joke here is that Randall chose to write 140 as the denominator, when 137 is much closer to reality and just as many digits (although 137 is a less "round" number than 140, and Randall writes in the table that he's "had enough" of it). At one point the fine structure constant 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" Eddington}} who argued very strenuously that the fine structure constant "should" be 1/136 when that was what the best measurements suggested, and then argued just as strenuously for 1/137 a few years later as measurements improved.
|-
|align="center"|Fundamental charge
|align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
|align="center"|1.59895121062716 × 10−19
|align="center"|1.602176565 × 10−19
|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
|-
|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"|0.2024561414932
|align="center"|202-456-1414
|align="left"|
|-
|align="center"|Jenny's constant
|align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
|align="center"|867.5309019
|align="center"|867-5309
|align="left"|A telephone number referenced in {{w|Tommy Tutone}}'s 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number is not only prime but a {{w|twin prime}} because 8675311 is also a prime.
|-
|align="center"|World population estimate (billions)
|align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
|align="center"|2005 — 6.5 
2006 — 6.6 
2007 — 6.7 
2008 — 6.7 
2009 — 6.8 
2010 — 6.9 
2011 — 7.0 
2012 — 7.0 
2013 — 7.1 
2014 — 7.2 
2015 — 7.3 
2016 — 7.3 
2017 — 7.4 
2018 — 7.5 
2019 — 7.6 
2020 — 7.6 
2021 — 7.7 
2022 — 7.8 
2023 — 7.9 
2024 — 7.9 
2025 — 8.0 
2026 — 8.1 
2027 — 8.2 
2028 — 8.2 
2029 — 8.3 
2030 — 8.4 
2031 — 8.5 
2032 — 8.5 
2033 — 8.6 
2034 — 8.7 
2035 — 8.8 
|align="center"|
|align="left"|Grows by 75 million every year on average (100 million every year, except for a pause every leap-year). As of 2024, still a bit too small. (7.8B)
|-
|align="center"|U.S. population estimate (millions)
|align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
|align="center"|2000 — 280 
2001 — 283 
2002 — 286 
2003 — 289 
2004 — 292 
2005 — 295 
2006 — 298 
2007 — 301 
2008 — 304 
2009 — 307 
2010 — 310 
2011 — 313 
2012 — 316 
2013 — 319 
2014 — 322 
2015 — 325 
2016 — 328 
2017 — 331 
2018 — 334 
2019 — 337 
2020 — 340 
2021 — 343 
2022 — 346 
2023 — 349 
2024 — 352 
2025 — 355 
2026 — 358 
2027 — 361 
2028 — 364 
2029 — 367 
2030 — 370 
2031 — 373 
2032 — 376 
2033 — 379 
2034 — 382 
2035 — 385 
|align="center"|
|align="left"|Grows by 3 million each year. As of 2024 the actual number is ~10 million smaller.
|-
|align="center"|Electron rest energy (joules)
|align="center"|<math>\frac {e} {7^{16}}</math>
|align="center"|8.17948276564429 × 10−14
|align="center"|8.18710438 × 10−14
|align="left"|
|-
|align="center"|Light year (miles)
|align="center"|242.42
|align="center"|5,884,267,614,436.97
|align="center"|5,878,625,373,183.61 = 9,460,730,472,580,800 (meters in a light-year, by definition) / 1609.344 (meters in a mile)
|align="left"|{{w|42 (number)|42}} is, according to {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
|-
|align="center"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
|align="center"|<math>\frac{e}{\pi}</math>
|align="center"|0.8652559794
|align="center"|0.8660254038
|align="left"|
|-
|align="center"|<math>\sqrt 3</math>
|align="center"|<math>\frac{2e}{\pi}</math>
|align="center"|1.7305119589
|align="center"|1.7320508076
|align="left"|Same as the above
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac {1} {\sqrt 3}</math>
|align="center"|0.5773502692
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|Feet in a meter
|align="center"|<math>\frac {5} {\sqrt[e]\pi}</math>
|align="center"|3.2815481951
|align="center"|3.280839895
|align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac{2}{e} + \frac32</math>
|align="center"|2.2357588823
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|Avogadro's number
|align="center"|<math>69^{\pi^\sqrt{5}}</math>
|align="center"|6.02191201246329 × 1023
|align="center"|6.02214129 × 1023
|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. In 2019 the constant was redefined to 6.02214076 × 1023, making the Approximation slightly more correct.
|-
|align="center"|Gravitational constant ''G''
|align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
|align="center"|6.6736110685 × 10−11
|align="center"|6.67385 × 10−11
|align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''2/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
|-
|align="center"|''R'' (gas constant)
|align="center"|<math>(e + 1) \sqrt 5</math>
|align="center"|8.3143309279
|align="center"|8.3144622
|align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>6 \pi^5</math>
|align="center"|1836.1181087117
|align="center"|1836.15267246
|align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
|-
|align="center"|Liters in a {{w|gallon}}
|align="center"|<math>3 + \frac{\pi}{4}</math>
|align="center"|3.7853981634
|align="center"|3.785411784 (exact)
|align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches. The British imperial gallon would be about 20% larger (but the litre is the same thing as the US liter).
|-
|align="center"|''g''0 or ''g''n
|align="center"|6 + ln(45)
|align="center"|9.8066624898
|align="center"|9.80665
|align="left"|Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 km/h/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.

Randall used a letter ''g'' without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>\frac {e^8 - 10} {\phi}</math>
|align="center"|1836.1530151398
|align="center"|1836.15267246
|align="left"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
|-
|align="center"|Ruby laser wavelength (meters)
|align="center"|<math>\frac{1}{1200^2}</math>
|align="center"|6.9444 × 10−7
|align="center"|~6.943 × 10−7
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
|-
|align="center"|Mean Earth radius (meters)
|align="center"|<math>5^8 6e</math>
|align="center"|6,370,973.035
|align="center"|6,371,008.7 (IUGG definition)
|align="left"|The {{w|Earth radius#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 International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).
|-
|align="center"|<math>\sqrt 2</math>
|align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
|align="center"|1.4142200581
|align="center"|1.4142135624
|align="left"|There are recurring math jokes along the lines of, "<math>\textstyle{ \frac35 + \frac{\pi}{7-\pi} - \sqrt{2} = 0}</math>, but your calculator is probably not good enough to compute this correctly". See also [[217: e to the pi Minus pi]].
|-
|align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
|align="center"|<math>\frac12</math>
|align="center"|0.5
|align="center"|0.5 (exact)
|align="left"|This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}: when an angular measure does not specify units, radians are the assumed default.
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
|align="center"|0.5772154006
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
|align="center"|2.2360678094
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
|align="center"|<math>\ln(3)^e</math>
|align="center"|1.2912987577
|align="center"|1.2912859971
|align="left"|
|}

===Proof===

One of the "approximations" actually is precisely correct: <math>\textstyle{ \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = \frac12 }</math>. Here is a proof:

:<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>

Multiplying by 1 (or by a nonzero 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>

The <math>\textstyle{ 2 \sin\frac{\pi}{7} }</math> on the top of the fraction is multiplied through the original equation:

:<math>= \frac {2 \cos \frac{\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{3\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{5\pi}{7} \sin\frac{\pi}{7}} {2 \sin\frac{\pi}{7}}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin B = \sin (A+B) - \sin(A-B)}</math> on the second and third terms in the numerator:

:<math>\begin{align}
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \left(\frac{3\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{3\pi}{7} - \frac{\pi}{7}\right) \right] + \left[\sin \left(\frac{5\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{5\pi}{7} - \frac{\pi}{7}\right) \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}}
\end{align}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin A = \sin 2A }</math> on the first term in the numerator:

:<math>\begin{align}
&= \frac {\sin \frac{2\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{4\pi}{7} \right] + \left[\sin \frac{2\pi}{7} - \sin \frac{2\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} } {2 \sin\frac{\pi}{7}}
\end{align}</math>

Noting that <math>\textstyle{\frac{6\pi}{7} + \frac{\pi}{7} = \pi}</math> and that the sines of supplementary angles (angles that sum to π) are equal:

:<math>\begin{align}
&= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
&= \frac12 \quad \quad \quad \text{Q.E.D.}
\end{align}</math>

To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7)  is the real part of the k-th 7-th root of unity, exp(2 k i pi/7). The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:


:0 = cos(0 pi/7) + cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) + cos(8 pi/7) + cos(10 pi/7) + cos(12 pi/7)

But one of these roots is just 1, and all other root go by pairs of conjugate roots, which have the same real part (alternatively, consider that cos(x) = cos(2 pi - x)):


:0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7))

Hence


:cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2

which, because cos(x) = cos(pi - x), can be rewritten as


:cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2

Q.E.D.

==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.

:{| class="wikitable"
|-
|colspan="2" align="center" | Relation:
|align="center" | Accurate to within:
|-
|align="center" | One light-year(m)
|align="center" | 998
|align="center" | one part in 40
|-
|align="center" | Earth Surface(m2)
|align="center" | 698
|align="center" | one part in 130
|-
|align="center" | Oceans' volume(m3)
|align="center" | 919
|align="center" | one part in 70
|-
|align="center" | Seconds in a year
|align="center" | 754
|align="center" | one part in 400
|-
|align="center" | Seconds in a year (''Rent'' method)
|align="center" | 525,600 x 60
|align="center" | one part in 1400
|-
|align="center" | Age of the universe (seconds)
|align="center" | 1515
|align="center" | one part in 70
|-
|align="center" | Planck's constant
|align="center" | 1/(30πe)
|align="center" | one part in 110
|-
|align="center" | Fine structure constant
|align="center" | 1/140
|align="center" | [I've had enough of this 137 crap]
|-
|align="center" | Fundamental charge
|align="center" | 3/(14 * πππ)
|align="center" | one part in 500
|-
|align="center"|White House Switchboard
|colspan="2" align="center"|1 / (eπ√(1 + (e-1)√8))
|-
|align="center"|Jenny's Constant
|colspan="2" align="center"|(7(e/1 - 1/e) - 9) * π2
|-
|colspan="3" align="center"|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
|-
|align="center"|Electron rest energy
|align="center"|e/716 J
|align="center"|one part in 1000
|-
|align="center"|Light-year(miles)
|align="center"|2(42.42)
|align="center"|one part in 1000
|-
|colspan="2" align="center"|sin(60°) = √3/2 = e/π
|align="center"|one part in 1000
|-
|colspan="2" align="center"|√3 = 2e/π
|align="center"|one part in 1000
|-
|align="center"|γ(Euler's gamma constant)
|align="center"|1/√3
|align="center"|one part in 4000
|-
|align="center"|Feet in a meter
|align="center"|5/(e√π)
|align="center"|one part in 4000
|-
|colspan="2" align="center"|√5 = 2/e + 3/2
|align="center"|one part in 7000
|-
|align="center"|Avogadro's number
|align="center"|69π√5
|align="center"|one part in 25,000
|-
|align="center"|Gravitational constant G
|align="center"|1 / e(π - 1)(π + 1)
|align="center"|one part in 25,000
|-
|align="center"|R (gas constant)
|align="center"|(e+1) √5
|align="center"|one part in 50,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|6*π5
|align="center"|one part in 50,000
|-
|align="center"|Liters in a gallon
|align="center"|3 + π/4
|align="center"|one part in 500,000
|-
|align="center"|g
|align="center"|6 + ln(45)
|align="center"|one part in 750,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|(e8 - 10) / ϕ
|align="center"|one part in 5,000,000
|-
|align="center"|Ruby laser wavelength
|align="center"|1 / (12002)
|align="center"|[within actual variation]
|-
|align="center"|Mean Earth Radius
|align="center"|(58)*6e
|align="center"|[within actual variation]
|-
|colspan="3" align="center"|Protip - not all of these are wrong:
|-
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
|align="center"|γ(Euler's gamma constant) = e/34 + e/5
|align="center"|√5 = (13 + 4π) / (24 - 4π)
|align="center"|Σ 1/nn = ln(3)e
|}

{{comic discussion}}
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[[Category:Math]]
[[Category:Physics]]
[[Category:Protip]]
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