1047: Approximations

This is comic number 1047. It was posted on April 25, 2012.[1]

Image 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.

A twin prime is: "Per wikipedia: A twin prime is a prime number that differs from another prime number by two"

"“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.

Incidentally, 75^4 overstates the number of seconds in a year by 29 hours."

Jenny’s Number = (867)-5309, “please don’t change your number on me” But since this is *explain*xkcd, let’s add that it’s from a song by Tommy Tutone: http://www.dailymotion.com/video/x20zsk_tommy-tutone-867-5309-jenny_music

The complicated formula for the White House switchboard yields 0.2024561415. (202) 456-1415 was (at least during the Bush administration) the phone number for the White House switchboard.

All these approximations actually work astonishingly well. There are re-occuring math jokes along the lines of, “3/5 + π/(7-π) – sqrt(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[π], which disproves the equality.

Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “π 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^(π^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).

Jenny’s number and the White House switchboard have already been explained. The other constants are either self-explanatory or simple physical constants that have decimal values you may google in about 1 sec. I am not going to explain what these constants mean. Three things to note: near the bottom, there is a constant which you may easily confuse for a 9. Instead, it’s a g, the standard gravity.

Now to the accuracy values: The fine structure constant is 0.007297something, which is approximately 1/137. Randall’s point here is that 1/137 is not a very useful approximation: Firstly, 0.007297 is input into a calculator as fast as 1/137, and it’s more accurate. Secondly, if you do not have a calculator, 1/137 earns you nothing, for 137 is a prime and therefore does not ease further computation. That’s why Randall stated he’s had enough of this crap. 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. Theoretically, it should be possible to measure the distance from the center for any point on the Earth’s surface and compute the mean integral, but practically it’s not, so geodesy has defined some sets of radii to take the mean of, yielding different mean radii. If you are interested in details, ask Wikipedia. http://en.wikipedia.org/wiki/Earth_radius#Mean_radii Randall’s value lies somewhere in between, thus actually being a possible definition for Earth’s mean radius.

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. π is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it’s not supposed to occur anywhere in an exponent (unless you deal with complex numbers). Thus a sensible use of the π-th root would be as if you found an English-speaking extraterrestrian community – not really impossible, but of such low probability that nobody would believe you. Same goes for the e-th power: e only appears in the basis of a power, not in the exponent.