Euler the sailor?
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Todays' xkcd:
Mouseover title: "It works because a nautical mile is based on a degree of latitude, and the Earth (e) is a circle."
The (upper) equation is correct (and I'm shocked that I never noticed this before). Wikipedia:
A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute (
1/60 of a degree) of latitude at the equator, so that Earth's polar circumference is very near to 21,600 nautical miles (that is 60 minutes × 360 degrees). Today the international nautical mile is defined as 1,852 metres (about 6,076 ft; 1.151 mi). The derived unit of speed is the knot, one nautical mile per hour.
Given the official International nautical mile at 6076 feet, this gives us 6076/5280 = 1.150758.
The Geographical Mile, defined as one minute of arc at the equator, is apparently 6087.08 feet, which would give us 6087.08/5280 = 1.152856.
To the same six decimal places, π is 3.141593 and e is 2.718282, for a ratio of 3.141593/2.718282 = 1.155727.
So comparing, we get either
(3.141593/2.718282)/(6076/5280) = 1.004319
which is off by 0.4319%, or
(3.141593/2.718282)/(6087.08/5280) = 1.002491
which is off by only 0.2491%. Certainly <0.5% in both cases…
And it's true that a nautical mile is one degree of latitude at the equator — but having e stand for "earth" is just a pun, and the history of the Statute Mile being equal to 5280 "feet" is entirely arbitrary in this context.
Still, once we've got pi and e together in the conversation, Euler's identity can't be far away. And for me at least, that brings up the limerick version:
I used to think math was no fun,
'Cause I couldn't see how it was done.
Now Euler's my hero,
For I now see why 0
Is e to the pi i + 1.
Gregory Kusnick said,
December 12, 2024 @ 5:23 pm
I suppose the astronomer's version might go something like this:
Diameter of Earth's orbit = 1000 light-seconds within 0.2%.
Earth's orbital speed = 10^-4 c within 1% (or e * 10^4 m/s within 10%, but that's stretching it a bit thin).
So 1 year = pi * 10^7 seconds within 0.5%.
(It turns out, though, that for back-of-the-envelope calculations, 1 year = 10^7.5 seconds is more useful, since t^2 appears frequently in equations of motion, whereas pi hardly ever does.)
Phillip Helbig said,
December 12, 2024 @ 5:34 pm
Not directly related, but Digital Equipment Corporation in some documentation described the value of some system parameter on the VMS operating system in units of microfortnights, which were approximated by seconds in the implementation.
Mark Liberman said,
December 12, 2024 @ 9:01 pm
@Phillip Helbig: "Digital Equipment Corporation in some documentation described the value of some system parameter on the VMS operating system in units of microfortnights, which were approximated by seconds in the implementation."
I believe that the VMS TIMEPROMPTWAIT micro-furlong joke grew out of a broader FFF ("furlong-firkin-fortnight") culture at MIT in the 1960s. (Though of course the furlong never achieved the fame of the smoot…
rosie said,
December 13, 2024 @ 1:39 am
For an explanation of the smoot: the bridge which is measured in smoots: https://www.youtube.com/watch?v=-scs_yF59YE
The phrase "official International nautical mile at 6076 feet" made me read again; Wiki's right; it's 1852m, so we have 3.14159265*5280*.3048/(2.71828183*1852)=1.00429961.
Peter Taylor said,
December 13, 2024 @ 2:58 am
@Gregory Kusnick, you then run into the claim that astronomers use 1 as a "good enough" value of pi…
(I'm not an astronomer, so I can't speak to the popularity of the use of 1 in actual practice).
Philip Taylor said,
December 13, 2024 @ 5:26 am
In re Victor's limerick, I would have cast the last line as :
but I can't explain why …
David L said,
December 13, 2024 @ 7:37 am
you then run into the claim that astronomers use 1 as a "good enough" value of pi
In my days as a practicing cosmologist, I had a colleague who liked to say that most of the time powers of ten were all you needed, but if a more accurate number for some quantity was required, it was three — that being logarithmically halfway between adjacent powers of ten.
And in the odd coincidences department, the speed of light is, to better than 2%, one foot per nanosecond.
KeithB said,
December 13, 2024 @ 9:08 am
As a joke, when I wrote a program to display RF power, I included Watts, dBm and horsepower.
At Los Alamos they coined "shake" to be 10 ns. They had some others, but the notebook I wrote this down in had classified information, so I had to turn it in when I left Sandia.
Andreas Johansson said,
December 13, 2024 @ 9:43 am
I somewhere saw the opinion that the only numbers you really need are zero, one, and infinity.
Rounding everything to the nearest power of ten is something I've seen in the wild. And it's frankly something we should do more often at work – too often we multiply a number that's accurate to a couple of digits with one that's derived along the lines of "it's got to be more than one but less than a hundred, so we'll call it ten" and then act like the result is accurate to three digits.
J.W. Brewer said,
December 13, 2024 @ 9:44 am
1.2 seconds is such a good approximation of a microfortnight that using 1.0 seconds seems inexcusably lax.
anonymous said,
December 13, 2024 @ 10:31 am
one of my favorites: to a good approximation, pi seconds is a nanocentury
I forget exactly how good, but it's surprisingly close
@Andreas Johansson: regarding zero, one, infinity,
I learned it as "zero, one, many", but I think we're on the same page.
Typically when checking computer programs that involve a loop, we need to look at three situations, and make sure they happen as planned and work as intended: zero (we never execute the code inside the loop at all), one (we execute it exactly once, then exit the loop), and many/infinity (we go around the loop repeatedly until/unless some exit condition is met)
Robert Coren said,
December 13, 2024 @ 10:43 am
Randall slipped up (and Mark repeats his error in the penultimate paragraph) when saying that the nautical mile is "one degree" of latitude, whereas the Wikipedia article correctly specifies it as "one minute". It would be small world indeed if the distance from pole to pole were 180 nautical miles.
Robert Coren said,
December 13, 2024 @ 10:46 am
As long as we're digressing into other mathematical coincidences, musicians can be grateful that 2^12 and (3/2)^7 are so close together that equal temperament works pretty well.
Matthew Juge said,
December 13, 2024 @ 2:06 pm
@Robert Coren: Those exponents should be 1/12 and 1/7.
Andrew Usher said,
December 13, 2024 @ 10:49 pm
Or, by cross-multiplication, 7 and 12. 2^7 ~ (3/2)^12 (closer than to 5^3), the logarithmic difference in either case being about 0.3%.
A better 'astronomical' analogy to an identity like this one is (AU in a light-year) ~ (inches in a mile), the latter being a number I can recall instantly. This not only has some real use – demonstrating the size of space – but is not entirely arbitrary, as relation between the AU (mean distance between Earth and Sun) and light-year depends only on the speed of light (natural constant as much as pi or e) and the Sun's mass, also not arbitrary. The error is smaller than Randall's equation.
k_over_hbarc at yahoo.com
Robert Coren said,
December 14, 2024 @ 10:23 am
@Andrew Usher: Yeah, I had the exponents reversed. Thanks.
Christian Weisgerber said,
December 14, 2024 @ 2:28 pm
Also, 1 mile = ln(5) km. Accurate to four decimal places.
/df said,
December 15, 2024 @ 7:50 pm
Maybe it's rather the sailor's version of π² = g (± 0.7%)? Because originally (Huygens) the met/er/re was defined as the length of a pendulum with period 2s.
Andrew Usher said,
December 16, 2024 @ 12:29 am
Yes, those are equivalent. But I don't know any such actual definition is use, though some scientists proposed the seconds-pendulum as a standard ef length. But by the 18th century, that was surely known to differ across the Earth,
But Randall's equation still, as I remarked, has no conceivable utility, being just a trivial fact, while yours also has use. (But I expect 10 m/s/s is must more used, of course pi^2 ~ 10 is also a practical approximation).
Mark Liberman said,
December 16, 2024 @ 4:21 pm
@anonymous: "one of my favorites: to a good approximation, pi seconds is a nanocentury. I forget exactly how good, but it's surprisingly close"
365*24*60*60*100 = 3153600000
(10^-9) * 3153600000 = 3.1536
3.1536/pi =1.003822
So within 0.3822%
Robert Coren said,
December 17, 2024 @ 10:34 am
Even if you use 365.25 as the year length (not perfectly accurate, but closer than 365) the ratio comes out to 1.004520, which is still reasonably close.
Philip Taylor said,
December 22, 2024 @ 5:15 am
Regarding VAX/VMS's "microfortnight", which I remember well, I also remembered as I lay in bed last night that there is another "unusual" unit embedded in Donald Knuth's TeX, the "rsu" ("ridiculously small unit)" :
[Source: Cahiers Gutenberg 1989, pp.~2–15]
Philip Taylor said,
December 22, 2024 @ 5:17 am
For $5.36 x 10^{-3}μ$ please read $5.36 × 10^{-3}μ$.