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Rationalizing Pi
While torturing
myself on the elliptical exercise machine in my basement, I often do mental exercises
to help pass time during the utterly boring, albeit beneficial, endeavor. Often
the routine is no more complicated than dividing elapsed calories shown on the counter
display by elapsed time to get a burn rate, then figuring how much longer it will
take until my goal is reached (usually between 1,200 and 1,800 calories), all without
the benefit of paper, pencil, or calculator. Of course by the time I finally come
up with a number, I'm way past the point where the calculations began, so I start
over. That reminds me of the old episode of Beverly Hillbillies where Jethro is
showing off his new watch to Uncle Jed. Jed asks what time it is, and Jethro says,
"Well, the big hand is on the 2 and little hand is on the 12 and the second hand
is on the 10, so that makes it uh...., shucks, its not that time anymore." He went
through the routine three or four times with no indication that he would ever stop.
Finally, Jed asks him, "What's the pertnear time?," to which Jethro responds, "It's
pertnear 2:13." ... but I digress.
December 2021 Update:
In July 2012, after I posted this article, an Internet search on the topic did
not turn up any similar discussion about equating pi being with a full rotation.
Now when I do the search, there is quite a bit out there that includes my idea.
Did I start a meme? The
Wayback Machine captured this page for the first time on January 18, 2013.
Notice how different the webpage layout was at the time.
During my last session on the cursed machine, for some reason I was contemplating
pi (π
 not yet cancelled by
WHO). Pi
has been an enigma in the realm of mathematics and physics since it was first recognized
as being irrational The fact that the ratio of a circle's circumference to its diameter
is an inexact number has caused enormous amounts of consternation for dogged investigators
of the aforementioned phenomenon. Pythagoras is believed to have first noticed the
irrationality of certain numerical ratios when even something as basic as the cornertocorner
diagonal of a unit square could not be calculated to a finite precision. In some
religious circles even contemplating such thoughts caused souls to be burned at
the stake for daring to assert that such an imperfection could exist in a perfectly
created world. Recall that Galileo was excommunicated for asserting that the earth
was not the center of the universe. I kid you not.
For anyone not familiar with irrational numbers, they are numbers with nonzero
decimal places that do not end and do not have a repeating sequence. The "do not
end" part is what bothers me about pi. Consider that we normally measure rotation
in angular units of radians, and that 2π radians is defined as one full rotation.
If we never had to count more than a single rotation, then stopping at something
other than an inexact number is not so bothersome... although it actually
is bothersome since the rotation stops
exactly at some angle even if it cannot be measured in terms of pi. It doesn't asymptotically
approach a exactly quarter of a turn at but never actually stops. Rotation can stop
at exactly a quarter turn (exactly 90°) even if
π/2 radians itself is inexact.
Rotation can stop at exactly half a turn (exactly 180°) even if
π radians itself is inexact.
That just doesn't seem right, does it?
To really make the point, progress thorough a full rotation. At some point the
first full rotation is exactly completed and the next rotation ensues. We routinely
equate exactly 2π radians (an
irrational number) with 360° (a rational number). So, how can an irrational number
be exactly equal to a rational number? The concept seems... irrational.
Mathematically, we are comfortable with taking the limit of a function as its
variable approaches some value asymptotically, and declaring that the result can
be rational. An example would be , exactly. However, we
never speak of a limit when using pi to define angles. 2π radians is precisely one full rotation,
π radians is exactly half a
rotation, as are 360° and 180°, respectively. Do you see the logical conflict that
beguiles me?
Therefore, not through application of rigorous mathematical manipulations but
through elementary application of reductio ad absurdum I assert that pi must in
fact be rational  or at least it cannot be irrational. Maybe it is pseudorational
or pseudoirrational. Either way, I boldly declare quod erat deomonstrandum (QED).
Quando omni flunkus moritati (Red Green, look it up).
This article is written without doing an Internet search to see whether someone
else has cogitated similarly. Surely someone has. I go out on a limb here publically
demonstrating either brilliance or idiocy. If it is the latter, oh well, I've done
it before and I'll consider it further confirmation that I am, myself, irrational.
If it is the former, then feel free to nominate me for the Nobel Prize in mathematics
for having proved that pi is rational after all.
Update: I posted this proposition on
LinkedIn and have received a few comments. It's probably not allowed
to copy them here, so I'll just post my reply to the opinions:
****
Thanks again for the comments. My primary point is that it makes no sense to
equate an irrational value like 2 pi radians to a rational number like 360 degrees,
which is what we routinely do. According to Merriam Webster, a radian is "a unit
of plane angular measurement that is equal to the angle at the center of a circle
subtended by an arc equal in length to the radius." A degree is "a 360th part of
the circumference of a circle." So, you can rotate exactly 360 degrees to complete
exactly one full rotation since a degree is defined as exactly 1/360th of a rotation,
thus closing the circle. Can you also rotate exactly 2 pi radians and complete exactly
one full rotation since a radian is defined by what always works out to be an imprecise,
irrational value? For any arbitrarily minute, exact step size you can increment
exactly from 360 minus [step size] degrees to a full circle
by adding one [step size] degrees. Can you do the same exact sort of operation using
pi and radians? Using pi and radians means you can only close the circle after reaching
the last digit in pi, which if pi is an irrational number, can never be reached.
Is there another set of values, not related to pi, where we exactly equate a rational
number to an irrational number? (8/2/2012)
Posted December 1, 2021 (updated from original post on 7/21/2012)
