December 2018 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
Wayback Machine captured this page on January 18, 2013.
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 pert-near time?," to which Jethro responds, "It's pert-near
2:13." ... but I digress.
During my last session on the cursed machine, for some reason I was contemplating
π. 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 corner-to-corner 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 non-zero 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 July 21, 2012
A huge collection of my 'Factoids' can be accessed from my 'Kirt's Cogitations'
table of contents.
Topical Smorgasbord, another manifestation of Factoids,
are be found on these pages:
| 2 |
4 | 5
| 6 | 7
| 8 | 9
| 10 |
11 | 12 |
13 | 14
| 15 |
16 | 17 |
18 | 19
| 20 |
21 | 22
| 23 |
24 | 25 |
26 | 27
| 28 |
29 | 30 |
31 | 32
| 33 |
34 | 35 |
All pertain to topics that are related to the general engineering and science theme
of RF Cafe.