More than 10,000 searchable pages indexed.
•−• ••−• −•−• •− ••−• •
RF Cafe Morse Code >Hear It<
These original Kirt's Cogitations™ may be reproduced (no more than 5, please) provided proper credit is given to me, Kirt Blattenberger.
Please click here to return to the Table of Contents.
Cog·i·ta·tion [koj-i-tey'-shun] – noun: Concerted
reflection; meditation; contemplation.
Kirt [kert] – proper noun: RF Cafe webmaster.
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 by 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 per-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 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 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 is actually 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 quarter of a turn at π/2 radians but never actually stops. It can stop at exactly a quarter turn even if pi 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 with 360°, which is 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?
Therefore, not through application of rigorous mathematical manipulations but through elementary application of reductio ad absurdum I assert that pi must 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'm 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.