Thursday, December 31, 2009

The Death of the Snipe

Among the general readership and the very authors of this blog, the concensus seems to be that--well, the posts have regrettably grown too deep and profound, to a point where it just fails to have any superficial meaning. And we apologize for that. To compensate for the madness that is the other posts before this, we dedicate the following to just pure, superficial meaning and making sense. In fact, never has so much sense and lack of deep cognitive requirements been packed into a single piece of writing. So take a break! It's been the end of a long 365 days.

The blog is the logbook of the internet. It has colonized the web, and at the same time the web has colonized it. Unlike the natural log, the blog was the faultless second child, the projection of perfection onto the real line. The natural log had its humble beginning as a tree and its humble end as a log cabin, from whence stemmed forth a nation mighty. The blog was born. While the root of the natural log was its base, the blog escaped the dreadful litmus of the conservatives. It conquered and was conquered; it tried to transcend but was transcended. Its' inherent logness could not be changed, and the blog fell to the curse of the metalogs.

But herein lies the paradox. It is understood that the existence of complex life on earth is vastly dependent on oxygen. For example, the final electron receiver in the electron transport chain in cellular respiration is the lowly, diatomic oxygen molecule. However, oxygen is a highly reactive molecule, very capable of damaging living organisms through uncontrolled oxidation. And this is indeed an unresolvable paradox, the untangling of which is akin to wrestling a cuttlefish. And this is in itself a paradox. The paradox of paradoxes is that while the vast majority of paradoxes requires the use of nonsense, nonsense is a highly confusing concept that would bewilder scholars by leading them to believe that what they are reading is nonsense rather than a paradox.

And so to this extent, the experts have proposed an ingenious solution: text boxes. That piece of modern art not cooperating in your text document? Simple. Slap it into a text box. From extensive testing in our laboratories, even the most existential of images will be reduced to an unmoving piece of ether with respect to an external text box. Unfortunately, we have yet to develop a method of capturing such images; at the same time, images have yet to develop a method of escaping from text files. So hunt them down vigilantly. They can run, but they can't hide, and their ability to run is limited. A win-draw situation.

The second theorem guarantees the existence of fireworks, the awesomeness of which is defined in the axiom of explosion. The propagation of awesomeness, then, is governed by a divergent generating function. As a corollary of this, there must exist the elementary particle awesomine, with spin +3. While such particles have not been detected in ordinary fusion, they are conjectured to be produced in massive quantities when the proportionality symbol is transferred via a flash drive. This idea has excited the scientific community, as if the conjecture is true, a perpetual motion machine of the second kind can be constructed by attaching fireworks to flash drives containing nothing but the proportionality symbol.

Henceforth, we must be thankful of the art of proof. It is a testament to the crumbling of society today that the developed nations are continually dissatisfied with what they have, and in dissatisfaction lead the expansion, crushing the weak and defenceless. General Hilbert's plans to invade physics may have been forestalled, but for only a finite time. In the time of great mourning, we must stand firm and remember who our true enemy is. It is not triviality. It is the radian, the steradian, the percent, all the imposters among the group of units. It is not the evil quadratic residues that we should fear, but the fools pretending to be sums of squares. As Fernando Westrick said, "Ask no questions--but partake of the proof".

Happy New Year!

Saturday, December 19, 2009

Degrees: The Menace of Mathematics

The method of measuring a rotation around a circle has vexed many a high-school student. For the general public, and in the education system before trigonometry is introduced, the unit of measure is known as the degree. The degree certainly has its historical roots; it is believed that the degree originated from the Babylonians and their base-60 number system (Bringhurst 2002), and first appeared written in English in medieval times (Miller et al. 2009; Chaucer 1386, quoted in Miller et al. 2009). Yet for the trigonometry student, and for one in calculus, another unit of measure is introduced: the radian. It has appeared in more recent times (Cajori 1919, quoted in Miller et al. 2009; Thomson 1873, quoted in Miller et al. 2009; Muir 1874, quoted in Miller et al. 2009), but is clearly more suited for measuring angles.

However, none of these authors actually deal with the importance of the degree measure. This brings to mind the question: is degree measure really necessary at all, and if not, can it be removed from the academic curriculum and from general society? I will analyze the extent of the usage of the degree and radian measures to discuss whether it is feasible to replace all occurrences of the degree with the radian. In this essay, I will argue that due to the extensive usage of the radian measure in mathematics and physics, and the inherent confusion from learning radians after many years of learning degrees, it is more convenient for students to do away with the degree measure entirely. To conclude, I will determine whether it is plausible to do away with the degree measure entirely.

The typical North American young student would encounter the subject of "geometry", as a sort of extension to math. The types of angles, in degrees, are taught: less than 90° is an acute angle, 90° is a right angle, 90° to 180° is an obtuse angle, 180° is a straight angle, and between 180° and 360° is a reflex angle. Transversals: these are vertical angles, these are corresponding angles, these are alternate interior angle, these are alternate exterior angles, these are consecutive interior angles, and these are consecutive exterior angles. There is the fact that a university student is highly likely to have forgotten these angle names, and thus the issue of whether transversals even need to be taught arises, but this essay does not deal with this, and further research is required to deal with this issue. Nevertheless, all these angles are expressed in degrees, and the concept of pi is left to being a irrational (weird) number that has something to do with the radius and circumference of a circle. If a student were to read, perhaps, a high-school or university textbook of an older sibling and ask the teacher what a radian was, the teacher would almost certainly reply that the radian was something else further away down the road of education, in high school.

Thus, it is in high school where the radian is introduced. It is not to say, however, that the moment high school starts, the students are immediately told that 2π radians constitute a circumference of a circle; far from it, it is only taught at the grade 11 and 12 level. However, the degree is not completely left behind; in fact, trigonometry and calculus may deal with radians, but physics continues to use degrees unabated. The dual usage of degrees and radians continues to the end of high school, where the physics final exam uses degrees, and the math exam uses radians in addition to degrees (and may even include a question requesting a conversion between the two) (BC Ministry of Education, 2007). The taboo surrounding trigonometry and calculus as "hard" can thus be explained: since students have thought of a circle as 360 degrees, and a right angle as 90 degrees, it is a sort of shock for them to suddenly switch to thinking a circle has 2π radians and a right angle has π/2 radians.

However, in university the dual teachings of the degree and radian end. In both mathematics and physics, the degree measure is abandoned for the radian measure.

Except not. Diffraction gratings? No one can understand that confusing stuff in radians. See that little line there? The point of maximum constructive interference? That's at 45 degrees. Not π/4 radians. No, you're mistaken. Yeah, that's right. I took that calculator. AND I THREW IT TO THE GROUUUUUUUNNNNDDDDD

WHAT YOU THINK I'M STUPID
I'M NOT A PART OF YOUR SYSTEM
REAL PHYSICISTS USE GRADIENTS
DUHHHHH


One may argue that the degree/radian case is similar to the case of conventional current, where Franklin's arbitrary definition of positive and negative charge resulted in electricity, so to speak, "going the wrong way". It is common practice in high school and university to teach circuits with a charge coming from the positive terminal of a battery, rather than the negative terminal. Obviously, it would be optimal to rectify this, yet time has secured this way of thought, ergo the term "conventional current". However, the argument that this may also apply to degrees and radians is invalid. While degrees is the common unit of angle measure, the radian is also used frequently as argued above. This contrasts the "actual" current of electricity, which is rarely discussed in high school or university.


-prolly mention "Anderson Cooper 2π" for lulz

A small sacrifice for a switch to a much more logical and natural system.



Works Cited:

http://mathworld.wolfram.com/Degree.html accessed November 12, 2009
http://mathworld.wolfram.com/Radian.html accessed November 12, 2009
http://jeff560.tripod.com/d.html accessed November 12, 2009
http://jeff560.tripod.com/r.html accessed November 12, 2009
http://www.bced.gov.bc.ca/exams/search/grade12/english/release/exam/0711ph_p.pdf accessed November 12, 2009
http://www.bced.gov.bc.ca/exams/search/grade12/english/release/exam/0711ma_p.pdf accessed November 12, 2009
Bringhurst, R. The Elements of Typographic Style, 2nd ed. Point Roberts, WA: Hartley and Marks, p. 276, 1997.