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.