Hello, David, I don't think we've met... Perhaps you're relatively "new" around. In that case, pleased to meet you! :-)
But let's go to your post below, shall we? On Fri, 27 Sep 2002 17:43:28 david clayton wrote: >The application of the radian unit has introduced ambiguity and contradictions in >logic to mathematics. > >The removal of the radian by the formalised use of circular measure (using an >abbreviation "circ" in equations) takes away the source of these complications. > Firstly, I must mention that I'm no big fan of this radian stuff. However, please forgive me if I don't follow your rationale on this. Your argumentations below sounded somewhat confusing to me actually. So, let's tackle them. >Examples of ambiguity: >Example 1. >"If an angle x is small and measured in radians, sin x approximately equals x." The above, to the best of my knowledge, is ONLY true IF the unit of measurement of the angle is in radians! No other one (unit) would do. Therefore, I'm not sure your argument stands, unless you mean that the statement is missing this specific requirement I just outlined. In that case, then, ok, I can see where you're coming from. ... >Also, angles are converted to radians rather than "measured in radians", because of >the impractical size of the radian. > ? I'm sorry but I also don't follow you here. Angles CAN be measured in radians if one had the proper instrument for that purpose. Evidently though I'm not familiar with the existence of any such instrument... So, for practical purposes your comment does have a point in the end. ... >Examples of Contradiction: >Example 1. >The radian is given the dimensionless value 1.0. In dimensional analysis, David, angles do indeed get their dimensional equal to one! I.e. [alfa] = 1.0. This is a mathematical reality that can be *proven*! In such metrology theory physical properties that do not require unit names are "assigned" the value of 1. Otherwise several other physical properties would end up with wrong units of measurement. A typical example or application of D.A is with torque calculations, where results usually involve angles between the vector force and the vector distance in a vectorial multiplication: T = r x F (where all 3 entities are vectors, hence with the small arrow at the top, something I can't reproduce in my keyboard). In D.A. this is what we'd have: [T] = [r x F] = [r] * [F] = m * N, regardless of the spacial "location" of vectors r and F. >Comment: >This dimensionless value contradicts the ability of polar co-ordinates to describe >two-dimensional and three-dimensional space, using the dimensional value of angle >units and a single length dimension. > ? There actually isn't any contradiction on the dimensionless characteristic of angles. Whether they're involved in 2-D or 3-D calcs, their "contribution" to the final unit of the units involved in the calc would be... "zero", i.e. given that [alfa] = 1, they (angles) evidently are irrelevant in that regard. >Example 2. >When functions such as sin theta(Greek letter) are plotted, the label used for the >x-axis can vary disconcertingly when the radian is involved. Three labelling methods >have been noticed: >Method (a) The x-axis is labelled theta/rad. >Method (b) The x-axis is labelled theta. >Method (c) The x-axis is labelled theta(rad). The above has more to do with style of presentation of graphics than anything else. But with regards to the unit of measurement aspect of it, one would have a choice between, say, radians, degrees (Babylonian unit universally adopted), gons (or grades - the French decimal-friendly unit) or percentages. ... >Replacing the Radian as a base unit of the International System Unfortunately due to the mathematical reality of angles there simply is no chance for the radian "unit" to be "replaced" or discarded. Radians are actually just like a math... "reality", so to speak, just like the "golden rule" is the square root of two, if you know what I mean. Now, I totally agree with you that using radians is just... "unsightly" or cumbersome and that we do require something more... practical. Therefore, please consider my pitch in favor of the gon as fulfilling your requirements below. Since this e-mail is already too long, I'll leave that assessment up to you. However, I can advance to you that the gon would fulfill pretty much all items below. The added benefit of the gon is that it's evidently decimal! Marcus >Desirable features of a base unit include: >1. Convenience in use. >2. If not always of a convenient size, it should be easily divided or multiplied into >units of convenient size. >3. Should already be widely known. >4. Should constantly have angular dimensions. >(The radian had none of these features) > >An angle of one revolution seems a candidate... Is your boss reading your email? ....Probably Keep your messages private by using Lycos Mail. Sign up today at http://mail.lycos.com
