The case of the disappearing angle units, or "the dangle of the angle"

[jlind]

(08-13-2019 07:28 AM)ijabbott Wrote:  

(08-13-2019 01:56 AM)jlind Wrote:  You are confusing dimensionless with unitless and equating them. They're not the same. A plane angle, which is dimensionless, is a scalar value with a unit to reflect a quantity, be it radians, degrees, grads, quadrants, sextants, turns, or some other unit of measure. Dimensionless and unitless are two very different things. There are an enormous number of dimensionless scalars with units of measure. The SI unit for a plane angle is the rad, the abbreviation for Radian. The unit for a solid angle is the sr, the abbreviation for Steradian.

But is the distinction important mathematically or only for engineering purposes? From what I can gather, mathematicians (or at least pure mathematicians) tend to think of the trig functions as purely numeric functions, without units. For example, "trig substitution" may be used to make certain integrals more tractable.

ijabbott,

An "applied" mathematician would also like to be thought of as "pure" versus "impure" although a "theoretical" one might think his "applied" brethren have allowed themselves and their abstractions to become contaminated by the concrete world around them. ;-) It's semantics nit picking, but as a Physicist and Engineer, I am by necessity also an applied mathematician. Forgive me. I couldn't help myself, but freely admit having committed the Faustian act of selling my soul using the theoretical in practical applications. :-D I'm not devoid of the philosophical though as I'm a Formalist, not a Platonist (I'll let you look those up).

From a standpoint of deriving various trig identities, such as sin^2(theta) + cos^2(theta) = 1, or in dealing with theorems, the angle variable is always there and it's implicit that its units are consistent throughout (i.e. the angles are all in Radians, Grads or Sextants, etc.). The units used are arbitrary, but they're still there. It's implicit in the Pythagorean Theorem for a plane geometry right triangle, "c^2 = (a^2 + b^2)^0.5", (the solid geometry version: d^2 = (a^2 + b^2 + c^2)^0.5) that range and domain variable units are consistent (all in fermi, furlongs, rods, bohr, leagues, cubits, etc.). Otherwise they would be polluted with conversion factors, such as 0.9 deg/grad, or 1000 am/fm. Likewise with Einstein's equivalence of matter and energy, E = mc^2. If S.I. (aka mks) is used, "m" is in kilograms, "c" is in meters/sec and E is in Joules (Newton-meter, or kg m^2/s^2). In "cgs", E is in ergs, m is in grams and c is in cm/sec (g cm^2/s^2), 1 erg = 10^-7 J. In Newtonian Mechanics (i.e. before Einstein's Relativity modified it), Newtons Law of Universal Gravitation (for two bodies) is often expressed as
F = G * (m1 * m2 / r^2)
where G is the Universal Gravitational Constant, F is the mutual force, and m1 & m2 are the masses of the respective bodies, and r is the distance between their respective centers of mass. No units are given in physics texts, but you'd best be consistent regarding mass and length, and for the Gravitational Constant G (it has units and its value is units dependent), and what that means for the resulting units you get for the mutual Force. Not too bad if you're dealing with mks vs cgs as it shuffles the decimal point a few places, but it was a mess when some of us had to deal with it in FPS (Feet, Pounds and Seconds with someone always giving "r" in miles; see remarks below).

Theoretical mathematicians and physicists like to deal in general cases using range and domain variables independent of units. Doesn't mean they're unitless. It's implicit in practical application that units will either be consistent or conversion factors employed. It makes them cleaner looking for clarity of the relationships. Trig identities are often written without the "theta" but it's implicit.

If you want a real joy, start using common "English" aka "US Engineering" fps (foot, pound, second) units related to force, mass and energy, using Pounds, Poundals and Slugs for mass and force. Don't even try to use the former British Engineering System which lacked coherence and contained ambiguity regarding what a "pound" is (force or mass ?) that could only be hopefully resolved by usage context (not the monetary version of the Pound Sterling). Had several years of that to contend with in school. Gave the scalar values for some velocities once in Furlongs/Fortnight and Leagues/Lustra out of sheer frustration, as the units for the answer were not specified. Gave an area answer one other time in Barns (and there's a smaller one related to it called a Shed) versus square feet. Ever so glad when metric and S.I. supplanted English units in most engineering (Civil, Construction and Architectural in the US must still deal with English units).

Hoping this has been at least partially entertaining.

John