From: mathwft@math.canterbury.ac.nz (Bill Taylor) Subject: UNITS and their importance - another RANT! Date: 20 Apr 1999 06:56:25 GMT Newsgroups: sci.math As it seems to be the "in" thing right now to have RANTS on sci.math, and as I'm pretty good at ranting anyway, and as I feel strongly on the subject of UNITS in math, here goes. By units, we mean physical units; cm^2, amps/sec, joules etc; as in the math teaching RANT thread over the way there. One thing that always used to irritate me as a child, was that Scrooge McDuck's money bin was said to be 3 cubic acres. No doubt many others felt similar passing annoyance at this inappropriate 6th-power-length unit, (even if the intended meaning was clear). Well, it eventually occurred to me, while teaching stats one day, that it had an application there. Clearly, Scrooge's money volume was variable, with mean = M cubic miles, and VARIANCE = 3 CUBIC ACRES ! ============= Howbout that! While I'm on statistics, let me plug another pet peeve of mine. Half of stats text-books parametrize the Normal distribution by mean and standard deviation, and half by mean and variance. IMHO the latter are QUITE WRONG, and doing a BAD THING! The parameters should be IN THE SAME UNITS, which are also the units of the variable itself. Thus, one can make instant comparisons, which is great for common sense checks. Thus, it is nice to know men's average height is 5.35 ft with a std-devn of 0.23 ft; but what f****** use is it to see that the variance is .0529, of *square feet* no less! Pah! Units really come into their own in non-linear comparison situations. One of my favorite examples is the formula for finding the distance to the horizon from some given height above sea level. Most of us can remember the relation is Distance = k*sqrt(Height), but can never recall a benchmark for that constant k. My favorite mnemonic for this, (and quite accurate), is ... "HALF A YARD SEES HALF A LEAGUE". ============================== More seriously, and on the topic of how keeping track of units can prevent, or quickly pick up, many mistakes in a calculation, a topic well covered in the Math Teaching RANT, here are some more guidelines that teachers might want to keep in mind. If one gets to be adding quantities with *different* units, clearly something has already gone wrong, and you must backtrack. More subtly, it might involve adding pure numbers to ratios of units which don't cancel out, even though such ratios might be OK in other contexts. 2.7183 + A/c is no good if A is an area and c is a speed! Extending this idea further, sines, cosines and (especially) logs and exps can ONLY be taken of pure numbers. log(A/c) [as above] is a sure sign of a major mix-up. The case of trig functions is quite interesting:- strictly speaking, they don't take numbers to numbers at all, only our standard convention of 2pi radians = 360 degrees ensures this. But strictly, trigs take inputs in angular measure (with its own peculiar units), and produce outputs which ARE pure numbers, (length/length in the original definitions). Though this is straying from the topic a little, it can be quite useful in untangling mistakes where you mis-remember those nasty little square-root formulas for inverse trig functions. Incidentally, inverse trig functions DO act on pure numbers, though outputting in angular units, of course. -------------------- Which reminds me - did you know the current US president comes from inverse Kansas? -------------------- Anyway - here are some more useful mnemonics concernic units. They are very good in situations where your x-y graphs really might both involve lengths. For example, in the Newton-Raphson iterative-solution formula... x^ = x - y/y' ...for which you don't really need a mnemonic, (as you can quickly pic the little diagram for yourself), it may be useful to note that it can't be y'/y, as then the length units would be wrong. (Note that y' always is a pure number here!) This will save you dividing by something horribly close to zero very soon! Similarly, in Euler's equation in the calculus of variations, d F F -- y' = y it is often hard (for me) to recall dx just which is which of those two partial derivatives. A quick check on units will soon relieve you. It can't possibly be... d F F because this LHS has units 1/length^2 -- y = y' and the RHS has units zero. Neat, huh? dx As a final anecdote, let me mention a marvellous talk we had here recently, by Sir Michael Berry (may his house ever prosper!), titled "The Seven Wonders of Physics". Probably I'll mention this talk again in various contexts, but he is also very keen on keeping units straight, and one of his "wonders" was directly apposite. He noted the odd fact, previously posted here from time to time, that a car's fuel consumption actually has the units of 1/AREA. Quite odd. Once I worked out that my car does about 2 billion to the acre, which sounds VERY impressive, salesmen take note. Well, this thingy also occurred to Sir Mike, who soon came up with the proper interpretation of it. The units of fuel consumption are 1/area. And the area involved, turns out to be precisely the CROSS-SECTIONAL AREA of a little tube of fuel, that runs continuously alongside the road, that the vehicle can just keep going with, if it sucks up the fuel as it chugs along. IIRC, for a modern car, it turns out to be about (1/4 cm) squared. Efficient! What a great image - what a great interpretation! Now THAT'S understanding units!! As a coda, he mentioned that the human body is also quite efficient; it seems you can bicycle 100 miles on just 4 chocolate bars. In other words, if you spread out a chocolate bar every 25 miles, evenly spread out in a similar groove to the petrol, then the cyclist can keep going just by continuously lapping it up... in some suitable manner. :) Now *there's* efficiency! ------------------------------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------------------------- __ __ _________________ /\ \ / \ / _______________/\ \ \ \ / /\ \ / / ______ ______\/ This sig is the result \ \ \ / /\ \ \ / /_/__ \ \ \ \ \ \ / / \ \ \ / _____/\ \ \ \ of my not having kept \ \ \ / / \ \ \ / /\____\/ \ \ \ \ \ \/ / \ \ \/ / / \ \ \ my units straight. \ \__/ \ \__/ / \ \ \ \/_/ \/__\/ \_\/ -------------------------------------------------------------------------------