We, of course, agree on the end result. I also agree that when keying into a calculator, or doing it somewhat manually, one or two "guard digits" is plenty. We may not agree on some other points. In one direction or the other, the conversion is usually short and simple. 1 yd = 0.9144 m is both exact and convenient. My calculator is frankly indifferent to whether I multiply or divide (although I am not, by pencil and paper). The conversions between gallons and liters, or pounds and kilograms may be a little too long to use the exact values on a calculator, but I don't want to know a bunch of values for the same thing. Finally, in a computer, there is little excuse for storing a wrong value (for a conversion). It should be exact, or expressed to the arithmetic capability of the machine/compiler. The program may be written to round sensibly after the full-precision conversion. I believe conversion should be viewed as a two-step process: 1) Convert exactly, or with considerably more precision than required. 2) Round sensibly, using rounding rules applicable to the situation (mins, maxes, targets round differently). Certain fields such as athletics have slightly weird rounding rules of their own. If viewed as a two-step process, these are rarely a problem. As to your example, the relationship may also be stated via rational integers as 1143 m = 1250 yd, least common factors have been extracted. When used in the form of 1250 yd/1143 m, the decimals will repeat forever.
--- On Mon, 4/6/09, Bill Hooper <hooperb...@bellsouth.net> wrote: From: Bill Hooper <hooperb...@bellsouth.net> Subject: [USMA:44451] rounding and conversion factors To: "U.S. Metric Association" <usma@colostate.edu> Date: Monday, April 6, 2009, 3:34 PM On Apr 5 , at 10:24 AM, John M. Steele wrote: I disagree that we should round conversion factors willy-nilly to a "nice" round number when we know better. John is right that conversion factors should not be rounded off "willy nilly", but it does makes sense to round them off to the degree appropriate for the data to be converted. That's not willy nilly. For example, even if we know that 1.093 613 298 338 yd = 1 m * it would be foolish to use that overly precise conversion factor to convert my height of 1.81 m into yards. By simple multiplication, that would give my height to be: 1.979 440 069 991 yd. or 5.938 320 209 974 ft. Such overly precise looking results are foolish. My height (the data being converted) is specified to only three significant digits so the result cannot have more than three significant digits. That is, my height MUST be rounded off to 1.98 yd or 5.94 ft Consequently, if we're going to round off anyway, it makes sense to round off earlier and save an awful lot of work. Realizing that my data (1.81 m) is good to only 3 digits, I round off my conversion factor to four digits (three might be enough but four allows some room for round off errors**), thus, instead of the long value above, I use: 1.094 yd = 1 m Using this rounded conversion factor, I calculate my 1.81 m height to be (before rounding): 1.98014 yd. or 5.94042 ft. both of which I round off to three significant figures, thus: 1.98 yd. and 5.94 ft. These are exactly the same values produced by the earlier, overly-precise calculations. The answers are the same with a lot less work. Bill Hooper 1810 mm tall Fernandina Beach, Florida, USA =========================== * I don't know how many decimal places there might be in this value. I calculated it by dividing 1 by 0.9144 which is the EXACT value for the number of metres in one yard. The division led to a very long decimal. I am not sure if it is infinitely long (repeating decimal) or if it is exact once one gets far enough "out" into the decimal digits. Yes, I could have used the exact conversion factor 0.9144 yd. = 1 m but I wanted a factor that illustrated my point better. However, these consideration has nothing to do with the main point above. ** If I rounded off the conversion factor to only 3 digits, my conversion factor would be 1.09 yd = 1 m and my raw results would be height = 1.9729 yd. and 5.9187 ft. , which after final rounding would be height = 1.97 yd. and 5.92 ft. These differ from the previous values by a quarter to a third of an inch. Quite honestly, those values would be good enough for most practical purposes, but the example does show that keeping one more digit in the conversion factor gives a slightly better answer. However, more than one extra digit does not give any further changes in the rounded off final value. ========================== SImplification Begins With SI. ==========================
