Preferred Numbers

In our last post, we discussed how standards can be somewhat arbitrary. A standard is a single number chosen from a span of values, any of which could be made to work. There’s no special reason, other than history, for a rail gauge of 1425 mm; it could easily be 1400, 1300, 1500, or anything in between.

How do we choose a number, then? Well, there’s a standard way of doing that too. All numbers are created equal, and are an ultimately arbitrary assignment of scale; once that scale is assigned, there are preferred numbers within that scale. How this works is interesting, and provides insight into the nature of standards.

The first preferred numbers are the numbers of the scale itself.  Already we’ve made a decision, and that decision will effect how we proceed. The scale provides a unit, which is a degree of difference (be it length, temperature, mass or otherwise) that is considered equal to a single integer. The scale also provides a base, and in many ways, this is more important.

The SI system of units comes with two bases in the present form, though it was born with one. It is fascinating that there is a binary base for SI, but not particularly meaningful in this discussion as it is seldom used for anything but bits and bytes. The decimal base is far more familiar, and is the system we all learned in school: centi-, milli-, micro-, kilo-, mega-, giga-, and everything in between, and past in both directions.

This is simple enough. A liter has a thousand milliliters in it, a thousand liters is a kiloliter, and Bob’s your uncle. What of a gallon? Well, a gallon is four quarts, each of which is two pints. Each pint is two cups, each cup is 8 fluid ounces,  and each ounce is two tablespoons. Each tablespoon is three teaspoons. Got all that?

This may appear totally insane, but realize: We have defined a gallon as 256 tablespoons, 128 ounces, 16 cups, 8 pints and 4 quarts. These numbers should be familiar: they are powers of two. So what?

Here’s what. If you pour from a full quart bottle into a pint bottle, filling it, you have a pint left in the quart bottle also. This is the standard way to divide liquids for trade, and this system arose in a culture where that kind of convenience was important, far more important than the ability to do arbitrary calculation. The imperial volume system has the preferred numbers built in. What this gives us is easy conversion where it counts most, assuming you’ve learned all the quaint terms for the various units.

What we lose is the ability to do math on our units without driving ourselves bonkers from conversion factors. It’s not worth it, and SI is better.  So we’re going to sacrifice baked-in preferred numbers for the decimal scale. How do we get our preferred numbers back?

Say we have a decimal scale, Graham units, abbreviated Gr, and we are going to devise a range of widgets that provide from 1 milliGraham (mGr) to 1 kiloGraham (kGr) of awesome.  The first numbers we pick are clear enough: we want widgets in 1 mGr, 10 mGr, 100 mGr, 1 Gr, 10 Gr, 100 gR, and 1 kGr.

This produces a logarithmic scale: the ratio between units is a constant 1:10, just as the imperial system keeps a more-or-less constant ratio of 1:2 between units. But keep in mind: 10 times as much awesome is a lot more awesome. It’s like if we were selling awesome by the Altoids tin, the bank box, and the shipping container. Sometimes people might want an intermediate amount of awesome. How do we give it to them?

First instinct: lets provide even intermediates of awesome. Instead of just widgets that deliver 1 Gr and 10 Gr., we’ll make units at 1 Gr, 2.5 Gr, 5 Gr, 7.5 Gr and 10 Gr, so on, up and down the line. This is an easy guess for how to do it, and dead wrong: I’ve had professors and bosses tell me to do it this way and they’re wrong too, and I got an ISO standard and some math to prove it.

The nice thing about our units, before, was the constant ratio between them. This is a good thing in a standard, as it means the preferred numbers are distributed ‘evenly’ throughout the standard. The reasoning, basically, is that a small amount of a small measurement matters, but a small amount of a large measurement doesn’t.

Here’s what this means. 1 Gr and 2.5 Gr have a ratio of 1:2.5, and 2.5 and 5 Gr have a ratio of 1:2. 5 and 7.5 have a ratio of 2:3, and 7.5 and 10 have a ratio of 3:4. In decimal, we’re going from 0.4, to 0.5, to 0.67, to 0.75.

This is horrorshow; our ratio has gone from a nice constant 0.1 to fluctuating all over the map. Remember, if we were selling awesome by the cup, we’d have a nice even 0.5 ratio with the occasional 0.25 thrown in to spice it up.  Something must be done; enter Charles Renard, stage left.

Charles Renard designed a standard for use with decimal scales; a way of subdividing them by 5, 10 or more such that the ratios remain close to constant. Therefore, following Renard, we’re going to sell widgets delivering the following levels of awesome: 1 Gr, 1.6 Gr, 2.5 Gr, 4 Gr, 6.3 Gr and 10 Gr.

Now what are our ratios? In decimal: 0.625, 0.64, 0.625, 0.635, 0.63.  This is good: six and change all the way up and down the scale. If we need finer scales, these are defined also, and we can divide our decimal scales into 10, 20, 40, or 80 distinct gradations.

What does all this mean? If you are designing something, from an experiment to a widget to a presentation, choose preferred numbers. Like magic, your data will have a meaningful spread, your parts will be more likely to be interchangeable, and your ratios will have a mysterious, aesthetically pleasing quality: the Golden Ratio is 0.618, and is the basis of the series of preferred numbers. The Reynard numbers are a compromise between the aesthetic and mathematical ideal of harmonic proportion, and the realities of actually measuring and building stuff.

Basically, if you work with decimal scales, burn these numbers into your heart: 1.6, 2.5, 4, 6.3. Make up a little song. Have dreams about them. Use them liberally and wisely. These are your father’s preferred numbers; an elegant weapon for a more civilized age.

Published in: on August 25, 2009 at 7:17 pm  Leave a Comment  

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