If you remember taking science classes in school, or teaching them, you may remember that some emphasis was placed on carrying the units of measurement along with the numbers when solving physical problems. To take a very simple example, if a cyclist rides 9.2 miles in 30 minutes, then his speed is given by:
Speed = 9.2 miles / 0.5 hour = 18.4 miles per hour
Doing this is just an additional check that the calculation has been done correctly, and that the answer is in the expected units; that is, if we are calculating a speed, we expect an answer in units of distance / time. If we end up with an answer in cubic feet, something is wrong. (I’ve done a bit of science teaching, and one of my little brain-teasers for physics students is: a number is given in units of stone-barleycorns per milli-fortnight. What is the number a measure of?)†
Another reason to preserve the numbers and units together, of course, is to ensure that someone else reading them will interpret them correctly. There have been some fairly serious problems caused by not getting the units right:
Out of context, a number can be a dangerous thing. In 1991, for example, NASA’s $125 million Mars Climate Orbiter was destroyed because one team of engineers used imperial units of measurement while others relied on metric ones.
The quote is from an article in Technology Review, which describes a new semantic database technology for dealing with numeric data, developed by a Cambridge, MA start-up, True Engineering Technology . The idea is fairly simple: each number is stored along with its units of measurement, context, tolerance, and semantic tagging information.
To store a number in the system–creating what the company calls a “truenumber”–a user simply types a short phrase into a form on the website … [for example] a user might type, “The distance from New York to London in kilometers is 5,581.”.
The software is being made available in two ways: as a product that companies can install on their own servers and networks, and as a free (of charge) public Web service at the company’s site, something that the company envisions as a “Wikipedia for numbers”. The market for the commercial version is expected to be engineering companies, primarily, but potentially any organization that works extensively with numbers is a possible customer.
I will be interested to see if this becomes successful. I have a vivid memory of one personal experience of numbers being used by someone who didn’t understand them. I worked for a time, quite a few years ago, for a consulting company that, among other things, published regular reports on the state of the economy and financial markets. These included figures on the rate of return to various investments over recent periods. One of the returns reported was for investments in 30-day US Treasury bills. Now this should have been easy to get, since the prices of T-bills are reported every day in the Wall Street Journal and many other papers. Yet our firm’s numbers were consistently different from the ones calculated by everyone else.
I asked if anyone had looked into this, and was told that they had, and that the formula used for the calculations had been checked, and verified to be correct. So I went to see the guy that did the calculation, and asked him to humour me by walking through a sample with me. The formula, as it happened, was exactly right; it’s just that it didn’t apply for the data that were being used. Treasury bills are discount instruments. Unlike, say, a corporate bond that provides a periodic interest payment (typically semi-annually), the T-bills interest is implicit. If the bill will be redeemed for $100 at maturity, it will initially be sold at a discount, at a price less than $100. For example, if the bill sells originally for $99, and is redeemed for $100 at maturity, the interest rate over that period is [(100/99) -1] * 100 = 1.01 %. And therein was the rub: the formula being used assumed that the rate entered was the interest rate, but the number reported in the paper is the discount (which would be 1% in this case). So the numbers were consistently too low.
Fortunately, the mistake was easily fixed, and the historical data corrected, once the problem was found. But I can’t help but think that there are many more examples out there, perhaps not as spectacularly disastrous as the Mars Orbiter, but waiting to bite someone nonetheless.
† Oh, I almost forgot. A stone is a British unit of weight, equal to 14 pounds (I weigh about 10½ stone). A barleycorn is an old English unit of distance, equal to about one-third of an inch. And a fortnight is a unit of time: fourteen days. Now weight is a measure of force ( F = ma, from Newton), and we know from elementary mechanics that the definition of work is:
Work = Force · distance
So we have a measure of work per unit time; or, in other words, a measure of power. (Compare the definition of one horsepower as 33,000 foot-pounds per minute.)