Quantity – Different Kinds of Numbers: Scalars
We’re used to representing quantities by numbers. In this series we’ve been looking at different kinds of quantities, and how certain actions on the numbers are meaningful for some kinds of quantities and meaningless for other kinds of quantities. It isn’t meaningful to double your phone number, but perfectly meaningful to double the distance you walk every day. It is meaningful to subtract the year 1951 from the current year 2009, but not very meaningful to divide them – it isn’t even meaningful to add them: what would 1951 AD plus 2009 AD represent?
The kind of quantity most of us think of – when we link a quantity to a number – is one where each of the operations plus, minus, times, divide is meaningful. These quantities that are perfectly fungible, scalable, and smooth. The amount of water I use in my house in a give month is like that. I just open the faucet, a little bit or a lot, for a short time or a long time, and a corresponding amount of water comes out. Add up all the uses of water over a whole month, and you get the amount for that month. I can express the total amount in gallons or in cubic meters, in cup fulls or in truck loads of the water truck, and depending on the unit I will get different numbers. But regardless of the unit in which the quantity is expressed, I am not actually limited in the way I use water. When I open the faucet, it doesn’t come out in spurts of one cup full each, it comes out in a smooth stream that I can smoothly adjust. I can fill any container to any level.
Quantities like this I will call scalar quantities. They scale up and down. They can be joined and split at will. Mathematically, these quantities are convenient. This is one reason we often pretend that quantities are scalar even if they aren’t really. We often think of money as a scalar quantity even if we can’t actually pay amounts smaller than a penny. We just round to the nearest penny or drop the last penny and don’t worry about it. We use calculators all day long that are designed for scalar quantities, and use them in all kinds of circumstances where the quantities we deal with aren’t scalar, and we cope quite well. There hasn’t been a huge market demand for calculators that only return amounts in pennies, or calculators that only show whole numbers. We kind of automatically embed the problem at hand into one that is expressed in scalar quantities, use calculators to do the arithmetic for us, and then bring the answer back into something meaningful in the situation. For example, we pump gas into the tank of our car, notice that the price of gas is expressed in tenths of pennies per gallon, and assume the pump will do the rounding to a number of whole pennies. And we assume we actually pay for the amount of gas pumped, not the rounded amount that is shown on the pump in tenths of a gallon. Mostly, all of this happens below the level of our conscious thought, we simply think of both the amount of gas and the amount we pay for it as scalar quantities, and that works well enough.
We saw earlier that even strange-sounding scalar quantities can actually make sense: it makes sense to talk about houses in the USA having 1.7 bathrooms on average (I made up the number), even though you won’t be able to find any house with 1.7 bathrooms. “Number of bathrooms per house” just seems like the kind of quantity that it makes sense to compute averages for, and that number is probably meaningful and perhaps even useful to some group of realtors or public policy makers.
A similar phenomenon obtains if a teacher computers an average score by combining scores of several tests: even if each individual test is scored with a whole number, the average may come out as something other than a whole number. You see this also in Olympic figure skating: each juror holds up a sign with his/her score, which is a whole number; the skater ends up with a single score, which is obtained from the set of individual scores by dropping highest and lowest, and taking the average of the remaining scores. This average determines how well the skater ranks relative to the other skaters. As you may expect, that average score is only occasionally a whole number.
The useful practical criterium for scalar quantities is therefore often not whether the quantity can vary smoothly or varies with a bump (going up from 1 kid to 2 kids without any number in between) but whether actions like taking averages make sense and are meaningful.
The issue of whether taking averages makes sense is not as simple as it may look. Let me leave you with a question to ponder: Does it make sense to talk about the average size of earthquakes, as measured on the Richer scale? If you say “yes”, why do you say “yes”? If you say “no”, why do you say “no”?