Well….it makes sense that “3 liters” is a quantity, but what is it really? Suppose you had a bowl of applesauce. That’s a quantity too. There may be 3 liters of applesauce in the bowl. But when we refer to a quantity are we referring to the “3 liters” or the actual applesauce in the bowl?
Answer: (You guessed it) Depends upon whether we are thinking colloquially or mathematically.
In common speech, I can certainly have a quantity of applesauce, maybe to share at a Thanksgiving dinner perhaps. But quantity in mathematical modeling has a precise, non-collquial, meaning. It is the precision of the definition that allows us to generalize quantity to far more interesting objects. (And unfortunately, Textbook School Mathematics blurs the uses of the word together, hoping the colloquial definition is “good enough,” which it isn’t.)
Some of these interesting objects appear in school mathematics. While this post is not about rates per se, let’s point out an interesting dilemma about rates that shows that quantity isn’t a definition that can be easily glossed over. Clearly, a rate should be a quantity. If you are walking 3 mph and you increase your speed by 3 mph to a run, you are now running at 6 mph. But what is the unit, “miles per hour?” It’s easy enough to compute with, for example, if you walked 6 miles in 2 hours at a constant speed, then you’ve averaged (6 miles)/(2 hours) = 3 mph during that trip. But unlike the physical distance of 1 mile, there is no physical way to divide the length given by 1 mile by the time it takes for 1 hour to pass, so there is no physical unit “miles/hour.” If mph is not a physical unit, then what is it and why do we have no problems calculating with it?
The answer is that a quantity, when using it in a mathematics, is an element of a set that models a type of measurement. Model is a keyword for it implies that the things we are measuring do not have to physically exist. Also, the model can handle very large or very small quantities that are not easy to observe physically, like an object that is 12,022,403 miles long.
Now we are ready to get precise. We base our definition on the work of Hassler Whitney, who described the idea of quantity in two articles in the American Mathematical Monthly some 50 years ago (Article 1 and Article 2). We encourage you to read these two articles to see how he puts the definition of quantity on a solid mathematical foundation. Here is a brief, non-technical summary:
- A quantity structure is a set that models a type of measurement and satisfies certain properties (see Hassler Whitney for descriptions of the properties). The quantity structure that is the set of all lengths is an example. Other examples: the set of all volumes, the set of all masses (weight), the set of all times, etc. Note the term “model” in the first sentence—we are not talking about physical measurements, just a model of such measurements. For mathematicians: The simplest quantity structure is mathematically the set of real numbers thought of as an ordered, 1-dimensional real vector space with R multiplication. (See Hassler Whitney for details.)
- A quantity is an element in a quantity structure. For example, in the set of all lengths, the quantity represented by “3 meters” is an element of that set. Note that the quantity is independent of how it is measured, so that same quantity can also be represented by 300 cm.
- A unit in a quantity structure is just a choice of a (usually positive) quantity that all other quantities are compared to. Examples include cm, m, in, ft, liters, sec but these are just standard choices, any choice will do. For mathematicians: a choice of a unit is really a choice of a basis element in the vector space.
- With a choice of a unit, all other quantities can be represented by a number times that unit. Therefore, “3 meters” is a representation of a length quantity. The description, “3 meters,” is short for “3 x (1 meter).” The numerical part of this expression is 3 and the unit part of the expression is 1 meter.
Note: the “type of measurement” in the definition of quantity structure above is not as important as the fact that the set satisfies certain properties. It is the properties that define the quantity structure. Therefore, since the set of all velocities satisfies the properties outlined by Hassler Whitney, we can rightfully work with velocity as a quantity structure and think of 5 mph as a quantity.
We end with this question: Can numbers be thought of as quantities?
CHANNEL: Engineering School Mathematics