Why do units (from physics) behave like numbers?

What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view?

I've made the observation that units "behave like numbers". For example, we can divide them (as in $m/s$, which is a unit of speed), and also square them (the unit of acceleration is $\frac{m}{s^2}$). In addition to that, we can cancel units: $$s = v\cdot t$$ If for example $v=4\frac{m}{s}$ and $t=5s$, then $$\require{cancel}s=4\frac{m}{\cancel s}\cdot 5\cancel s=20m.$$

Note that $\frac{m}{s}$ can also be written as $ms^{-1}$. This is another example where units "behave" like numbers.

So why can we cancel units, why do units behave like numbers?

I want to get an answer that can be understood by highschool students.

Suppose that there is a set of really natural units: a truly fundamental amount of length that we could count all lengths in, a fundamental amount of time, a fundamental amount of electric charge and so forth -- "God's units", if you will. Then every quantity in physics would just be unitless, and there would be no need for keeping track of them.

Unfortunately, different gods favor different sizes of the fundamental units, so if we buy a set of instruments that show results in Zeus units, the numbers we get wouldn't agree with another set of instruments that use Odinn units. But we want to write down our formulas and measurements such that we don't need to redo everything just because we switch instruments.

Now, algebra to the rescue! We know how to make letters stand for yet-undetermined numbers, so let us decide to use, for example

• the letter $m$ to stand for how many god-units-of-length there are in the length that our old non-divine system called one meter
• the letter $s$ to stand for how many god-units-of-time there are in the time that our old non-divine system called a second
• the letter $C$ for how many god-units-of-charge, etc etc etc.

Now, when we say that, for example, a certain distance is $1.435m$ what we mean is "I don't know what your instrument will show when you measure this length, but I do know that it will be $1.435$ times the $m$ that works for your set of instruments".

In this way, the letters $m$, $s$, $C$ and so forth can be thought of as standing for actual numbers that we might multiply the numeric parts of the measurements by. As such, they follow the same algebraic rules as any other algebraic unknown does -- in particular they can cancel.

What makes this work is the implicit assumption that our unit systems are at least coherent -- so the if the Zeus instruments measure speeds in Zeus-lengths per Zeus-time, so the Odinn instruments had better measure speeds in Odinn-lengths per Odinn-time rather than in some completely unrelated unit that has nothing to do with the size of an Odinn-length.

Electrical tensions are not numbers but vectors in a one-dimensional vector space of tensions. Choosing the unit "volt" means choosing a basis in that vector space. In this way each tension is then a scalar multiple of the unit "volt", and may be "identified" with that scalar, i.e., considered as a number.

Same thing for (directed) lengths along a line with chosen origin, as studied in the analysis of linear motion. Such lengths are vectors, and only the choice of a basis vector "meter" turns them into real numbers.

These numbers (= coordinates with respect to the chosen basis) transform according to the rules learnt in linear (or tensor) algebra.

It is unfortunate that the prevailing teaching of elementary physics has not come up with a definite and "canonified" handling of this dimensional aspect of physical description.

At any rate I cannot endorse the view that a physical "unit" behaves like a number.

Without just imagining being a teacher, but as a former high school math teacher speaking from experience, I have successfully explained units to high school students as follows:

(Note that this is a very interactive format; in general you can help someone learn better if you coax them to ask and answer questions—in other words, to look—than if you just talk at them.)

"What's something that you can measure?"

"I don't know. Um, a table."

"Okay. What about it would you measure?"

"What do you mean?"

"Well, let's say you measure this table. What measurement do you get?"

"Maybe five feet."

"Okay. So what about the table are you measuring?"

"How many feet it is?"

"Okay, great. So what quality of the table are you measuring?" (I usually avoid the word "attribute" unless dealing with someone fairly bright.)

"Its length!"

"Good! Now, length is a distance, right? If you measure its height or its width, you would still measure those using 'feet,' right?"

"Sure...."

"What else can you use to measure a distance?" (The time it usually takes for a student to answer this question is surprising. Keep at it, be patient. At first they tend to only think of metric units vs. imperial units. You will not get "light years" or "nanometers" as your first answers.)

(After much coaxing) "Feet, yards...oh, meters! Um, inches. Centimeters. Millimeters."

"Okay, how about you think BIG?"

"Oh, miles! And kilometers." (Meanwhile I'm writing down all of these on a sheet of scratch paper under the heading "DISTANCE.")

"Okay, that's fine. All of these things are UNITS. Each of these is a specific AMOUNT of distance. Right?"

"Yeah, okay."

"All right. So these are units of distance. Can you think of any other type of unit? Something else you can measure, besides distance?"

"Um...not really. Length? Wait, no, that's distance. Height...um...oh! You can measure weight!"

"Good!" (Write another heading, "WEIGHT," next to "DISTANCE.") "What are some units you would use to measure weight?"

"Pounds, ounces, kilograms. Um, grams also." (DON'T get into an argument about weight vs. mass. More confusion than you need at this stage.)

"Okay, good." (Writing them all down under "WEIGHT.") "Now, you can convert between different units of the same type of thing, right? Like for instance, how many feet in one yard?"

"Three."

"Good, and how many inches in a foot?"

"Twelve."

"Good. How many grams in a kilogram?"

"One hundred. Oops, I mean a thousand!"

"Right. So, how many inches in one pound?"

"Uh...what? That doesn't make sense!"

"Right! You can't do that. It just doesn't make sense. You can only convert between units of the same type of thing, whether it's distance, or weight, or...what other kinds of units are there? What other properties of things can you measure?"

"Temperature?"

"Sure! And the units?"

"I can only think of degrees."

"Yep. But there are two kinds of degrees, right? Celsius and Fahrenheit. Actually there's another kind, also, but we don't need to get into that right now." (If they question about how many Celsius degrees in a Fahrenheit degree, then I explain that the thing you are really measuring here is heat, and so the zeros don't line up because you aren't really counting something. And then move on.) "There's something else you can measure. But first, can you check how much time we have left?"

"Um, thirty minutes to lunch time. Hey! Time is something you can measure!"

"Good!" (Writing down the header "TIME" and the unit "minutes.") "And what other units can you use to measure time?"

"Hours, and also seconds. Oh! And days, weeks, months, years. Centuries."

"Good! And decades, millennia. Anything smaller than a second?"

"Yeah, milliseconds."

"Okay." (Writing it down.) "Now what are some other things you can measure? There's a lot of things. How about the surface of this table? How much surface it has?"

"Yeah, it's about five feet, like I said."

"Okay, but remember your geometry? It's five feet long, but how wide is it?

"About three feet."

"Good, so five feet by three feet is...?"

"Fifteen square feet. You can measure its square feet!"

"Okay, good! But square feet is just another unit. What kind of unit is it? What are you measuring? It's not really just distance; it's...?"

"Area!"

"Right! Square feet is a unit of AREA." (Writes it down under its header.) "How about another unit for area? Do you know how property is measured, like how big a field is?"

"Football fields?"

"Sure, that's a good unit for area. But if you're going to buy a house—maybe you didn't know—you can find out how much area the property has, and it's usually measured in acres."

"Oh, right, acres."

"Now what about a really small area? Like a sheet of paper? It's smaller than even one square foot."

"You can use square centimeters, right?"

"Yep! Now, centimeters are a unit of what?"

"Distance."

"Good. So DISTANCE times DISTANCE equals AREA." (Write it down under "AREA." Let them look it over.) "So you can use any unit for distance, times a unit for distance, and get a new unit for area."

"Square miles, square meters, square kilometers?"

"Sure. It doesn't even have to be the same unit twice. What if I have an area that's one foot wide and one yard long?"

"Three square feet."

"Right—or, one foot-yard." (Write down "1 foot x 1 yard = 1 foot-yard.") "Why not? It's distance times distance, right?"

"Yeah...hmmm. Okay."

"Or what about if you're making a spaghetti farm, so you want to buy a property with an area of one mile-inch? Yeah, that's a joke. It's out of Garfield. But it's a valid unit of area."

"So you could convert that to square feet?"

"Exactly!"

From there I would cover volume as the next logical step. (And don't forget to include gallons and liters amongst your volume units.)

Next after that I would cover speed.

Then I would discuss how you can get volume from distance times distance times distance, or you can get it from distance times area.

Then I would discuss changing speeds on the freeway, or when going onto the freeway or off the freeway, or when coming to a sudden stop. The student would bring up a time he was in a car that was coming to a squealing halt and everything fell on the floor. Then I would ask him how fast was the car going (roughly), then how long did it take to stop.

Then I would go into the fact that a change in speed from (say) 40 mph to 0 mph in 5 seconds can be shown using units of SPEED over (divided by) TIME. And write out "40 mph / 5 seconds."

Then I would bring up the idea of change of an amount as distinct from an amount itself. I would stand up and ask:

"How far away am I from you?"

"Three feet."

"Okay, now how far?"

"About ten feet."

"Good. How long did it take me to get here?"

"About a second."

"Okay, so that's ten feet per second—or is it?"

"Yeah. Wait, no...you didn't go ten feet."

"But I'm ten feet away from you now, right?"

"Yes. But...."

"Now it's been another second; how far away am I?"

"Fifteen feet."

"So that's fifteen feet per second, right?"

"No! You started from ten, so it's only five feet per second!"

"Good!" (Sit back down.) "The point is that change of distance is different from distance, even though you measure them with the same unit. Change happens across time. So the position now is 10 feet, then it changes to 15 feet in a one second time period, that's only five feet per one second because it's a change that I'm counting here. Got that?"

Next I would discuss how acceleration is a change in speed that happens across time. And look at the formula ACCELERATION = SPEED / TIME, and then point out that properly speaking, the formula is ACCELERATION = CHANGE IN SPEED / TIME, or ACCELERATION = (CURRENT SPEED - ORIGINAL SPEED) / TIME IT TOOK TO CHANGE SPEED. But that ACCELERATION = SPEED / TIME is an acceptable way to write it, and get the student's agreement that this is acceptable and makes sense.

Next comes the jump into "square time" which confuses so many students. I would point again to "distance times area" for the "volume" formula, and that area is distance times distance. Then I would show that since speed is distance (change in distance) over time, acceleration is:

(distance/time) / time

And then write it as (d/t)/t and make the student simplify it algebraically.

They would get d/t^2, and then I would write:

distance / (time x time)

Then I would emphasize that it's really change over time of the rate that distance itself is changing. Not just the rate of change of distance—but the rate of change of speed.

From there it's a short leap (albeit an important one) to get that: "If your position (distance) is changing at 5 miles per hour, and you wait 2 hours, how much will your distance have been changed?"

"Ten."

"Ten what, ten gallons? Ten chickens? What's the unit?"

"Ten...miles!"

Then write out the algebra for it.

(5 miles / hour) x 2 hours = 10 miles

Definition of a unit: Anything you can count.

Non mathematician teachers sometimes argue about this definition, but it's true. Since you can count poops, "poops" is absolutely a valid mathematical unit.

I dare you to do the above with a high school student and NOT wind up with them understanding units. It'll be hard work.

And after they've been through the above, always, always, always insist that your students include the correct units in their answers to their math problems.

If I imagine I am teaching high school students, I will explain it in the following way:

The unit is a short-hand to tell us how should the number before it change if one changes the definition of units of some fundamental quantities like L, T, M.

For example, why is the unit of area is m$^2$? Because if you change the unit of length by a factor, by the definition of area, the area will be changed by the "square" of that number.

If we consider units as a shorthand to determine how the number should change when one changes L,T,M, then it is clear that when you multiply two quantities, and denoting the unit of their product by the product of their units will do the job correctly. One can therefore apply the same algebra as numbers to units.

For example, why is the unit of velocity m/s? Because when one change the definition of units of L by a factor, and the units of T by another factor, the velocity changes by a factor which is the quotient of the two factors. Why one can do cancellation in m/s$\times$s=m? Because the product is clearly independent of the definition of unit of time, and in fact can be correctly represented by the shorthand m.