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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.

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    [Of possible interest](http://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/). –  Oct 22 '16 at 15:16
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    What you are referring to as "unit", is in fact a **number**. You do not "divide meters by seconds", you divide **x** (meters) by **y** (seconds), and you get a speed of $\frac{x}{y}$ (meters per second). – barak manos Oct 22 '16 at 15:20
  • Also of interest: http://web.mit.edu/2.25/www/pdf/DA_unified.pdf – John11 Oct 22 '16 at 15:21
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    @barakmanos that doesn't answer why 4 seconds times meters per second equals 4 meters – Stella Biderman Oct 22 '16 at 15:24
  • @barakmanos: We have the two units "metres" and "seconds" and get the new unit "metres per second". It is just a question of terminology if one wants to call this new unit a "quotient" or the "division" of the two initial units. I find this terminology natural. –  Oct 22 '16 at 15:24
  • @user170039: Thanks! It seems like this is the "professional answer" to my question. Now the challenge is to give a satisfactory answer (of course, one would have to simplify things a bit) that a highschool student could understand. –  Oct 22 '16 at 15:26
  • Also of possible interest: the Buckingham $\pi$ theorem: https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem – awkward Oct 22 '16 at 17:44
  • I might define 1m/s as the velocity needed to cover 1m in 1s. This is similar to defining $x/y$ as the number you multiply by $y$ to get $x$. – Ben Oct 22 '16 at 17:47
  • @barakmanos No, a physical quantity equals a number multiplied by a unit. $1 \text{ m}$ and $1000 \text{ mm}$ are equal, even though the corresponding numbers for each are not equal. –  Oct 23 '16 at 10:13
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    In principle, dimensions are not numbers but something resembling a Free Group - unfortunately, not so easy to translate into high school language. In any case, Buckingham's theorem was already mentioned. – Captain Emacs Oct 23 '16 at 12:34
  • Putting aside physical quantities this happens because you are working with ratios hence fractions. If I have 100 candy and I give 2 candy per child then you have enough for 50 children by doing: $100candy\div\frac{2candy}{child}=50child$ – Ian Miller Oct 23 '16 at 14:34
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    As a physicist, [this recent question](http://physics.stackexchange.com/questions/286964/should-zero-be-followed-by-units) on Physics.SE may be of interest. Every top answer there disagrees with every top answer here. – knzhou Oct 24 '16 at 05:56
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    @StellaBiderman "4 seconds times meters per second" doesn't equal anything. Four seconds times *one* meter per second equals four meters. – OrangeDog Oct 25 '16 at 09:22
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    If it helps, think of **units** as *dimensional scaling factors*. This is how I've thought of them since I was in middle school and it's always worked for me. You cannot mix units of dissimilar dimensions, but you can rescale to different units if they are of the same dimension. – RBarryYoung Oct 25 '16 at 12:00
  • @OrangeDog that was meant to be read as 4 (seconds meters per second) with the parenthetical as a unit. – Stella Biderman Oct 26 '16 at 17:10
  • @lastresort: $1000$ mm is $1000$ millimeters. A "milli" means $1/1000$, so the corresponding numbers **are** in fact equal (in other words, "milli" is a number, not a measurement unit). – barak manos Oct 29 '16 at 06:43
  • Note that sime units are dimensionless, and they may bite your (dimensional unit) calculations if you forget about that. Example: radians (see: [a related question](http://math.stackexchange.com/questions/316419/arbitrarily-discarding-cancelling-radians-units-when-plugging-angular-speed-into)) – Rolazaro Azeveires Nov 20 '16 at 11:03

14 Answers14

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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.

hmakholm left over Monica
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    This is very interesting, I actually never thought of it this way before. Do you know any references that explain it like this? – littleO Oct 23 '16 at 03:41
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    @littleO: No. To the extent it's not implicit in the notation itself, I think I came up with it by myself, long ago. – hmakholm left over Monica Oct 23 '16 at 03:48
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    There is such a set of units. But they are not called "God's units" but "Planck units". But the fact that it simply doesn't make sense to add a length to a mass doesn't change if you decide to assign pure numbers to them. – celtschk Oct 23 '16 at 07:49
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    Pulling gods into a discussion of physics (with high school students!) might not be the most profitable move from a scientific point of view. – ilkkachu Oct 23 '16 at 09:46
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    @ilkkachu: I guess he used this word "god" just for illustrative purposes, it's not something that has to be taken verbatim. –  Oct 23 '16 at 12:37
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    @celtschk: And the fact that it doesn't make sense to add a length to a mass is reflected in the fact that even _equality_ of the values created that way depends on which instrument conventions we use. Namely, if length A plus mass B equals length C plus mass D in Zeus-units, the same is not necessarily the case in Odinn units -- so (a) there'd be no way to convert between units, and (b) a theory or method of computation that depends on comparing A+B to C+D has a _hidden dependency_ on the measurement system being used. – hmakholm left over Monica Oct 23 '16 at 12:43
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    @celtschk: Which is also why I'm _not_ using Planck units for the example. In order to make the analogy work I need to have a setting where there are _multiple_ arbitrary "fundamental" systems of measurement. – hmakholm left over Monica Oct 23 '16 at 12:46
  • @Nullachtfünfzehn, I'd be very careful with counting on an audience of a _school class_ adding the necessary grain of salt, esp. re. the context of god. Even a high school. (Unless you want them to soon learn arguments like "Because God made it so.") As for differing units for e.g. length, it should be enough to invoke an image of neighboring kingdoms having different lengths of a "foot", simply because their kings aren't physically similar. – ilkkachu Oct 23 '16 at 12:52
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    @HenningMakholm: Thanks (+1). I really like your explanation, and it also matches with my observation that units "behave like numbers". Your answer makes clear that one could interpret units really as numbers, namely as *unspecified* scalars, which we can multiply with our magnitude (measured in our system) to get the quantity in the "ultimate unit" that works for all quantities. –  Oct 23 '16 at 14:29
  • @HenningMakholm: Although I got your point, I wonder why you use the plural "a set of really natural units". Shouldn't it be just one "fundamental God's unit"? –  Oct 23 '16 at 16:17
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    @Nullachtfünfzehn: To a certain extent that's mostly a matter of which words to choose -- I prefer units in plural because the god needs to choose how _long_ his fundamental unit of length is, how _much time_ his fundamental unit of time is, and so forth. There's a choice to make for each basic dimension, even though he may _write_ all of the measurements as raw mathematical numbers. – hmakholm left over Monica Oct 23 '16 at 16:22
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    @ilkkachu Honestly you're being silly. Nobody is going to be confused by this. People refer to 'the gods' in speech all the time, nobody actually thinks that that means that they are polytheists. – Miles Rout Oct 23 '16 at 21:38
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    You still have to have units to represent what dimension you're talking about. 4 masses + 4 lengths doesn't work, because **they're not the same type of thing**. The fact that there are multiple units we can use to describe each type of thing complicates the issue, but it's not the cause of the issue. – MichaelS Oct 24 '16 at 02:24
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    This isn't how physicists think of units. For example, you can experimentally measure the displacement of some object over some period of time to be zero meters within experimental error. Under your logic, this is the same as measuring it to be zero days or zero Coulombs! – knzhou Oct 24 '16 at 05:52
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    I think this answer is conflating units (which do behave a lot like numbers) with dimensions. You can measure some quantity with given dimensions in many different unit systems, even a very natural unit system. But that doesn't change what the dimensions are. – knzhou Oct 24 '16 at 05:54
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    This answer seems very harmful to me. Using units has nothing to do with scale and everything with "categories" or "types" of numbers. A length is a length, a temperature is a temperature. No matter which scale you apply to each other, it never ever makes sense to add or subtract their numbers. Arguing that there could be a "god" a.k.a. "Planck" scale which somehow makes the most important constants fold over to "1" so you can leave them off, then applying that to school, is very misleading indeed. That is just a notational trick. So unfortunate that this answer was high-voted and accepted! – AnoE Oct 24 '16 at 09:09
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    @MilesRout, I just take it seriously, as I know there are people who take their religion seriously. I find them incompatible, since one relies on observation and independent thought, the other on blind belief in what some other entity said. And this still doesn't answer the fact that from an everyday pov, lengths and masses can't be added together. Sure, we could create systems where they are in some sense equivalent, but even Planck units are pretty far from practical. Perhaps worth a mention on an advanced high school course, but not on the level where we're discussing $s = v \cdot t$ – ilkkachu Oct 24 '16 at 09:15
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    @MichaelS It's worse than that. Sometimes even things with equivalent units can't be combined. E.g., torque and energy can't be simply added, even though Joules are equal to Newton-meters. – jpmc26 Oct 25 '16 at 04:31
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    I totally agree with @MichaelS: keeping track of dimensions is useful beyond just getting numeric answers. [Dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis#Examples) can often tell you something about what form that answer must be, just by looking at the dimensions of the thing you're trying to calculate and the variables you know. (e.g. whether there's a $v^2$ or just a $v$ in the formula for something). There are cases where it's not obvious, or useful as a sanity check or quick estimate of how something scales without working out the constant factors. – Peter Cordes Oct 25 '16 at 10:08
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    @celtschk Consider e.g. https://en.wikipedia.org/wiki/Natural_units , which lists at least 6 different systems of units in a similar spirit as Planck units. So, this actually reinforces the point in the answer that “different gods favor different sizes”. – Emil Jeřábek Oct 25 '16 at 13:12
  • Actually, the god you believe depends on what you are measuring as much as the instrument you use. For example, if you are measuring electric fields you have one god, and if you are measuring magnetic fields, you have a different god. Or maybe it is one god with two faces like Janus. And in some branches of physics, there are even more gods. Some might even have three components, but this must never, ever, be discussed with students if you life in Saudi Arabia or Iran. – richard1941 Oct 25 '16 at 19:20
  • @AnoE In general, you have a point. But for example adding (squared) time and (squared) length makes perfect sense in a 4-dimensional space-time. Using $c=1$ is not just a notational trick, having to use $c$ at all is just an unfortunate remainder from times when we didn't know length and time are pretty much the same thing. – JiK Oct 26 '16 at 09:16
  • For the time vs length thing: Compare this with a strange civilization who use don't realize that you can rotate things and don't think of horizontal and vertical distances as the same thing. They might use miles for horizontal distances and feet for vertical distances. Then they learn to rotate things, start doing some advanced physics, and find that a constant $d=5280 \text{ft}/\text{mi}$ seems to appear in a lot of formulas. Would you argue that actually $d$ does equal $1$, or would you say that using $d=1$ is a useful notational trick but nothing more. – JiK Oct 26 '16 at 09:21
  • @jpmc26 The torque vs. energy thing is actually related to the question [whether angles are dimensionless](http://physics.stackexchange.com/questions/252288/are-units-of-angle-really-dimensionless). Multiplying a torque by an angle gives the work done by the torque. Saying that angles have dimension solves the problem of not confusing torques (Nm/rad) and energies (J=Nm), but creates difficulties with some formulas. – JiK Oct 26 '16 at 09:25
  • @JiK "when we didn't know length and time are pretty much the same thing." I argue they are not the same thing. Time, at least local time, has also a relation with the speed of processes and thermodynamics - related to the rate of entropy increase if you like. Expressing time in terms of length may make sense as a reverse of light-years - i.e. a mathematical trick - but the best example of non-sense of equalling time with space is "Millenium Falcon made the Kessel Run in less than twelve parsecs". – Adrian Colomitchi Oct 26 '16 at 12:53
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    @AnoE: See the comment of the user "knzhou" under the question. He noticed that a similar question has been asked over at physics stackexchange, with the quite remarkable result that the top answers there completely disagreed with this top answer by Makholm here at math.SE. I guess the reason for that is that physicists like to think in terms of units and argue that a physical quantity should always consist of a magnitude *and* a unit (and the unit shouldn't be forgot!), whereas from a mathematical point of view the interpretation of Makholm is more fruitful, –  Oct 26 '16 at 15:46
  • which reduces physical quantities to mathematical objects (i.e. real numbers). I really love the answer by Makholm, that's why I accepted it. The many likes suggest that *many* other people think so too. Isn't it my right to accept the answer then? –  Oct 26 '16 at 15:49
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    Also, nobody claims that the phycisists approach is worse than the "unit-less" approach. Of course, both have there pros and cons, and it's always good to know many different ways of looking at a problem (each viewpoint increases the understanding and emphasizes different aspects!). But I think that it would be hard to exactly pin down the formalism that encodes the phycisist-approach (that is, the "unit"-calculus), especially if highschool students should understand it. Makholm's answer can be understood by every highschool student, thus it perfectly answers my original question. –  Oct 26 '16 at 15:58
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    @Nullachtfünfzehn, I did not try to take your right away, but was excercising my own right to post my opinion... – AnoE Oct 26 '16 at 18:34
  • @HenningMakholm: With "What makes this work is the implicit assumption that our unit systems are at least coherent", do you mean that if one says that velocity equals distance divided by the time needed, and one measures distance is some unit a and time in some unit b, then velocity should be measured in a/b? –  Feb 01 '17 at 21:24
  • @user7280899: Yes. – hmakholm left over Monica Feb 01 '17 at 21:28
  • @HenningMakholm: Ahh thanks. I'm sorry that we didn't came along well with each other well on the other thread. –  Feb 01 '17 at 21:30
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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.

Christian Blatter
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    Speaking for the OP. The question makes it clear what the analogy to numbers is supposed to mean: there is an algebra (or arithmetic, if you prefer) of units. Your answer says nothing about the algebraic manipulation of units (e.g. multiplying meters per second by seconds), which is very common in physical reasoning. The question isn't what units are but why they can be manupulated in this manner (hence the words "behave" and "like"). – symplectomorphic Oct 22 '16 at 16:39
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    See also here for very slightly different formalization than Tao's. http://math.stackexchange.com/questions/571050/what-mathematical-structure-models-arithmetic-with-physical-units . In your last sentence do you mean there is something unsatisfactory with Tao's formalization? – zyx Oct 22 '16 at 19:52
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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.

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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.

velut luna
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    This sounds like a completely roundabout way, only useful if your goal is to confuse your students. – AnoE Oct 25 '16 at 21:36
  • OK this is probably easier to just show, than it seems to be to describe in prose. – uhoh Oct 26 '16 at 12:15
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Dimensional analysis is often used in order to claim that a certain formula can't possibly be correct. For example, suppose I proposed that the period of a pendulum is equal to its length times its mass. Most physics students would simply tell you that this can't possibly be right, because time can't equal distance times mass. But what is the proof that it can't be right?

Concretely, I'm making the following empirical prediction:

  1. Measure the mass of the pendulum in grams.
  2. Measure the length of the pendulum in centimeters.
  3. These are both numbers. Multiply them together.
  4. I claim that this will be equal to the period of the pendulum in seconds.

Notice that I'm never asking you to do anything undefined (somebody once tried to tell me that adding different units was like adding different sizes of matrix, but this is not a valid analogy). How can one prove, only on philosophical grounds, that my empirical prediction can't possibly be correct?

The answer is that, even if my prediction were correct, it would become false if I used different units. Suppose I had a pendulum with a mass of 10g, 50cm long, and, exactly as predicted by my formula, a period of 500 seconds (over 5 minutes - already my formula is looking less plausible, but just pretend). What if I'd measured length in inches rather than cm? That's about 20 inches, so I would have gotten $10\cdot20\neq 500$. Thus my formula can't possibly be correct for all choices of units.

On the other hand, the actual (approximate) formula for the period of a pendulum:

$$T=2\pi\sqrt\frac L g$$

Where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity, works in all choices of units, because if I switch from centimeters to inches, I multiply both $L$ and $g$ by the conversion factor, and so it cancels out.

Dimensional analysis is justified by the following philosophical principle:

Nature doesn't care about the metric system. More generally, Nature has no preferred choice of units. If an equation is a physical law, then it must be true no matter what units we use.

You can of course question this axiom. You certainly can't prove it mathematically. But it's still quite reasonable. The rules for dimensional analysis, therefore, are the answers to the (mathematical) question: under what conditions is an equation invariant under changes of units?

Jack M
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Here's an attemps to describe a mathematical formalism that accounts for the physical units. I will try to distinguish the physical requirements from the mathematical features. The most fundamental physical input is the following :

we need a finite number of fundamental units to build a system of units covering all physical quantities.

Usually these are chosen as mass [M], length [L], time [T], temperature [$\Theta$], electric charge [Q], quantity [N], luminscence [I] but other choices are possible (for instance one can distinguish horizontal and vertical lengths).

Different units are used to quantify things that cannot be added. You can add a distance and a length and you will obtain another length so distance and length have the same physical nature. Thermodynamics states that you can add a physical work and a heat, so these two concepts are of the same physical nature, now called energy.

As is does not make sense to add different units, but only to multiply them, the complete unit system is a multiplicative structure, with identity element [1] (the dimension of dimensionless quantities) and the seven fundamental units as independent generators. We can denote this structure as $(\mathfrak U,\cdot)$, with $$\mathfrak U=\left\{[M]^m[L]^{\ell}[T]^t[\Theta]^\theta[Q]^q[N]^n[I]^j,(m,\ell,t,\theta,q,n,j)\in G^7\right\}$$ where $(G,+)$ is a group such that $G\supset\mathbb Z$. In elementary physics, it is usually enough to take $G=\mathbb Z$. Brownian motion requires to use $G=\frac12\mathbb Z$ and critical phenomena, in which critical exponents appear, or dynamical systems use real valued exponents, so it is best to chose $G=\mathbb R$ from the start.

Furthermore, one needs a real number (sometimes a complex number can be used but that's probably far beyond the scope of the question) to represent the amplitude of the physical quantity. As a final construction we have the structure $$(\mathbb R\times\mathfrak U,\cdot)$$ equipped with the canonical product $(x,u)\cdot (y,v) = (xy,\,uv)$.

The physical units are of great importance in physics, since this is used to determine dimensionless quantities thanks to the $\pi$ theorem.

Tom-Tom
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    Thank you for providing a mathematically correct and very clear/clean answer which shows plainly why you can work with unit-ful numbers in some ways and not others, and why units are not simply used as "naive" numbers; albeit far removed from the highschool level of the OP. Your answer belongs right at the top, together with the more school-appropriate answer from @Wildcard. – AnoE Oct 25 '16 at 21:53
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This question is closely related to the influential essay

Wigner, Eugene P. The unreasonable effectiveness of mathematics in the natural sciences [Comm. Pure Appl. Math. 13 (1960), 1–14; Zbl 102, 7]. Mathematical analysis of physical systems, 1–14, Van Nostrand Reinhold, New York, 1985.

Historian Grattan-Guinness published a response to this in

Grattan-Guinness, Ivor. Solving Wigner's mystery: the reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. Math. Intelligencer 30 (2008), no. 3, 7–17.

Grattan-Guinness's point is that the question is backwards. The following is perhaps not exactly his point but is related. When the Greeks or other ancient philosophers first delved into science and mathematics they typically dealt with magnitudes or quantities. These were always tied to the physical entities being studied, whether objects being counted or frequencies arising in music theory.

Eventually magnitudes and quantities were abstracted into what we know today as natural numbers, which were ultimately extended to broader number systems such as the hyperreals. The point is that the effectiveness of numbers in describing reality stems from the fact that the numbers themselves have magnitudes and quantities as source. The puzzle, according to Grattan-Guinness, only arises if we forget their source.

Thus what the OP refers to as the "units" were the source of the numbers, rather than vice versa. A philosopher would therefore describe such a problem as a self-inflicted pseudoproblem.

Mikhail Katz
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    There's a big difference. Those magnitudes abstracted from physical entities have a ring structure that dimensional quantities do not (or is not useful to add it). – zyx Oct 23 '16 at 23:31
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Because the unit (the number 1) is a number

A good book overviewing the philosophical aspects of unit analysis (e.g., how it relates to Aristotle's notion of unity and his understanding of mathematics as being derived from the physical world) is:

Summary:

This work provides the means for re-establishing the unity of science by interpreting the whole of modern experimental science from the perspective of analogous transfer of the metaphysical principle of unity rather than in terms of efficient causality.

Another:

Dealing with the metaphysical foundations of modern physical science, this book demonstrates that not only is classical metaphysics not in conflict with the principles of modern experimental science but that, when analogously transferred to the different divisions of modern science, the metaphysical principle of unity makes intelligible all the laws of modern science. This revolutionary book provides the means for reestablishing the unity of science by interpreting the whole of modern experimental science from the perspective of an analogous transfer of the metaphysical principle of unity rather than in terms of efficient causality.

Geremia
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    This sense of ‘unit’ has little relation to the sense meant by the OP. – Anton Sherwood Oct 22 '16 at 21:32
  • @AntonSherwood Dimensional units are what we consider to be 1. Just as all numbers are based on 1, so all measurements are based on a unit. – Geremia Oct 22 '16 at 22:31
  • These sorts of essays tend to have religious motivation or content. I guess it can be judged on its merits, but you are pointing to something that is primarily of interest to Aquinas or Aristotle scholars and not a well defined formalism in modern language for understanding units. – zyx Oct 22 '16 at 23:01
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    Metaphysics and religion both involve arguments that go beyond what we can establish using purely scientific and mathematical methods, but that does not make them equivalent. – Michael Kay Oct 24 '16 at 09:52
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Simply put, units are not separate from numbers but rather extensions of numbers - specifications for how they are to be used or interpreted.

Side note: in my opinion, numbers without units really aren't numbers at all.

When units cancel, it is because the 'usage specifications' are the same for two numbers being used and can thus be safely ignored, or 'factored out,' if you will, from the equation. This leaves the remaining units, or specifications, to be considered in the equation.

So, to answer your question, canceling units is really the same as canceling or reducing numbers. Different units are related in their own way, just as different numbers are. They are not independent of one another.

Here's an analogy that should be easy for high school students to understand: Picture the equation as a finished model car. All of its parts come together in a certain way to create the working model. If done correctly, the model is useful for something - for example, being used as a way for car manufacturers to make sure the actual car they are building looks like and works like the model. But in order to correctly put together this model, two things are needed: labeled parts and instructions on how to use those parts. In an equation, the parts themselves are the numbers, the labels on the parts are the units, and the instructions are the formula. The labels assist in the correct use of the parts and the instructions. The labels are used to help build the manufacturer's understanding of the model in the same way the parts are used to help build the actual car.

Conclusion: Units and numbers are not separate, but parts of a cohesive whole, which is incomplete without either the units or the numbers. When you cancel numbers in an equation, you can cancel the units too because the units are the numbers. When no units are listed (which is bad practice everywhere outside of a boring algebra book) the unit assumed is just null, and this unit cancels with the numbers as well but is just not written, as it was not written in the original problem.

Sorry for the long-winded answer, but I hope this helps clear it up!

Plato
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    "numbers without units really aren't numbers at all." what – Miles Rout Oct 23 '16 at 21:39
  • @MilesRout: I assume [tinlyx' answer](http://math.stackexchange.com/a/1982253/265963) is along the same lines as that statement. I.e., a number without units really doesn't make any physical sense. Even a ratio (mathematically considered unitless) only makes sense if we know what it's a ratio of. – MichaelS Oct 24 '16 at 02:49
  • That's not what @Plato said. – Miles Rout Oct 24 '16 at 04:24
  • To clarify my point, I did say that numbers without units do not make sense in the realm of reality; however, I did say additionally that there are times when units aren't necessary, such as in textbook practice problems or illustrations of abstract concepts where units limit the concept rather than define it - such as the statement 'if a = b and b = c, then a = c' (in this case, adding units would limit the concept's scope when it should be universal) - but overall, I did mean that units are extensions of numbers and help clarify the 'purpose' and usage of those numbers. – Plato Oct 25 '16 at 00:33
  • @MilesRout, sure. With [my definition](http://math.stackexchange.com/a/1983842/276406) of a unit as "anything you can count," a number for counting something doesn't make sense without saying what you are counting. Whether it's chickens or square feet or foot-pounds per second, you're counting *something.* And the number isn't really a number without the unit. (Put another way, numbers are numbers *of something* if they are to be useful.) – Wildcard Oct 25 '16 at 19:00
  • @Wildcard No, you aren't. You're on the mathematics stack exchange for christ's sake, you should know very well that you don't have to be counting things for numbers to be numbers. – Miles Rout Oct 26 '16 at 06:31
  • @MilesRout, good thing I'm not obstructed by a necessity to be rigorously consistent in my beliefs. :) Truth is a relative quality, not an absolute. It is modified by viewpoint. The statement, "Numbers without units aren't really numbers at all," is a relative truth. The statement, "Numbers are entirely abstract and don't have any units," is another. Depending on what you want to *do*, one of these is more true, or more useful, than the other. – Wildcard Oct 26 '16 at 07:10
  • @Wildcard. No. Just no. That's just not the case. The first one is *objectively incorrect*. Numbers without units are entirely meaningful. – Miles Rout Oct 26 '16 at 08:21
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The units, as a formal system and their algebraic manipulations are, how shall I put it, "tautological" in that we freely (and formally) create units, overcoming the "apples and oranges" problem. And so we say $3m/s^2 \times 2kg/m^2=6kg/ms^2$ whatever these units are. In other words $3 apples \times 2 oranges=6 apple oranges$, so we do the usual arithmetic we do with numbers and accept whatever cloned units come up. So this is not something to fret about, at least for "scalar" units, but vector units would likely work, with vector operations.. Now what MIGHT be more interesting is to find the intrinsic MEANING to the units we play with or come up with.

Of course, before we ever had these discussions, some tricksters did those manipulation secretly and got the random clones named after them, or got Nobel prizes for them. For instance, here is how Einstein came up with "his" energy formula: He was sitting in his office one day trying to solve a quadratic equation $ax^2+bx+c=0$. Being rather poor at math he of course had his wife Mileva Maric do all things mathematical for him. She used the quadratic formula as a matter of automation, but it was way beyond Albert. All he could do was doodle with those coefficients while writing his and Mileva's initials (her small, his capitalized, as he was such a Narcisus). Mileva noticed that of the three doodles he did, $E=ma^2, E=mb^2, E=mc^2$ one of them could, with right dimensional analysis represent something "meaningful." She laconically mentioned this to Albert and a couple of days later, Albert was showing this to all his friends, as his own invention, never once mentioning Mileva. The rest is history -- he divorced Mileva to marry his first cousin Elsa, and got a Nobel prize, but was later FORCED by a court order to share that prize with Mileva. Such is life. This hopefully describes the issues with units.

Why did I use a qualifier MIGHT above? I believe actually that mathematical meaning as I hinted at it is the best intricacy we will get. I am of course aware that it did not stop many big heads even before Aristoteles to the modern age, who have been and who are still seeking this intrinsic meaning.

p.s. The humorous part of the explanation is as good an explanation as the rest of my argument (the actual facts of that part are not too far removed from my depiction). The argument I give here is what I believe is the essence of the issue.

Rado
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In my opinion, arithmetic with units are more natural/fundamental than with unitless numbers.

In your example, it's right to have units cancel out, e.g. when multiplying velocity and time, in the first place.

I think the unitless numbers can be viewed as a special case of arithmetic with units where the units of the terms are the same and there is no need of mention. And the better question probably is

When is it meaningful to/Why can one use numbers as if the quantities involved had no units?

For example, if $x$ is length in geometry, is it really meaningful to talk about $x^2 - x$? Literally, $x^2 - x$ has an inconsistent unit as the first term is area and the second is length, unless you assume, e.g., that the second term is $x * 1 \space length \space unit $. Or maybe, $x^2 - x$ is about counting the number of trees in a stand. In that case, the unitless expression may or may not make sense.

The numbers without units may be based on a certain assumption that we can ignore units, which may or may not hold, like many other assumptions/axioms. Computation with units are the more primitive/original form, IMHO.

tinlyx
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  • Multiplying two real numbers yields a real number of the same kind; by contrast, multiplying two lengths gives an area, which is a quantity of a different kind. – supercat Oct 25 '16 at 14:40
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To get what they are mathematically, you may have to teach the students some abstract algebra.

That will require more than I can cover in a single stackexchange post. I'll briefly describe what a group and field are (but a real lession will be longer), then talk about how we stitch together the group of units with a scalar field.

From a mathematical point of view, a g is a generator of a free group, as are m.

A group is a collection (or set) of stuff $S$, an identity element $e$, a binary (two-argument) operator $*$ such that a few rules (or axioms) hold. First, $a*b$ is in $S$ if $a$ and $b$ is in $S$. Second, for all elements $a$ there is another element denoted $a^{-1}$ such that $a*a^{-1}=e$.

In this case, the group commutes, so that $a*b = b*a$. In addition, it is a free group, which means no $a$ and $a*a$ and $a*a*a$ etc. are distinct elements -- the only way to get back to $e$ is to multiply in inverses.

We take each of these unit groups and we take their cartesian product and get a new free group with many generators. The identity of this "product" group is the element $e,e,e,e,e,e,e$ (the identity of each group we multiplied together).

We then attach this to a field of scalars, usually the reals. A field is a group over the operator $+$ with idenity $0$ attached to another group over the operator $*$ with identity $1$, such that the second group does not include $0$ but otherwise shares the same element. We then extend the second group with the rule $a*0=0$ and $0*a=0$, and the requirement that $a*(b+c) = a*b + a*c$ (distribitive law). (This is very brief).

In this merging of the unit group and the scalar field, we define $+$ only when the unit groups are equal. We define $*$ to multiply both the scalar field and the unit groups together.

Yakk
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  • In between groups and fields, there are also rings, such as the integers. I also think it's helpful to recognize the concept of a "universal Zero", such that multiplying *anything* by Zero yields Zero, and adding anything to Zero yields itself. – supercat Oct 25 '16 at 14:42
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Units can be thought as contextualized numbers.

Numbers and units do sometimes behave similar. However, they don’t do so most of the times. Contrary to what has been said before, they do not necessarily "follow the same algebraic rules as any other algebraic unknown does". They just do so sometimes. For start sum and subtraction do not apply to units of different quantities(*); Ex: 5 bees (or meters) - 5 Universes (or seconds)=?.

Numbers are the cardinal or count of something in a collection of similar things. A complete description of that collection should include its count plus what it is, for example: 5 cars; 5 trees; 5 atoms of hydrogen; 5 degrees Fahrenheit; 5 inches, etc.

In algebra, one removes the what it is part, so it shouldn’t be reasonable to expect that a partial description contains all possible behaviors of the total description. Still, the total description may conserve some of its partial description behaviors, so in these cases, they may behave similar.

Care should be taken when comparing units and numbers.

J. Manuel
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If units had no relations and could not be multiplied, units would be a transcendental extension of a field, usually $\Bbb{R}$ (which is also a vector space). Alas, there are relations and we can multiply units, so the result is an (associative, unital, graded) algebra over the ring $\Bbb{R}$.

Eric Towers
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