Why do units (from physics) behave like numbers?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

Because the unit (the number 1) is a number

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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:

• Crowley, Charles B. 1996. Aristotelian-Thomistic philosophy of measure and the international system of units (SI): correlation of international system of units with the philosophy of Aristotle and St. Thomas. Lanham: Univ. Press of America. (review)

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.

Answer 5

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!

Answer 6

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.

Answer 7

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.

Answer 8

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?

Answer 9

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.

Answer 10

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.

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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.

Answer 11

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.

Answer 12

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 $\Bbb{R}$.

Answer 13

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!"

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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."

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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?"

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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.

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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

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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.

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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.

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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.

Answer 14

Here's an attempt 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.