Formally, why is it meaningful to multiply, but not add, different units?

Question

I know a bit of dimensional analysis from physics.

Simple example, if you have

$10J\cdot m^{-1} \times x = 20J\cdot kg\cdot m^{-2}$

then you can deduce that $x=2kg\cdot m^{-1}$, the units being determined in the obvious, natural way.

You cannot add Newtons units to metre units, but you can multiply them to get a new unit of Newton-metres.

This is intuitively meaningful to me and requires no further explanation. But in the formal, axiom-and-theorem sense, what allows me to multiply arbitrary units, and what forbids me from adding different units?

Answer 1

This is definitely an interesting thought. Nothing forbid you in any sense. Mathematics is an abstract art where you can do pretty much anything your want. The only reason you don't see units being added is because the interpretation of the result is not meaningful in any immediate way. But there is no axiomatic rule saying you can multiply units but not add them.

A similar question students often ask if why when we multiply matrices, we do that weird way instead of just multiplying them component wise? The same answer holds; we don't do it because the result isn't meaningful as far as elementary mathematics goes. However in some cases it is (see Hadamard product). In that sense, if you can find a way to derive some meaning from adding units, then perhaps it would be useful to add different units.

There is actually an inherent difference between mathematics and how we choose to interpret that mathematics. In the field of pure maths, the concept of units is in some sense non existent; saying $5$ newtons is an interpretation of the abstract number $5$ into a meaningful real world thing, namely, the magnitude of a force.

If you wanted you are free to say, I am now going to define the addition of units. Consider a $5N$ force and an area of $10 \ m^2$. I will define the unit sum (denoted $\oplus$) as:

$$ 5\ N \oplus 10 \ m^2 = 15 \ N\oplus m^2 $$

where the units of the answer is $N\oplus m^2$. See what you can discover about such a system and whether you can derive any meaning from it. Do share your findings!

Answer 2

The important thing is to realize that standard units are fundamentally multiplications of real-number quantities. From Wikipedia:

Any other quantity of that kind can be expressed as a multiple of the unit of measurement... For instance, when referencing "10 metres" (or 10 m), what is actually meant is 10 times the definite predetermined length called "metre".

From this, we can apply basic field axioms of real numbers.

Since units are multiplications, they play nice with other multiplications, which are fully commutative and associative with other multiplications. E.g.: $a(bc) = abc = cab$ and any other order you want to express it.

On the other hand, the interaction between multiplications and additions is ruled by the distributive property. Writing it in reverse, we can apply the fact $ca + ba = (c + b)a$ to combine like terms. But to be true in general, this requires the common factor/unit (here, $a$). If you try to write a similar simplifying equality with say, four variables and no common factor/unit, then it's trivial to find a numerical counterexample.

Answer 3

If you have two piles of apples, each with 7 apples, it’s easy to think: 7 apples/pile × 2 piles = 14 apples. But if these piles grow vertically along the Z-axis, say into 5 layers, then the 14 apples effectively shift units, becoming 14 apples/layer. At that point, it becomes: 14 apples/layer × 5 layers = 70 apples. So, I think this isn’t really an issue of dimensional analysis but rather one of operations. The shift in operations elevates the problem, much like moving from addition to multiplication—it’s essentially an “operational upgrade,” transitioning from one dimension to two dimensions or even higher.

Even for quantities that seem inherently directional, like force, velocity, or acceleration, if we place them inside a Schrödinger’s box, from outside the box, their direction could point anywhere. In other words, from the perspective outside the box, all vector calculations could essentially be viewed as scalar calculations.

Answer 4

Addition is for accumulating measurements. Multiplication is for forming a new type of composite measurement.

Addition accumulates measurements

If I have a ruler, I can walk it along the boundary of my room, and if I'm careful to keep it level and make successive measurements end to end, I'll have a guarantee that the sum of the individual measurements is a physically meaningful quantity, namely the total perimeter of my room. You can do the same for time, or temperature, or pressure.

Adding mixed units such as meters and seconds is not invalid; it's just that addition is supposed to be for accumulating successive measurements, and we don't have a real world process (like walking the ruler) that corresponds to accumulating measurements of different types. Certainly if you add the numerical values of measurements with different units, you'll get nonsense.

If you like, you can say you are allowed to make sums out of mixed units, it's just that (a) they don't simplify (they have to stay as "5km + 3sec") (b) they don't systematically correspond to any real combined quantity.

Multiplication creates composite units

In contrast to addition, which accumulates measurements, multiplication creates new sorts of measurements.

For example, if you measure the rate of energy consumption of an appliance to be around 5kW, and you run it for three hours, the total energy consumption is 15 kW*hr. This is not a sum of several little measurements and it's not new information; it's a way of combining two related quantities you've measured to get at the value of a third quantity (like energy) without needing a special instrument.