Why can you multiply different units but not add them?

It’s usually an error to add quantities that have different units. If you have an equation that adds meters and kilograms, for example, something is almost certainly wrong. A formula that multiples meters and kilograms might be correct, but not one that adds them.

Why is this? It’s because the laws of nature don’t depend on our choice of units. For example, F = ma whether you use English, metric, or any other system of units.

Changing units introduces a multiplicative factor. You convert inches to centimeters, for example, by multiplying by 2.54.

If you were to add a length to a mass, and say you cut your unit of length in half but left your unit of mass the same, this wouldn’t be a multiplicative change (unless the mass were zero). But when you multiply or divide quantities with different units, the result does scale multiplicatively.

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23 thoughts on “Why can you multiply different units but not add them?

  1. Good observation, not as straight forward as it seems. There’s an even weirder case. You can subtract two dates, but you can’t add them together.

  2. Two thoughts:

    First: Since you discuss computer languages often… think of units/dimension as a “type system” for Physics. And a type system are no more than restrictions on all the possible operations of more fundamental objects (bytes) that help the abstraction.

    Second: To take your concern to an extreme… sometimes is not even ok to add two quantities with the same units. Because one can be as restrictive as one wants in “inventing new units”. For example you say that two distances can be added. Ok, what if I invent two new units/dimensions: vertical and horizontal distance (vertical- and horizontal- meter)? Is it still ok to add them? (remember they are unrelated dimensions in this definition) Depends on the context!, for example if we are talking about architecture it doesn’t make sense to add them because no usual architectonic operation turns things by 90 degrees from vertical to horizontal. So as you see in this context “it makes sense” to invent more units to *avoid* certain operations. Units are no more than a priori constraints on the operations that can be done with quantities. And therefore units are very arbitrary in this sense, it depends in great degree in the context and the paradigm. Sometimes one needs to overcome a self imposed constrain, for that one can invent a new constant that allows to transform one unit into the other.

    Another example to illustrate this and to finish, back in the 1800, energies and masses where measured in different units. This was in a simplistic view to “accidentally” add an energy with a mass in a derivation. However later Einstein discovered that energy and mass are fundamentally equivalent E = mc². Suddenly it made sense to add energy and mass, because they are the same things. But we are most of the day in a non-relativistic environment. If our brains where used to relativistic speeds we wouldn’t find a reason to not to call c = 1 (with no units) and energy and mass would be the same thing. Moral of the story, units is a physiological/psychological/cultural construct. It tells “math” (mathematical equation describing physics) what operation cannot be “usually” done with the current understanding of the problem.

  3. @joost, good example, in physics it is useful (again, depending on the context) to give different units to the difference, specially when the original quantity only makes sense as a positive quantity. For example, temperature. T1 = 100K, T2 = 10K, DeltaT = T2 – T1 = 90 diffK. This has the advantage that entropy (Joule/Kelvin) has different units than specific heat (Joule/diffKelvin). Again, (like in my other post) how useful this is depends on the context, if one want to prevent your self from adding entreopies and specific heat this is useful, otherwise it is just a burden.

    For example, in the computer language context, common C language style distinguishes between the types `size_t` and `ptrdiff_t`.

  4. Locations are like dates too – you can subtract, but not add. Actually, in both cases I think affine combinations work as well, e.g. “Halfway between now and a day from now” is 1/2(now)+1/2(day from now) is 12 hours from now.

  5. Having common units is necessary but not sufficient to meaningfully add two things. Force and torque have the same units, but they’re different kinds of things.

    Someone said of dimensional analysis that it’s trivial, but it’s not trivial why it’s trivial. :)

  6. @joost You can’t really add OR subtract dates at all, in the way you can add and subtract masses. The difference between two dates isn’t a date – it’s a duration, which has different units than a date. Locations are similar, of course. You need to remember the distinction between position and distance. This can be confusing, because you typically describe position as the distance from some reference point, but positions and distances are fundamentally different.

    @alfC energy and mass do NOT have the same units, which should be obvious from Einstein’s equation E=mc^2 (c has units of velocity)

  7. I suspect that the need to match units actually arises from the conservation of energy: if you’re “doing something to twice as much stuff” (pushing twice as much mass, applying a force over twice the distance, etc.), you typically need twice the force / twice the energy, at least in some base domain (i.e. once you integrate this quantity over the domain to come up with a more familiar formula like 1/2 m v^2, it might become a quadratic relationship). Effectively, twice as much “stuff” requires the energy to be spread twice as thin. Note that if you reduced all the equations in physics to energy equations, you *could* add them all.

  8. @Tom, “energy and mass do NOT have the same units”, that again depends on the context. In particle physics masses and energies are measured in electronvolts (which in other contexts are just units of energies). This is a necessary simplifications, since in this context energy transforms into/from mass very often.

    @John, you probably mean “work and torque” (not “force and torque”). But again, that depends. If you introduce “radian” (for angles) as an unit they don’t have the same units. Do you see the pattern? the more you want to “restrict” operations the more units you have to introduce.

  9. @alfC Yes, I meant to say “work and torque.”

    I agree that adding radian as a unit works, and at first that seems sensible, but then it seems a bit post hoc. I could imagine deciding that we shouldn’t add work and torque, then adding radians as a dimension after the fact. But it seems less likely that someone would discover that one shouldn’t add work and torque on the basis of dimensional analysis.

  10. @alfC Electron volts are a unit of energy, not mass. I can understand why a physicist might describe a particle’s mass in terms of its equivalent energy, but that doesn’t make electron volts a unit of mass – sorry.

    This reminds me of a story they used to tell back in one of the labs I worked in as an undergraduate. Apparently there was a grad student in plasma physics who was so wrapped up in his field that when he was quizzed in his thesis defense about some basic laws of physics, he insisted that F=mv, which is generally true in plasmas (if I understand correctly), but not in general, where F=ma. The point of the story was to impress on us not to let the common assumptions and approximations appropriate to our area of study make us forget the fundamentals. Energy is not mass.

  11. @Tom When I was a student at University of Texas, I heard that it was routine practice to keep physics grad students humble by stumping them on some elementary physics problems at the end of their oral exams.

  12. Interesting post.
    How about a 2d location in polar coordinates ? It could be interpreted as a distance travelled, plus a rotation about the origin. Though in once sense you can consider them added together, in the normal sense they maintain their separation in that the expression cannot be simplified to have a single numerical element. OR CAN IT ? ;-)

  13. @Tom, you say you “can understand why a physicist might describe a particle’s mass in terms of its equivalent energy, but that doesn’t make electron volts a unit of mass.” It precisely does, because in that context there is no difference. For example the mass of the muon is 105.7 MeV, yes, you can say it is 105.7 MeV/c but what for? so you can convert things to your every-day intuition (which is perfectly valid). In that context, MeV/c becomes another symbol equivalent to MeV.

    And don’t confuse the “fundamentals” with the “every-day”. That is what physics is mostly about, to discover that the fundamentals are beyond the everyday. And that sometimes it mean to suvert the unit-system because in more general context one thing can naturally convert into another.

    For example, in a relativistic context, you can “convert” energy into mass by simply changing the reference system. That is without doing any specific action on the system you can “convert” energy into mass. Of course you are not converting anything because in this context the two things a the same. (another example is time and distance in relativity).

    What I am saying that an abstract discussion of units doesn’t make sense, it all depends in the context.

  14. @alfC
    To have any meaningful discussion of units the context must be first “cancelled out”, to get purity of unit. For instance one cannot when discussing energy insist on measuring energy by meters (for a given quantity mass, in a given gravitational field, for a variable elevation in meters), then claim that the unit “meters” is in some way equal to the unit “joules”. It is not. This is what you are doing with the electron volt. It is only a “unit” of mass when you assume that the other units which must be multiplied to genuinely get a “unit” of mass are implicitly there (as a fixed constant). But it does not change the fact that a unit of energy is not a unit of mass.

  15. I don’t know if it quite applies, but perhaps you *can* actually add different units. If I have 2 apples and 3 oranges, and I add them together in a basket then I have a basket with “2 apples and 3 oranges” (or “5 fruit”, because apples and oranges are related, but this is not my point). If I add to the basket (the accumulator) another 2 apples, the total will be “4 apples and 3 oranges”. This is a vector addition where apples and oranges are considered orthogonal in some sense. Similar to the way you can add imaginary and real numbers together. Likewise you could add a length to a mass – the resulting quantity would have both length and mass. 5m + 10s + 2m = (7m + 10s)

  16. In that example, you’re not actually adding apples and oranges. You’re adding fruits. There’s a “forgetful functor” mapping both oranges and applies to a common type. But there’s not a natural way to convert kilograms to meters, for example.

    When you add real and imaginary numbers, in a sense you’re multiplying. You’re forming a cartesian product or two-dimensional vector space in which both components retain their identity. They’re not really added in some sense.

    Ditto with the length and mass example. Going back to the fruit example, you’re not really adding mass and time but functions of mass and time. You have a function that maps 5 meters into the pair (5 meters, 0 seconds), and a function that maps 10 seconds into (0 meters, 10 seconds). And it’s these pairs that are being added, not the basic measurements per se.

  17. @John, nevertheless @Michael’s example is a good one. Whether you can add apples and oranges depends on the context, if you see them as the part of the same class (fruit) it does. Yes, you may think that it is just arbitrary to group the two very different classes, but it is not. Apples and oranges are related to a common ancestor, that was a fruit, it made sense to add oranges and apples when there was no or small distinction between two classes that later would give raise to either apples or oranges (I know, I am simplifying the complex topic of evolution of trees/flowers and fruits). The point I am trying to make is that units are not set in stone, it depends in the context.

    Let me give a theory of what (in a given context) gives rise to a unit system. In a given paradigm (for example Galilean/Newtonian mechanics) there are multiplicative laws, like F = ma, additive laws (like x’ = x + v*t, change of reference system or M = m + m’ addition of intertia) and maybe others (involving non algebraic functions). My view is that a corresponding (and useful) unit systems encode what are these basic multiplicative and additive laws.

    An no discussion can go without mentioning other examples of units that only make sense because of the context, for example lumens (why is this different from irradiated energy?), or the Boltzmann constant (why Kelvin has units different from kinetic energy per atom). Again, the answer is that in the every-day only circumstantial mechamisms convert thermal energy into mechanical energy or energy into luminosity. While in other context they are the same or almost the same thing (astrophysics and statistical mechanics respectively.)

  18. The number of fundamental quantities and of the corresponding units is largely arbitrary, we can even conceive systems of quantities where there is only one fundamental quantity (e.g. time), or where each quantity is fundamental (a real nightmare!). Indeed, in quantity systems of the former type, dimensional analysis would be virtually useless, whereas in those of the latter type all the equations would contain fancy constants. The current number of fundamental quantities, 7, is thus a trade-off between simplicity and usefulness.

  19. “For instance one cannot when discussing energy insist on measuring energy by meters (for a given quantity mass, in a given gravitational field, for a variable elevation in meters), then claim that the unit “meters” is in some way equal to the unit “joules”. It is not.”

    Tell that to my doctor, who measures my blood pressure in mmHg.

  20. My question is on a much more basic scale:

    How do I explain to middle schoolers the reason as to why i can multiply different units, yet I can’t add or subtract them. I’m an assistant principal at a middle school, and I overheard one of my teachers tell a student today, “because you can’t; don’t ask why.” What a horrible response! In a sense, she was telling the students to stop thinking. I know the basic answer, which is related to the apples and oranges in a basket comment, but how can I do deeper with the students and the teacher, while remaining on a level is understandable by 11-14 year olds? Reading the comments section here reminds me of reading my Scientific American magazines… it’s amazing, yet dizzying!

  21. There’s a line in Fiddler on the Roof that comes to mind: A bird may love a fish, but where would they build a home?

    If you add 5 pounds and 3 yards, what do you get? Something in pounds? Yards? Some other unit?

    If you change 3 yards to 9 feet, do you get a different result? Maybe ask the students this. Press them for answers, then show the problems with each answer. For example, what happens when you change yards to feet?

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