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In school we were taught that multiplication is "repeated addition." Of course, that idea breaks down when asked to add 4 to itself -3 times. I have the same intuition that multiplication is arbitrary when dealing with units. I'll do my best to explain with examples:

If we were to take 3 meters ($m$) and multiply that by 4 times, I would simply add 3 meters 4 times and get $12 m$ total. The result is one dimensional.

Now multiplying 3 meters by 4 meters is extending the 3 meters into a perpendicular dimension to a length of 4 meters to get $12 m^2$. Addition no longer makes sense here as we are not adding 3 meters to itself 4 meters times. The result is 2 dimensional.

What about when multiplying different units? Multiplying 3 Newtons by 4 meters gives 12 Newton-Meters = 12 Joules. Furthermore, the space we are working with is defined by our definition of the units. 12 N-m seems 2 dimensional (Newton dimension and meter dimension) but 12 Joules is 1-dimensional.

Is there any established way of interpreting unit multiplication or is it pretty much arbitrary?

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  • $\begingroup$ Or even add $4$ to itself $1/2$ times. $\endgroup$
    – Thomas Andrews
    Jan 13 at 19:59
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    $\begingroup$ Multiplication of units makes sense if they represent counts: $3$ dozen by $4$ dozen is equal to $12$ dozen${}^2.$ But multiplication once we get to even rational numbers is more than repeated sums. You just have to get over that - the repeated sums view of multiplication really only works for natural numbers. $\endgroup$
    – Thomas Andrews
    Jan 13 at 20:02
  • $\begingroup$ It gets even better than that. To convert Fahrenheit temperatures to Celsius you have to add $0°C-32°F$ and then multiply by $1°C/1.8°F$. (I'm not making this up!) $\endgroup$
    – Oscar Lanzi
    Jan 13 at 20:12
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    $\begingroup$ Just as an example, let's take the joule, the unit of energy, AKA a watt-second. If a power source of 1 watt delivers energy for 1 second, the amount of energy is 1 watt-second. In 2 seconds it is of course 2 times as much: 1 watt times 2 seconds is 2 watt-seconds. But what happens when you multiply kilograms with meters? You multiply mass with distance. You get some strange physical quantity that probably has no name because it's not a very useful quantity. $\endgroup$
    – Accelerator
    Jan 13 at 20:29
  • $\begingroup$ What it means to multiply things in the most abstract of senses is probably best described by the product in Category Theory, or the product in Type Theory. A classic example is the cartesian product, which is a tool we can use to construct mathematical formulations of dimensions. If this all sounds very vague, it's because it is. It's a purely geometric concept which happens to have a variety of uses, both theoretical and physical. $\endgroup$
    – Graviton
    Jan 15 at 23:39

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Multiplication of different units of measurement is never arbitrary but considered in a larger context involving adding/subtracting/powers, we talk about operating with rational expressions. In extension of fractions which we can not properly add or subtract unless they were expressed in the same denomination, the units of measurement need expressed in a coherent manner.

The apparent discrepancy between the 2-dimensional nature of Newton x meter and one-dimensional Joule comes from the wrong interpretation of Joule as one-dimensional. Imagine meter as unidimensional unit of length and meter x meter as bi-dimensional unit of area.

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