# What does it mean to multiply units?

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?

• Or even add $4$ to itself $1/2$ times. Jan 13 at 19:59
• 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. Jan 13 at 20:02
• 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!) Jan 13 at 20:12
• 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. Jan 13 at 20:29
• 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. Jan 15 at 23:39