From discussion with Michael:
I love this! Your multiplication of negative numbers explanation is one of the best I’ve ever heard.
One thought on your addition stuff:
I think it’s an awesome insight that combing some quantities means simply adding the numbers, but other times that doesn’t make sense.
But I don’t think the way to tell is by asking “are these quantities in the same dimension”. I think it’s something else.
Let’s consider weight, volume, and density:
Total weight of q1 and q2 is weight(q1)+weight(q2)
Total volume of q1 and q2 is volume(q1)+volume(w2)
Total density of q1 and q2 IS NOT density(q1)+density(q2)
But, density(q1) is in the same dimension as density(q2). It’s just that density is defined as weight / volume, so when you combine two densities, you have to compute the total weight and total volume and then get the new density from there.
I think keeping track of units is the critical piece here. Some units are inherently addable: weight, volume, years, people, kinetic energy, distance, time.
Others are rates, so you can’t just sum up the individual densities (or whatever):
density is weight/volume
age is years/person
temp is kinetic energy/molecule (or something like that)
speed is distance / time
Combining two trips that were both 5 miles and 1 hour long gives you double the distance and double the time, so the speed remains the same.
I think the important question is “is this measure an addable one or not?”. I think most people can figure that out, maybe with some help.
If not then you have to think about it in terms of the units that make it up.
Just my two cents (if two people give their two cents, do you end up with four cents?)
Great point, I think this would be a fun topic for another article. You’re right, some items are “add-able” and others are simply “average-able”, i.e. you directly combine volumes but need to average (perhaps weighted) different densities.
Another fun one is directions: North and East can’t directly be combined. We could write them as a vector using (1, 1), aka a single mile in each direction. We can add these vectors with special rules, like (1, 1) + (2,3) = (3,4). However, they can be combined when we are measuring “total distance”, and the shortcut is to multiply items by themselves and take the square root (the Pythagorean theorem). My intuition is that while North and East can’t directly be combined, the amount of interaction North has with itself can be compared to the amount of interaction East has with itself. We’ve shifted to a “self-interaction” scale, which can be compared, vs. the quantities, which can’t.
I also like this application for various types of averages (http://betterexplained.com/articles/how-to-analyze-data-using-the-average/) as well – the way items interact determines the type of average to use (or addition, or other types of arithmetic).