Understanding and Avoiding Mistakes in Unit Calculations

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Mistakes people make when thinking about units, published by Isaac King on June 25, 2024 on LessWrong.
This is a linkpost for Parker Dimensional Analysis. Probably a little elementary for LessWrong, but I think it may still contain a few novel insights, particularly in the last section about Verison's error.
A couple years ago, there was an interesting clip on MSNBC.
A few weeks later, Matt Parker came out with a video analyzing why people tend to make mistakes like this. Now I'm normally a huge fan of Matt Parker. But in this case, I think he kinda dropped the ball.
He does have a very good insight. He realizes that people are treating the "million" as a unit, removing it from the numbers before performing the calculation, then putting it back on. This is indeed the proximate cause of the error. But Matt goes on to claim that the mistake is the treating of "million" as a unit; the implication being that, as a number suffix or a multiplier or however you want to think of it, it's not a unit, and therefore cannot be treated like one. This is false.
So what is a unit, really? When we think of the term, we probably think of things like "meters", "degrees Celcius", "watts", etc.; sciency stuff. But I think the main reason we think of those is due to unit conversion; when you have to convert from meters to feet, or derive a force from mass and acceleration, this makes us very aware of the units being used, and we associate the concept of "unit" with this sort of physics conversion.
In reality, a unit is just "what kind of thing you're counting". Matt uses two other examples in his video: "dollars" and "sheep". Both of these are perfectly valid units! If I say "50 meters", that's just applying the number "50" to the thing "meters", saying that you have 50 of that thing. "50 sheep" works exactly the same way.
So what about "millions"? Well, we can definitely count millions! 1 million, 2 million, etc. You could imagine making physical groupings of a million sheep at a time, perhaps using some very large rubber bands, and then counting up individual clusters. "Millions" is a unit![1]
So if millions is a perfectly valid unit, why do we get an incorrect result if we take it off and then put it back on again after the calculation? Well, because you can't do that with other units either! 100 watts divided by 20 watts does not equal 5 watts. It equals the number 5, with no unit.
This is a somewhat subtle distinction, and easy to miss in a casual conversation. But it makes sense when you think about the actual things you're counting. 50 sheep is certainly not the same thing as 50 horses. And 50 sheep is also not the same thing as the abstract number 50; one is a group of animals, the other a mathematical concept. If someone were to say something to you involving the number 50, you would not simply assume that they're talking about sheep.
This perfectly solves the problem. If 100 watts / 20 watts equals only the number 5, with no "watts", then 100 million / 20 million also equals only the number 5, with no "million".
But what about Matt's example? 80 million sheep - 50 million sheep = 30 million sheep; not just 30. That's because this is subtraction, not division. Units work differently depending on what operation you're performing! If you're doing addition or subtraction, the units are preserved; you can take them off at the beginning and then put them back on at the end. But for multiplication and division, this is not the case.
Division cancels out the units, removing them entirely, and multiplication gives you a new unit, equal to the previous unit squared.
This seems kind of arbitrary, right? Why do they work differently depending on the operation? To understand this, let's go back to a different example that Matt used in his video. Near the beginning, when he's performing the ...