Oi! Well, the winter break is over (which is my excuse for Winter mute-ness), and I've hit the ground running again. As the Cake is a Lie, Syllabus Day is a myth.
Today I heard the best formulation for what significant figures are:
The mathematical method for keeping track of the least accurate measurement.
From this least accurate measurement in a series of measurements, you can tell just how well you know the "true" value of a given measurement. I don't know if this just went over my head in early chemistry classes, but I'll remember it now because it solidified all the abstruse rules I've been utilizing to no purpose aside from it being something... necessary... because, like, yeah...
It's not as if the rules are terribly difficult, unto themselves, but I don't recall ever knowing why I was using them. Obviously they were significant (harhar), but to what end... eh. However, this helps me understand why the addition of significant figures only depends upon the number with the least decimal places. If you add what you measure to be 2.5 Liters to a measured .0532 Liters, the relatively large uncertainty in the first measurement will "wash out" the relatively finer accuracy in the second measurement, giving you "'bout 2.6 Liters".
This leads perfectly into Uncertainty in measurements, which is related but something we utilize in the more formal sense in our everyday life. If you're getting off the clock in 16.3 minutes, you'll likely think of it as "15 minutes". And, for the purposes at hand (having a feeling for when you're going to get off), that level of accuracy is perfectly acceptable. If, then, someone asks you to stay for 5, you'll probably realize they don't mean "5.00 minutes", but rather there is some variance (or whatever it might mean in your particular social context). This is us utilizing the uncertainty in their measurement (their feeling for how long it will take until an extra task is done) that can be modeled mathematically: This could take plus or minus such-and-such an amount of time. "5 minutes" doesn't mean " 5.00 minutes", but maybe 5.0 plus or minus .5 minutes (or whatever). While you know from the clock that you have 16.3 minutes, the most number of significant figures you can keep is determined by their measurement of 5.0 minutes plus or minus .5 minutes, so you add the two measurements together and obtain that you could be on the clock from 21 minutes to 22 minutes after rounding off with significant figures. Perhaps a better model of conversation and on the fly estimates would actually use 5 plus or minus 4 minutes. From this you could conclude that you could be on the clock between 20 to 30 minutes, because then you would only have 1 significant figure. I mean, sure we don't actually go through the step-by-step analysis, but we tend to have a "feeling" for these things in conversation despite not knowing how we can conclude that our "feeling" may or may not be correct. I think the above expressed well what I always intuited about significant figures but never concretely expressed.