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PaperclipPerfector  4d ago (text post)   13628 thread views

Culture War Roundup for the week of November 17, 2025

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I'm definitely wanting LaTeX support, because even though it's just a derivative, it would look soooo much prettier.

I've found it interesting to think about it marginally. What's the time savings of each marginal additional mph? A derivative gives a very good approximation of this, and particularly for highway speeds, it's a good enough approximation to be reasonable over at least a handful of individual mph changes without recalculating.

Governing equation, which everyone should know from high school: d = rt, distance = rate*time.

We care about time. t = d/r.

dt/dr = -d/r^2

(Yes, this would be a nice place for LaTeX, because the d's for the derivative should be visually different from the d for distance.)

One has to be careful with units, because we're often talking about changes in time in terms of minutes, but we have rates in terms of hours. What we ultimately want is something mixed, like minutes/mph.

We're starting with [miles]/[mph]^2, so we need to correct by [60min]/[1hr], meaning if we just plug miles and mph into our formula, we need to multiply the result by 60 to get the units we want.

The obvious mental math version to just get your bearings on the magnitudes of things is to consider r=60mph, because then one of our denominator terms will cancel with our unit correction. It's also not too far off from the nominal speed of many state highways. This case gives us that a quick approximation, with inputs in terms of miles and mph and output in terms of minutes/mph is -d/r. That is, again, kind of assuming r=60, then if your trip is, say, 150 miles, you're saving (note that it's negative, because this is a reduction in time spent) about 2.5 minutes per mph you increase.

Some things to note. Time savings is linear in distance. I personally don't think it matters much until we're getting to pretty significant distance trips (I think trying to speed a bunch to save time on your 20mile commute is kinda dumb). It's also a 1/r^2 in rate. That is nasty in terms of diminishing returns. It's also why I'm kinda fine with the nominal 60mph mental math to make the unit change "free"; yes, I'm slightly overestimating the value of speeding if the nominal speed limit is 70, but it's probably not huge error. I haven't bothered actually quantifying the error; this is all just to quickly get into a ballpark.

I think the economist would probably want to slap a utility on this derivative and set it equal to something representing your estimation of the likelihood of getting pulled over. I think it's easy enough to handwave that a little bit and just think a little about the trip you're planning and this marginal rate of improvement and come to some approximation that you're comfortable with.

Time savings is linear in distance. I personally don't think it matters much until we're getting to pretty significant distance trips

Time savings per hour spent speeding is constant. The shorter distance also has a shorter speeding time.

LaTeX also doesn't interpret repeated * symbols as a request to hide them both and italicize the text between them. ;-)

Oof, yes. Fixed.

That one's bitten me too many times before, but these days using Unicode for x·y·z when I want an explicit symbol works well and is almost second nature, when I'm thinking in "math mode". If I'm thinking in "programming mode" then I default to x*y*z but the whole expression is inside backticks and markdown doesn't ruin it.

What markdown still wrecks for me is the use of "~40" to indicate that the number is a loose approximation, which then sometimes turns into a big strikethrough if I use it a second time. Unicode gives us ≈, but that doesn't feel right to me when there's nothing to the left of it.

I like your preferred method of analysis, BTW. It reminds me of the old argument that the US should be expressing vehicle efficiency in gallons per mile (or per 1000 miles, whatever) rather than in miles per gallon, because the latter leads us to overestimate how large the additional savings are when we further improve already-high-efficiency vehicles and to underestimate how important improvements are for the remaining gas guzzlers.

Oh man, approximately still gets me every time. I just write it, then when I got to paste it in, I see that it breaks, then I consider whether I want to spend a few minutes re-figuring-out what I want to do. I've given up in some comments not too long ago; pretty sure I just wrote "\approx", not caring. I'll be saving @ToaKraka's comment, and hopefully I remember I have it the next time it happens.

And indeed, at the time I came up with this way of thinking about it (I'd say about a decade ago), I was thinking about the debate on mpg/gpXm. I almost mentioned that here, because, yeah, a lot of people think in terms of having a fixed number of miles they're traveling, but don't intuitively grok 1/x relations. This speeding analysis is at least linear in distance, but it still has a 1/x relation, in a way. Yes, it's actually 1/x^2, but I guess what I'm kinda thinking is that one can lean on the 60min/1hr conversion, you have (d/r)*(60/r), and almost do it in two stages in your mind. If your nominal speed is 70, then you still have a d/r relation, but it's slightly modified by a 60/70, and I at least have a bit of an intuition as to how far off that's going to make my approximation, rather than having to just brute calculate the whole thing, actually thinking about the squared term.

What Markdown still wrecks for me is the use of "[tilde]40" to indicate that the number is a loose approximation, which then sometimes turns into a big strikethrough if I use it a second time. Unicode gives us ≈, but that doesn't feel right to me when there's nothing to the left of it.

Fun fact: You can use the code charts (or software like BabelMap) to find official alternatives for problematic characters.

Alternatives to 007E ~ tilde (HTML named character reference Tilde) (not including combining characters):

  • 02DC ˜ small tilde (HTML named character reference tilde)

  • 2053 ⁓ swung dash

  • 223C ∼ tilde operator (HTML named character reference sim)

  • 2E1B ⸛ tilde with ring above

  • 2E2F ⸯ vertical tilde

  • 301C 〜 wave dash

  • FF5E ～ fullwidth tilde

Obviously, HTML named character references are a lot easier to remember than hexadecimal representations, so & sim; (without the space) probably is the best alternative for use on this website.

How do you use the hexadecimal ones?

& #x223c; (without the space) = & sim; (without the space) = ∼

HTML documentation

Oh, wow, that's a good idea. ～ is actually a better "pay attention, this is not a blurry minus sign, this is approximation" character, at least on my screen.

Obviously, HTML named character references are a lot easier to remember than hexadecimal representations

I'm a big X Compose Key fan, especially with custom sequences. Need the real numbers? Win, m, b, R (mnemonic "math bold R") gives me ℝ. Defining the square function x↦x² with "bar arrow" notation? My "Bar arrow" is Win, |, >, and "squared" is Win, ^, 2. I'm already using Win, , for ≈ ... but Win, b, ~ should work for a "bold" tilde, if I could ever remember to use it.

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