Hours, days, who’s counting?

Yesterday’s post contains an error so embarrassing that I was briefly tempted to yank the whole thing. But of course That Would Be Wrong. What’s more, the error supports the larger point I was trying to make before I derailed myself.

I was talking about Bret Victor’s notion of explorable explanations, which he illustrates on a page called Ten Brighter Ideas. I’d looked at it before, but when I revisited it yesterday I had trouble believing that the following claim could be true:

If every US household replaced 1 incandescent bulb with a compact fluorescent bulb, we’d save 11.6 TWh (terawatt hours), which is the energy equivalent of 1.5 nuclear reactors or 9.5 coal plants.

Some people intuit what these units and quantities mean. But most of us — me included — don’t. And even experts are prone to error. A few months ago I spotted one such error. A Ph.D. economist wrote an editorial that consistently used billions of barrels of oil rather than, as intended, millions. The column was syndicated to hundreds of newspapers, and so far as I know nobody noticed until I happened to check.

What prompted me to check? My friend Mike Caulfield, who’s been teaching and writing about quantitative literacy, says it’s because in this case I did have some touchstone facts parked in my head, including the number 10 million (roughly) for barrels of oil imported daily to the US.

The reason I’ve been working through a bunch of WolframAlpha exercises lately is that I know I don’t have those touchstones in other areas, and want to develop them. Having worked a few examples about global energy, I thought I’d built up some intuition in that realm. But in this case the intuition that prompted me to check Ten Brigher Ideas was wrong.

When I did check, things went completely off the rails:

If 111 million households each swap out one 75W bulb for a 25W bulb, saving 50W each for 180 hours (i.e. half of each day for a year), we’re looking at 100,000,000 * 50W * 180hr = 999GWh. We’re off by a factor of about 1000.

As Pasi points out in a comment:

Hmm, “half of each day for a year” is not 180 hours, but 365*24/2=4380 hours?

My brain thought days, my fingers wrote hours. I think I’m slightly dsylexic when it comes to units, and so I’m prone to that sort of error. It’s another reason why I use WolframAlpha to check myself. When I do that, I try to take advantage of WolframAlpha’s marvelous ability to automate conversions. For example, during an earlier exercise I needed to visualize the gallon equivalent of the energy released by combustion of one kilogram of gasoline. Normally this would entail looking up the density of gasoline, 0.726 g/cm3, applying that constant, and then converting to gallons. But in WolframAlpha the phrase density of gasoline is meaningful and can be used directly, like so:

http://www.wolframalpha.com/input/?i = ( 1 kilogram / density of gasoline ) in gallons

Similarly, here’s what I could have done to check the Ten Brighter Ideas claim:

http://www.wolframalpha.com/input/?i = (1/2 year) * 111,000,000 * 50W as TWh

That comes to 24 TWh, which is in the ballpark of the claimed 11.6. Maybe Bret assumed lights are cumulatively on 1/4 of the time, I haven’t checked, but if so that would nail it.

Why didn’t I write the WolframAlpha query that way in the first place? Because, I think, we still expect to do a lot of basic computation ourselves. You want the answer in hours? Put hours in. How many? You can figure that out. But should you?

I think it depends. It’s good to exercise your inboard computer — not only to calculate results but also to store and retrieve certain touchstone values. But it’s also good to delegate calculation, storage, and retrieval to outboard computers that can do these things better than we can — if that delegation can be smooth. WolframAlpha points to one way that can happen, Bret Victor’s simulations point to another.


  1. The old simple trick of dimensional analysis seems like a useful addition to this type of problem. If you are not familiar with it, the complicated version is in Wikipedia, but simply just enter the dimensions, hours, kilograms, etc and then solve the equation with those units converting where appropriate. It has helped my kids distinguish between miles per gallon and gallons per mile on family trips.

    1. Historically it was, I’d say, not an addition but a prerequisite. It’s what you have to do to reason through a sequence of conversions and transformations. And it’s a bit of a chore, so much so, in fact, that most of us would tend to lose the direct connection (as Bret Victor would say) to the question we were considering, if that were a question like: “Hey kids, we’re going to burn 10 gallons on this trip, how much energy is that? How long would the equivalent amount of energy power our house?”

      The reason I keep coming back to these examples is that we’re starting to have tools that can do that dimensional analysis for us so that we can reason more directly about, among other things, sources and uses and quantities of different forms of energy.

      1. “…we’re starting to have tools that can do that dimensional analysis for us so that we can reason more directly about, among other things, sources and uses and quantities of different forms of energy. …”
        Yep, the Internet as a whole contains incredible sources of information that speed up the process, keep in mind that I can reflect back 60 years ;-). Using that information with all you say being true requires some diligence in the insertion of the data assuming your formula is correct. There is probably a location on the Internet where you can even test the dimensional integrity of your solution (I have not investigated that, but I will – have not really used the WolframAlpha). Handy when you do not have a good intuition or experience (kind of like your ‘touochstone facts’ suggestion) with what the size of the answer should be.

  2. My first intro to Dimensional Analysis was as a ChemE undergrad. I really did enjoy the ability to do the unit conversions as part of the calculation of the final answer. It was also supposed to help figure out the units your final answer should be in Ft/lb vs. Joules, etc. Sometimes you had to multiply by 1 to get the units to cancel each other out. But it was just a fancy form of fractions IMHO.

  3. A minor, but important and related point in this post is it never hurts to run even a simple conversion through WA. Just this morning I did a calculation that came out to 0.45% and wrote out 4.5 out of 1,000. I then quickly did this:


    To make sure I hadn’t botched it. I hadn’t, this time, but here’s the thing. I will at some point botch it, and the numbers that you botch end up being the ones that propagate. It would be a good habit for all journalists to get in — conversion errors cause no end of problems once released into the wild (did you hear that 35% of California in-state college students are illegal aliens? I guess it started life as 0.35%, but who’s counting?).

    People run spelling through a spell check, despite the fact that no real harm usually comes from letter transpositions. But we don’t machine check our numbers near enough.

    1. Great point.

      It’d be nice to build a list of the kinds of common English idioms — like your ‘4.5 out of 1000’ — that WA will handle. There are lots you wouldn’t expect or might not think of, like your example, that actually work. But some that you might think of don’t work.

      Maybe you’d like to propose a tag for things belonging on that list?

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