83333

The other day while driving to the office, the odometer of my car turned to 83,333. Pretty typical. My car is 8 years old and I commute nearly 20 miles each way to work, so I’m putting on about 10,000 miles per year. But I liked that number. It reminded me of something. What? Was it the cosine of some angle? I’m a physicist and I like to work with numbers, so I do run into cosines all the time. That didn’t seem to ring true. Was it some simple fraction? That probably was it. I decided to see if I could figure it out. Here’s my approach.

Consider the infinitely repeating decimal 0.8333333…. . Is it some simple fraction? Well I know that 0.333333…. is just 1/3 and this looks almost like it – at least the repeating part. Can I manipulate this to make it look like that? First there this “0.8” in front that doesn’t belong. So let me write my decimal at 0.8 + something else. Subtracting off the 0.8 I get 0.033333…. So my decimal = 0.8 + 0.03333…. The 0.8 is just 8/10 which equals 4/5 by cancelling the common factors of 2 at the top and bottom. The other part is almost the 1/3 but it has a 0 in front after the decimal. I know that you put extra zeros after the decimal by dividing by 10. So 0.03333…. = (0.3333….)/10 or (1/3)/10 = 1/30. So my fraction is 4/5 + 1/30. I can combine these together by putting them over a common denominator. If I multiply the first fraction by 1 = 6/6 (you can always multiply something by one without changing its value – but you can change the way it looks to help you) I get 4/5 = (4x6)/(5x6) = 24/30. I can now add this to the 1/30 to get 25/30. This is 25/30 = (5x5)/(5x6) so I can cancel the common factor of 5 to get 5/6. A nice simple fraction as I had expected (and should have remembered). Then I see that the complement of my decimal – it’s difference from one – is just 0.166666…. (You have to make the two of them add up to 0.99999….. which actually is just equal to 1 – but that’s a discussion for another entry.) If I had seen that I would have gotten it right away. I work a lot with fractions and I know that 1/6 is about 0.16 and is a repeating decimal.
I know, I know! DWDM (Driving while doing math) – or maybe it should be “DWA” (Driving while academic) – is probably as dangerous as talking on a cell phone while driving. Not a good idea! But the exercise illustrates some lovely principles about number and education.

First, one of the things I have been trying to do with my work recently is to teach non-physicists (biologists, actually) to think a bit “in the way that a physicist does.” They often tell me this when I ask what I should teach. But what does this mean? I think my example, although it’s just about number and not about physics itself, tells me something. As a physicist I see numbers as real things, not just abstract relations. They’re things with structures and connections and properties. I can work with them, turning them twisting them, multiplying by 1 in various forms (5/5 or 6/6 for example) or adding 0 in various forms to change the way they look and give insight into what they are. Being able to see them in different ways gives them a solidity and a reality that I suspect most of my students don’t feel. Seeing that I can do the problem in the two ways discussed – manipulating it into a fraction by taking it apart and looking at the complement and recognizing a familiar number – gives me the comfort that arithmetic and all the complicated stuff about fractions and decimals isn’t just something I have to remember. It’s something that makes sense. It’s reliable and consistent. I can look at things in a variety of ways and confirm my answer and catch my mistakes, though, luckily in this case, I didn’t make any. I often do, especially when doing math in my head, but having the multiple perspectives on number (and equations) usually helps me nail down the correct form pretty quickly.

I showed this analysis to one of my students who came in to my office for help on this week’s homework. She was flabbergasted. She said she had never seen anything like that. How sad! This is the sort of thing I would love to see taught to every 5th grader. We seem to be making a useful transition in our teaching of arithmetic – from doing tedious rote math (adding long columns of 4 and 5 digit numbers) to using calculators to eliminate the tedium. But learning to use a calculator shouldn’t just get rid of the tedium but also give students the sense that numbers are somehow magic and can’t really be thought about.

[If you can find it, I recommend Isaac Asimov’s 1939 short story, “The Weapon Too Dreadful to Use”, which describes a world in which number had been automated by computers so much that people didn’t realize you could figure things out. Asimov isn’t usually predictive, but this one hits the nail on the head.]

Calculators can in fact be used to help students develop this “sense of number” very effectively, though they often are used in exactly the opposite way. For some references to this, check out the literature review in my paper with my student Tom Bing, “Symbolic manipulators affect mathematical mindsets,” (Am. J. Phys. 76, 418-424 (2008)). Interesting books on the sense of number, how people develop it and build it into math include:

Stanislas Dehaene, The Number Sense – a very readable book about the neuroscience and psychology of basic math.

George Lakoff and Rafael Nunez, Where Mathematics Comes From: How the embodied mind brings mathematics into being – this one is quite a bit more technical and takes one deep into sophisticated math.