Thoughts on Scientific Notation
Scientific Notation is one of those topics that I find most disappointing in math texts. The topic is often presented as four rules, one each for converting between decimals and scientific notation. This is the way a computer would be programed. These rules are all about the direction that the decimal point moves and how many decimal places. Here are my thoughts.
1) A good exploration of a topic can be accomplished with the words and images of pros. I own a copy of Powers of 10. These days, it is free, in its entirety, on the internet. What a great way to think about how useful it is to be able to talk about scale. The short extract of COSMOS is one of those moments that has stuck with me from childhood. Morgan Freeman's excellent narration and more personal approach to the same topic is also great. I like all three.
2) Like many topics in essential mathematics, English is a huge part of scientific notation. In mathematics, the words for big and small are all used to discuss the relationship of two numbers on the number line. In spoken usage, and in the physical world, big things take up a lot of space and small things take up a little space. Negative numbers are noticeably absent from this interpretation. My discussions of scientific notation begins with a review of the mathematical use of words for big and small... and then asks what we should call numbers near zero and numbers far from zero. A colleague proposed "weak" and "strong" numbers. I use the the phrases "near zero" and "far from zero."
3) You do not need rules if you know that a number with negative exponent is a number near zero. Similarly, a number with a positive exponent is a number far from zero. So, instead of rules for numbers near zero that sometimes move the decimal point to the right and other times to the left, this knowledge unites the two ways of representing numbers near zero.
4) Read a math text carefully and you notice that the rules for exponents often appear in the same order as you would derive and generalize these rules. So, is there a place for knowing why negative exponents were defined the way they are? For me, I do not address this in essential math courses unless I am asked. The answer provides a good insight into the way math works. What would your answer be? We are so familiar with the rules that it is tempting to say that the rules must be that way. But what does "must" mean in this context? To me, "must" means that math can be generalized. If we have exponent rules for positive exponents, then why not define a new type of exponent so that the existing rules work for them too? To me, this underlines one of math's astonishing strengths... it is perpetually generalizing its results to a larger domain. At each step, we take the assumptions from one domain to a larger one. Generalization (along with abstraction) characterizes mathematics.
5) "When am I ever going to need this?" Is scientific notation merely a science topic? My answer to this is that the topic reminds us that there is a lot going on at different scales. Galaxies and supernovas or blood cells and atoms are both pretty far removed from the scale of day to day life. Scientific notation allows you to fill the galaxy with marshmallows or count the lamps needed to create as much light as a supernova. By relating these things to something we have a sense of, perhaps we can explore the world outside our our experience? What do you think?
6) Incomplete operations... I was lucky enough to have had two classes from Bill Thurston. He was amazing! When he learned that some of his students were interested in teaching, he started adding reflections on his own experience of mathematics. Pencils would be lowered and for a moment a Field's metal recipient was talking like an algebra student. One day he mentioned when mathematics caught his attention: fractions. That caught my attention. What would he say that was so interesting about fractions that drew a genius to math? In his young mind, they weren't finished. He saw that by formalizing incomplete division, fractions are more useful than if we went to the effort of performing the division. If I may characterize scientific notation in a similar way, we can see that not moving the decimal point a bunch of times and leaving an exponent that in two ways tells us a lot about the number is more useful than making a decimal too. Both fractions and scientific notation are in some way incomplete operations.
--Ben
(As a side note: http://www.math.cornell.edu/News/2012-2013/thurston.html)