Professor Steven Strogatz of Cornell has a new post, “Division and Its Discontents”, available on his Opinionator blog at the New York Times Web site, continuing his series on some of the basic elements of mathematics. In this installment, he talks in more detail about a theme I mentioned in my last post about his column.
There’s a narrative line that runs through arithmetic, but many of us missed it in the haze of long division and common denominators. It’s the story of the quest for ever-more versatile numbers.
We saw how introducing the idea of subtraction led naturally to the development of negative numbers, which obey the same rules of arithmetic as the positive integers, which in their use for counting are our natural starting point. In this column, Prof. Strogatz talks about the operation of division, and how it leads us inexorably to the use of fractions (that is, rational numbers).
Although I have already given away the main thrust of the “plot”, in some sense, I do urge you to read his column. He has some lovely examples of some of the ways people get themselves in a muddle when thinking about fractions, one from the movie My Left Foot, and one from a conversation Mr. George Vaccaro has with representatives of Verizon Wireless.
Later, he talks a bit about the representation of fractions as decimals, and the fact that if
1/3 = 0.3333…
then, multiplying both sides by 3, we get
1 = 0.9999…
Understanding why this is true will take one a long way in understanding the properties of numbers and their representations.
Finally, Prof. Strogatz introduces our friends the irrational numbers. I motivated their introduction by a geometric argument: what is the length of the diagonal of a unit square? He starts off with a number whose representation is irrational by construction; its decimal representation does not terminate or repeat:
All of these numbers, remember, are required to behave themselves in certain specific ways, so that the rules of arithmetic still hold. And even though we have wandered fairly far from the basic idea of counting rocks or birds or apples, we still find our numbers relevant to the real world.