Still Thinking About Numbers

As I mentioned in a blog entry last week, the value of pi slips away from you if you try to state it as a simple fraction, even in a good try such as 22/7.  If we represent numbers in decimal rather than fractional form we can express the strangeness of pi, its irrationality, in a different way. We can turn any simple fraction into one of three sorts of number: a simple integer, an integer followed by a string of decimals that comes to a definite end, or an integer followed by the string of decimals that doesn’t come to an end but that settles down into satisfyingly dull repetition.     

Here are some simple fractions: 9/3, 11/4, and the aforementioned 22/7.

The first, 9/3, resolves itself into the integer 3.

11/4 is 2 ¾, or in decimal form 2.75. That is where it terminates.

22/7 is 3 1/7 or in decimal form 3.142857142857.... It is a repeating decimal. That is: the sequence "142857" will continue repeating for as long as you have the patience to continue the long division.

Simple fractions, then, resolve themselves into one of those three equivalents. But the distinguishing feature of irrational numbers, as we said above, is that they can’t be represented by any simple fraction. Another way of putting this … they don’t resolve themselves into any of those three equivalents. If you try to represent pi or its kin decimally, you get an unending string of numbers that never settles down into a repeating pattern. 

Now let's think historically. We can’t claim that philosophy began with Pythagoras (c. 570 BC – c. 490 BC). Pythagoras was born while Thales of Miletus was in his 50s, and it is Thales, the everything-is-water guy, who by convention gets credit as the founder of philosophy.

Still, Pythagoras stands tantalizing close to that origin. (He had been dead for twenty years when Socrates was born.)

Pythagoras’ name is most closely associated in our time with a theorem about right triangles. That may have been due to a disciple of his, some member of the “Pythagorean school” who piously attributed his own work to its founder. At this distance we can’t straighten out questions of attribution. What we have are some fascinating stories, legends, about an early think tank, a community of ascetics devoted to numbers, shapes, music, and to learning all they could about each of those.

One of these stories tells us about a student of Pythagoras named Hippasus, who tried to figure out the square root of 2, and tried to represent it as a fraction. Not only did he fail in this effort, along the way he established that the thing can’t be done.   So this, √2, became history’s first known irrational number.

But to Pythagoras, the new discovery was quite unwelcome. Numbers are beautiful, rational things. Surely such a simple procedure as looking for the square root of two can’t lead us into a dark alley!  And, if by chance it can play that trick on us, we (imagining ourselves to be of the Pythagorean sect) have to keep that a well-guarded secret from the masses. It would disturb them to know so strange a truth!

Poor Hippasus met with an untimely end. The Pythagoreans put it out that he had drowned at sea. Some suspected that the gods were angry with Hippasus, and others suspected that the gods had human assistance in the matter.   

Clearly at this very early time in the history of both math and philosophy, the feeling was very strong that irrational numbers deserve that unflattering moniker – that they are poorly-behaved numbers, numbers about which people who know what’s good for them ought to be quiet!

Irrational numbers, like disadvantaged non-citizens in a body politic, fought their way to general acceptance only very slowly, and only by proving that they could be productive, helpful folk.

More than a century after Pythagoras’ death, Plato wrote a dialogue, Meno, in which the fictionalized Socrates discusses geometry with a slave. [Much else goes on in the dialog, but this is the passage that advances our story.]

We know this character only as “Meno’s slave.” We also know that, under Socrates’ prodding, he figured out that the square root of 2 cannot be stated as a fraction. Socrates (and accordingly Plato) used this as evidence that everybody really knows such things, that such knowledge is innate and Socratic questioning is only necessary to bring it out of storage.    

By this time, I submit, irrational numbers had won their struggle for admission to full citizenship in the world of numbers. And yes, it is fitting that the Meno is largely remembered not for the discussions with the title character but for the discussions with an anonymous slave of his. Human slavery, alas, would last a lot longer than had the disenfranchisement of those odd numbers.