The Pides of March

Yesterday was Pi Day — a particularly good one, 3/14/15 in American date format (for 3.1415) — and today is the Ides of March. So: the Pides of March.

Pi (that is, π) is a transcendental number (in a special mathematical sense of transcendental). Now, a few words about different kinds of numbers.

We start with the natural numbers, the ones we use for counting things: 1, 2, 3, 4, … Everything else is an extension from these: zero (0), fractions, negative numbers, imaginary (vs. real) numbers, complex numbers, irrational (vs. rational) numbers, transcendental (vs. algebraic) numbers, and more.

Most people deal with only a few of these types, and then usually in the context of calculating values for practical purposes, like calculating the area of a circle (A = πr2). For these purposes, we can restrict ourselves to non-negative real numbers, which will be dealt with in computations via decimal fractions.

The universe of these numbers:

1. rational numbers, expressible as the quotient p/q of two integers (q ≠ 0), with two subtypes as decimal fractions;

1a. terminating decimals, like .1 (for 1/10), .2 (for 1/5), and .5 (for 1/2);

1b. repeating decimals, like .142857142857142857… i.e. .142857, with an underline marking off the repeated part (for 1/7); for practical purposes in computations, approximations will be necessary (say, .14 for 1/7);

2. irrational numbers, not so expressible (so their decimal expansions will be non-terminating and non-repeating, and approximations will be necessary for practical purposes in computations), with two subtypes:

2a. algebraic irrationals; an algebraic number is the root of a polynomial equation with rational coefficients. For example, √2 ( = 1.414…), the positive root of x– 2 = 0.

2b. transcendental irrationals, ones that are not algebraic, like π ( = 3.1415…).

It took some considerable time for people to accept the existence of irrational numbers. Pythagoras balked at the idea. Now it turns out that most numbers are irrational, and indeed, nearly all numbers are transcendental. Most of us just don’t have to deal with many of them.

(Teachers often give approximations to irrationals for the purpose of computation; 22/7 or 3 1/7 is sometimes suggested as a approximation to π for these purposes, and then since 1/7 = .142857, you might want to approximate that, as 3.14 or 3.143.)

Leave a Reply

%d bloggers like this: