I am not a number theorist, but I’ve always had a distant fascination with p-adic numbers. I have a list of “neat math topics” that I want to write about on my blog, and the p-adic numbers are on that list. So I was happy to see an interesting article about them by Andrew Rich called “Leftist numbers” in the November 2008 issue of the College Mathematics Journal.
The usual construction of p-adic numbers is pretty complicated for the nonexpert. But, here’s the idea in a nutshell.
The rational numbers. The set of rational numbers is the set of numbers that can be written as a fraction. Examples of rational numbers are 4, 13, 2.1, 22/7, 0.333333… The set of rational numbers has a lot of “holes” in it, and there are different ways of filling these holes.
The real numbers from the rational numbers. Our usual means of filling these holes creates the set of real numbers. For example, we want the sequence of rational numbers 3, 3.1, 3.14, 3.141, 3.1415, 3.14159,… to converge, so we create a new number called to be the limit point of this sequence. To make sense of this we need a notion of closeness, and we all know that two numbers are “close” if their decimals agree for a long way to the right.
The p-adic numbers from the rational numbers. We play a similar game to construct the p-adic numbers except that we choose a new notion of closeness. (When we speak about p-adic numbers, p is some specific number, usually a prime, and the digits of the number are 0,…,p-1.) Now we say that two numbers are “close” if their digits are the same for a long way to the left! So the 10-adic numbers 0.03, 0.53, 6.53, 96.53, 196.53, 1196.53, 21196.53,… are getting closer and closer together.
The real numbers have finitely many digits to the left of the decimal point and possibly infinitely many digits to the right of the decimal point. However, as we will see, the p-adic numbers can always be written with finitely many digits to the right of the decimal point and possibly infinitely many digits to the left of the decimal point (that is why Rich calls them “leftist numbers”). For example 33.333333… is not a 10-adic number, but …333333.33 is. In particular, the sequence given in the previous paragraph converges to some new 10-adic number …21196.53.
Here are some cool consequences of this construction.
1. Addition. We can add any two p-adic numbers. Here is a 10-adic example—add as usual, carry to the left. (Note that since addition proceeds right-to-left, it is a lot easier to add p-adic numbers with infinitely many digits than real numbers with infinitely many digits.)
2. Multiplication. Just like addition, multiplication of two p-adic numbers is possible, and is easier to carry out than for real numbers.
3. Subtraction. There is no need for a negative sign (–) for negative numbers. For example, as a 10-adic number, we can express -16 as …999984. To justify, just observe that 16+(…999984)=0:
Similarly, it is possible to show that every p-adic number has such a “positive negation.” Thus we can always subtract by adding.
4. p-adic rationals. Every p-adic rational number can be written with finitely many digits to the right of the decimal point. For example, we usually think of 1/3 as 0.3333…, but in the 10-adic numbers we can express it as …666667. To justify, observe that (…666667)*3=1:
Moreover, as Rich shows in the article, a p-adic number is rational if, and only if, the digits are eventually repeating to the left of decimal point (which is a nice analog of the real case in which a number is rational if, and only if, it is eventually repeating to the right of the decimal point).
5. Division. What about division? As Rich shows in the article, it is often, but not always, possible to divide two 10-adic numbers. The trouble is that there can be two nonzero 10-adic numbers x and y such that xy=0. See the article for details. However, and here is the important point, if p is prime, then this cannot happen. When p is prime, every nonzero p-adic number has a reciprocal, and thus we can always divide two such numbers.
6. Ordering. Here’s a final curious fact about the p-adic numbers. We all know that if x and y are two non-equal real numbers then either x<y or y<x. However, there is no linear ordering of the p-adic numbers!
7. Mathematical gobbledygook. Here is a more mathematical way of saying all this. If p is prime then the p-adic numbers form a complete metric space containing the rational numbers (it is a completion of the rational numbers) and it is also a field. (Note that because of the division problem, when p is not prime, the p-adic numbers are not a field, but only a ring).
For more details, examples, and proofs, see Andrew Rich’s excellent article “Leftist numbers.”