Ramble, Part 37: A Different Distance
Nov. 2nd, 2007 11:49 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowIn an earlier Ramble post, I gave an argument for the assertion that 1+2+4+8+... = -1. Given the usual definition of the sum of an infinite series, this is clearly false. Let me now, though, present a context in which it would be nice to call this assertion true, and then a justification for doing so.
The standard notion of the sum of an infinite series comes from the viewpoint of analysis. To justify the assertion above, we need to turn to the viewpoint of number theory.
Suppose you have a polynomial equation you wish to solve, which happens to have a whole-number solution. One way of searching for that solution is this. First, solve the equation modulo 2. (This is easy, since you only need to look at 0 and 1.) Next, solve it modulo 4. If the mod-2 solution is 0, the mod-4 solution must be either 0 or 2, and if the mod-2 solution is 1, the mod-4 solution must be either 1 or 3; so, again, you only need to check two possibilities. Now proceed to modulo 8, to modulo 16, and so on. If the solution is positive, eventually you will find it - the solution modulo 2n is the solution, if n is large enough.
But what if the actual solution is negative? Let's take a very simple example, the equation x+1=0. Modulo 2, the solution is 1. Modulo 4, the solution is 3, or 1+2. Modulo 8, it's 7, or 1+2+4. Modulo 2n, it's 2n-1, or 1+2+4+...+2n-1. Would it not be nice to be able to think of these sums as closing in on the correct value - in other words, to say that 1+2+4+8+... = -1?
It turns out that it is possible to do so, but it requires using a different notion of distance. Let r be a nonzero rational number. It is possible to write r=2np/q, where n is a whole number (positive, zero, or negative) and p and q are odd whole numbers. We define the "2-adic value" of r to be |r|2=2-n. Thus, for example, |2|2=1/2, |3/2|2=2, and |12|2=1/4. We also define |0|2 to be 0. Now, define the 2-adic distance between two rational numbers to be the 2-adic value of their difference.
Under the usual notion of distance, 1+2+4+8+... does not converge. But under the 2-adic notion, it does - and it converges to -1!
This may seem to be a bizarre thing to do, but the notion of 2-adic distance (and similarly defined 3-adic, 5-adic, or in general p-adic distance, where p is any prime number) has proven tremendously useful in modern number theory. There is a great deal of very beautiful work built on those foundations; these notions were, for example, one of the (many) building blocks used in Andrew Wiles' proof of Fermat's Last.
(Just as a taste of what else can be done, there is a 2-adic series which converges to Sqrt[-7]. It begins 1+2+8+64+256+...; the precise pattern is not predictable, but there is an algorithm for constructing it step by step. For now, just note that 12 is congruent to -7 mod 8; 32 is congruent to -7 mod 16; 112, mod 128; and so on up the line.)
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The standard notion of the sum of an infinite series comes from the viewpoint of analysis. To justify the assertion above, we need to turn to the viewpoint of number theory.
Suppose you have a polynomial equation you wish to solve, which happens to have a whole-number solution. One way of searching for that solution is this. First, solve the equation modulo 2. (This is easy, since you only need to look at 0 and 1.) Next, solve it modulo 4. If the mod-2 solution is 0, the mod-4 solution must be either 0 or 2, and if the mod-2 solution is 1, the mod-4 solution must be either 1 or 3; so, again, you only need to check two possibilities. Now proceed to modulo 8, to modulo 16, and so on. If the solution is positive, eventually you will find it - the solution modulo 2n is the solution, if n is large enough.
But what if the actual solution is negative? Let's take a very simple example, the equation x+1=0. Modulo 2, the solution is 1. Modulo 4, the solution is 3, or 1+2. Modulo 8, it's 7, or 1+2+4. Modulo 2n, it's 2n-1, or 1+2+4+...+2n-1. Would it not be nice to be able to think of these sums as closing in on the correct value - in other words, to say that 1+2+4+8+... = -1?
It turns out that it is possible to do so, but it requires using a different notion of distance. Let r be a nonzero rational number. It is possible to write r=2np/q, where n is a whole number (positive, zero, or negative) and p and q are odd whole numbers. We define the "2-adic value" of r to be |r|2=2-n. Thus, for example, |2|2=1/2, |3/2|2=2, and |12|2=1/4. We also define |0|2 to be 0. Now, define the 2-adic distance between two rational numbers to be the 2-adic value of their difference.
Under the usual notion of distance, 1+2+4+8+... does not converge. But under the 2-adic notion, it does - and it converges to -1!
This may seem to be a bizarre thing to do, but the notion of 2-adic distance (and similarly defined 3-adic, 5-adic, or in general p-adic distance, where p is any prime number) has proven tremendously useful in modern number theory. There is a great deal of very beautiful work built on those foundations; these notions were, for example, one of the (many) building blocks used in Andrew Wiles' proof of Fermat's Last.
(Just as a taste of what else can be done, there is a 2-adic series which converges to Sqrt[-7]. It begins 1+2+8+64+256+...; the precise pattern is not predictable, but there is an algorithm for constructing it step by step. For now, just note that 12 is congruent to -7 mod 8; 32 is congruent to -7 mod 16; 112, mod 128; and so on up the line.)
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