For my 900th (thirty squared) post, here is the explanation of the p-adics I promised to give previously.
In this particular case, though, Lagrange in his advocacy of a prime base had a point (though he could not have known this), because the p-adics only work with all the desired properties for prime p. So instead of using decimal, I'm going to write the rest of this post in quinary.
{5} (default quinary)
We know how to count in quinary fairly well, of course: 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, and so on. Whenever we need to increment a 4, it turns into a zero and carries one to the place right before it. Every place corresponds to a different power of five: thus "143" means one times five squared, plus four times five to the first power, plus three times five to the zeroth power. Of course, five to the first power is just five, and five to the zeroth power is one.
We also know fairly well what finite quinary expansions mean. For example, 0.1 must be one fifth, as that position one place to the right of the radix point corresponds to five to the negative first power: that's the reciprocal of five, which is one fifth. Then we can keep going further and further to the right of the radix point.
Infinite expansions, however, are a little more tricky. What does 0.444... mean, with the fours going off to infinity? It must be the sum of 0.4, 0.04, 0.004, and so on to infinity, but what is that?
Well, we can look at the interval between 0 and 1. If I mark the point 0.4 on that interval, I've divided it into two parts of four-fifths and one-fifth each. Going to 0.44, we then take the smaller interval and divide it into two parts of four-fifths and one-fifth each. And then we go to 0.444, and divide that tinier interval again into two parts of fourth-fifths and one-fifth each. The differences between these numbers and unity, {0.1, 0.01, 0.001, ...}, dwindle exponentially to zero.
The sequence {0.4, 0.44, 0.444, ...} can be said to converge to 1, in the precise sense that you can get a number as close to 1 as you like by picking a number sufficiently far along in the sequence. Since the partial sums of the infinite series 0.4 + 0.04 + 0.004 + ... thus approach 1, the series itself has the sum of 1. Since by definition the sum of the series can also be written 0.444..., we must have 0.444... = 1.
Now, of course, this makes many people uncomfortable, I suppose because it seems strange that we should be able to allow the digit-1's in the sequence {0.1, 0.01, 0.001, ...} to go away by taking the limit. Some might point out that if we take the sequence {1, 10, 100, 1000, ...}, the digit-one also moves farther and farther away, so that eventually it should vanish. Of course, this doesn't work in the real numbers, since the numbers {1, 10, 100, 1000, ...} increase in absolute value without limit and so get further and further from zero.
But is there a metric on the rationals such that the numbers {1, 10, 100, 1000, ...} really approach zero as one would naïvely think they should?
{a}
Yes indeed! This is called the 5-adic metric, and it works if you replace the 5 with any other prime. The general idea is that you can express any rational number r as some power of five 5^{n} times another rational number p/q where neither p nor q is divisible by five and they do not share a common factor. Then we define the 5-adic absolute value |r|_{5} as follows:
|r|_{5} = 5^{−n}
|0|_{5} = 0
In the simplest case, this means that the powers of five really do approach zero in the 5-adic metric because |5|_{5} = 5^{−1}, |25|_{5} = 5^{−2}, |125|_{5} = 5^{−3}, and so on. The negative powers of five don't approach anything however, since |1/5|_{5} = 5^{1}, |1/25|_{5} = 5^{2}, and |1/125|_{5} = 5^{3}, and so on. (The values of the absolute value are interpreted as real numbers, not 5-adic numbers, of course, because the other choice results in a circular definition.) This means that numbers can now expand infinitely to the left, because they converge that way; but no longer to the right, because they don't converge that way.
Previously, the choice of 5 as our radix had nothing to do with the structure of numbers, and its being crowned "the king of numbers" was an empty title with no real power. Now, the lordship of 5 actually means something, and it is in the very fabric of this new metric on the rationals. (But it should also be noted that any other prime number could equally well take on that lordship.)
The distance between two numbers x and y in the 5-adic metric is then |x − y|_{5}, so that shift-invariance of distance is preserved.
Under this metric, the 5-adics (and indeed any of the p-adics) form a complete ultrametric space, meaning that if d(x,y) is the distance between x and y, then d(x,z) ≤ max{d(x,y), d(y,z)}; this is known as the "strong triangle inequality", since it implies the triangle inequality as well. This grants it a few strange and characteristic properties: all triangles are acute isosceles or equilateral, every point inside a ball is its centre, and interescting balls are contained wholly in each other.
Nevertheless, some of the familiar properties we know and love are retained. It is still true that the absolute value is multiplicative, so that |x|_{p}|y|_{p} = |xy|_{p}, and it satisfies the triangular inequality as previously stated, so that |x + y|_{p} ≤ |x|_{p} + |y|_{p}.
{5}
In the 5-adic numbers, there is no longer a need for a negative sign. If we consider the calculations {10 − 1 = 4; 100 − 1 = 44; 1000 − 1 = 444; ...}, we notice that the minuends approach 0, so that the differences must approach 0 − 1 = −1. Then if we take this to the limit, every place borrows from the place before it, so that ...000 − 1 = ...444. An infinite sequence of fours extending to the left of the quinary point is then the 5-adic expansion for −1.
Similarly, further negative integers are {...444, ...443, ...442, ...441}, and so on, all with an infinite number of fours trailing into the distance after a brief prelude. (This is the same idea as two's complement in binary, although here it should probably instead be called "five's complement", since this is quinary.)
The 5-adic numbers that do not need a radix point to represent are called the 5-adic integers, which are labelled Z_{5}. (For other primes p, we replace 5 with that p.) Z_{5} is also the set of all rationals r such that |r|_{5} is non-negative. Z_{p} is always a commutative ring with the two binary operations of addition and multiplication when p is prime. This means that addition and multiplication are both commutative and associative, that they both have identities (0 and 1 respectively), that every member has an additive inverse, and that multiplication distributes over addition. Its topology is that of a Cantor set.
Of course, all members of Z are members of Z_{5} as well. But that is not all! Consider the rational number one-half. Its simplest representation in the format given above is 5^{0} * 1/2, so |1/2| = 5^{0} = 1, which is non-negative. So one-half is a 5-adic integer, and so we should not need any digits to the right of the radix point to represent it. And indeed, we have 1/2 = ...22223. When we add ...22223 to itself, the units column produces a 6, which cascades a carry down all the columns to the left that produce fours to infinity; thus the result is ...00001.
We can similarly check that 1/3 = ...13132, with the sequence "13" repeating to infinity. In fact, all rational numbers are 5-adic integers as long as their denominators in simplest form are not divisible by 5. So {1/2, 1/3, 1/4} are all 5-adic integers, but 1/5 is not.
Allowing digits to the right of the radix point (but only finitely many of them) admits such rationals, and results in Q_{5}, the 5-adic numbers. When p is prime, this is a field: otherwise, we end up with zero divisors if it has multiple distinct prime divisors, or it reduces to the field generated from its constituent prime if it happens to be a prime power. Being a field means that in addition to the properties of a commutative ring, we also always have multiplicative inverses for every element except the additive identity. (We also demand for a field but not a ring that the additive identity is different from the multiplicative identity, but this is true for Z_{p} as well as Q_{p}, and is indeed true for the commutative ring Z as well.) The topology of Q_{p} is that of a Cantor set with one point missing.
However, the set of 5-adic integers is obviously uncountable, as there are infinitely digits to the left of the radix point. So the rationals cannot make up all of them. And that would be correct, but not every real number appears in the 5-adics either. The end digits of squares in quinary are limited to {0, 1, 4}, so the square root of 2 is not a 5-adic number. Even further, not every 5-adic number actually corresponds to a real number. Square roots of −1 actually exist in the 5-adics: one of them begins ...00301131300030330421304240422331102414131141421404340423140223032431212, and the other one is of course just its additive inverse. The different p-adic systems for differing p are all different: so there are square roots of −1 in the 5-adics, but not in the 7-adics, for example. (However, there are no square roots of 2 in the 5-adics, but there are two in the 7-adics.) In general, p never has a square root in the p-adics; but there are always p-adic numbers that are not in R for every p, because −1 is always the sum of four squares in Q_{p}. So, the p-adics are not just an overly complicated way of writing the reals, but something really new.
None of the Q_{p} are actually algebraically closed, and they are much further from being so than R is; R only requires a quadratic closure (adjoining a solution of x^{2} + 1 = 0) to become the algebraically closed C, but the closure of Q_{p} to form Q̅_{p} has infinite degree, and even that is not metrically complete. Its metric completion is called C_{p}, the p-adic complex numbers, and there we finally can stop. In fact, C_{p} for any p is just C, the standard complex numbers, with an exotic metric replacing the usual one; this is different from the relationship between Q_{p} and R, which are much more different.
Well, this is just a quick explanation, and I skipped over a lot of things, but I hope it gives a reasonable idea of what these are. Please feel free to comment if something's not clear or mistaken!
One can construct the chords of every polygon, whose sides divide (p+1)/2 or (p-1)/2. One can construct the cyclotomic numbers for CZ(p^n(p-1)/2). The powers of any number raised to the pth power, converge on the sevenite-tails. For 5, these are 000000, 000001, 4431212, 013233, and 4444444. These also represent the fifth roots of 1, to 6 places. Either one of '2' or '3' can stand for i, so 0013234 ^ 8 = 000031, as you should expect from (1+i)^8 = 16. D.S. gives ...00301131300030330421304240422331102414131141421404340423140223032431212 for this, Here is the other one!
Code: Select all
[D:\]rexxtry say translate("00301131300030330421304240422331102414131141421404340423140223032431211", "01234", "43210")
44143313144414114023140204022113342030313303023040104021304221412013233
........................... D:\save\cdata\batch\rexxtry.cmd on WIN32
We find in base 13, that i=01550155.
Ordinary long division works, if you progress from the right.
The rules of polygonal isomorphism are observed.