Valuations and Completions

In ordinary everyday life, there are several notions of closeness. There is for example, a physical notion of distance, and we say, for instance, that we are close to our next-door neighbors. But there is another sense of closeness, such that we can say that we are “close” to our relatives, or to our friends, even though physically they may be far away.

There is also a similar notion of “closeness” between numbers. The most basic method is provided by the familiar “absolute value“. Given three numbers $x$, $x_{1}$, and $x_{2}$, to say that $x$ is closer to $x_{1}$ than to $x_{2}$ means that $|x-x_{1}|<|x-x_{2}|$. So for example, since $|(-1)-(2)|=3$ and $|(8)-(2)|=6$, we therefore say that the number $2$ is “closer” to the number $-1$ than to the number $8$. In other words, the smaller the value of $|x-y|$, the closer $x$ and $y$ are to each other.

But there are also other notions of “closeness” for numbers, just as we have explained above, that with our relatives or friends we may be “close” even if we are far away from each other. Consider the numbers $1$ and $10001$. Simply by looking, they can perhaps be said to be “relatives” or “friends”, which makes them in some way closer than, say, $1$ and $18$. The same may be said for $5$ and $3000005$, that they are perhaps members of the same “family”. This is, of course, because their difference is divisible by a large power of $10$, and since we use the decimal system to write our numbers, there is some sort of visual cue that these numbers are “family members”.

But in number theory, $10$ is not really very special. Perhaps it just so happens that we have $10$ fingers which we use for counting, so we used $10$ as a base for our number system. What is really special in number theory are the prime numbers. So for our notion of closeness we choose a prime, and define our measure of closeness so that two numbers are closer together whenever their difference is divisible by a large power of that prime number. For our chosen prime $p$, we want an analogue of the absolute value, which we will call the $p$-adic absolute value, and written $|x-y|_{p}$, which is smaller if the difference of $x$ and $y$ is divisible by a large power of $p$. The “ordinary” absolute value will now be denoted by $|x-y|_{\infty}$.

We want to define this for rational numbers as follows. Given a rational number $a$, we express it as

$a=p^{m}\frac{b}{c}$

such that $b$, $c$, and $p$ are mutually prime, i.e. they have no factors in common except $1$. Then we set

$|a|_{p}=\frac{1}{p^{m}}$.

We can see that this definition gives us the properties we are looking for – the value of $|a|_{p}$ is indeed smaller if $a$ is divisible by a large power of $p$.

The absolute value (both the “ordinary” absolute value and the $p$-adic absolute value) is also called the multiplicative valuation. There is also a related notion called the exponential valuation, which, in the $p$-adic case, we denote by $v_{p}(a)$ for a rational number $a$. The exponential valuation is obtained from the multiplicative valuation by setting

$v_{p}(a)=-\text{log}_{p}|a|_{p}$.

In the case above, where $a=p^{m}\frac{b}{c}$ and $b$, $c$, and $p$ are mutually prime, we simply have

$v_{p}(a)=m$.

For the ordinary absolute value, we just set

$v_{\infty}(a)=-\text{ln}|a|_{\infty}$

where $\text{ln }$ of course stands for the natural logarithm.

The concept of “closeness” between numbers, even just the “ordinary” one, was used to discover something interesting about the number line. If it was merely composed of the rational numbers, then there would be “gaps” in the line. To make a “true” number line, one must fill in these gaps, and this lead to the construction of the real numbers by the mathematician Richard Dedekind in the 19th century.

We elaborate on the nature of these “gaps”, following closely the idea behind Dedekind’s construction. Consider the real number $\sqrt{2}$. It is known from ancient times that this number cannot be written as a ratio of two integers and is therefore not a rational number. However, we can construct an infinite sequence of rational numbers such that every successive rational number in the sequence is “closer” to $\sqrt{2}$, compared to the one before it.

The mathematician Leopolod Kronecker once claimed, “God made the integers, all else is the work of man.” We know how to construct the rational numbers from the integers (for those who would like to think of the natural numbers as being even more basic than the integers, it is also easy to construct the integers from the natural numbers), by taking pairs of integers, and considering sets of equivalence classes (see Modular Arithmetic and Quotient Sets) of these pairs; for example, we set $\frac{1}{2}$ and $\frac{2}{4}$ as equivalent, because “cross multiplication” on the numerators and denominators gives us the same result. So the rational numbers are really equivalence classes of pairs of integers.

The problem we face now is how to construct the real numbers from the rational numbers. We have seen that we can construct sequences which “converge” in some sense to some value that is not a rational number. By “converge”, we mean that successive terms become closer and closer to each other late in the sequence. Technically, we do not refer to such a sequence as a convergent sequence, since it is a sequence of rational numbers but it does not converge to a rational number. Instead, we refer to it as a Cauchy sequence.

And this gives us a possible solution to our problem above – we could simply define the real numbers as the set of all Cauchy sequences. Those that converge to a rational number “represent” that rational number, and those that do not “represent” an irrational number such as $\sqrt{2}$. However, there is still one more problem that we have to take care of. There may be more than one Cauchy sequence that “represents” a certain rational or irrational number.

Consider, for instance, the sequence

$\displaystyle 5,5,5,5,5,...$

which obviously converges to the rational number $5$, and consider another sequence

$\displaystyle 6,5,5,5,5,...$

which is different in the first term but similarly converges to the rational number $5$. They are different sequences, but they “represent” the same rational number. We would like to have a method of “identifying” these two sequences under some equivalence relation. In order to do this, we consider the “difference” of these two sequences:

$\displaystyle 1,0,0,0,0,...$

We see that it converges to $0$. Such a sequence is called a nullsequence, and this gives us our equivalence relation – two Cauchy sequences are to be considered equivalent if they differ by a nullsequence. The set of real numbers $\mathbb{R}$ is then defined as the set of equivalence classes of Cauchy sequences under this equivalence relation.

The process of “filling in” the “gaps” between the rational numbers is called completion. Note that a notion of “closeness” is important in the process of completion. If we had a different notion of closeness, for example, by using the $p$-adic absolute value instead of the ordinary absolute value, we would obtain a different kind of completion. Instead of the real numbers $\mathbb{R}$, we would have instead the $p$-adic numbers $\mathbb{Q}_{p}$. The $p$-adic numbers play an important role in number theory, as they encode information related to primes.

References:

Valuation on Wikipedia

Complete Metric Space on Wikipedia