May 1997 Presentation Topic (continued)
The algebraic structure of the integers is very rich and is the subject of Number Theory. Questions about the distribution of prime numbers are among the most difficult open problems in mathematics. One amazing result is that, for large numbers N, the number of primes from 1 to N is on the order of N / log(N); this fact (which can be stated more precisely) is known as the Prime Number Theorem.
The integers are adequate for describing whole quantities but not portions of quantities. For this purpose, a new concept must be defined: that of a rational number. A rational number can be defined to be a pair of integers r and s (with s non-zero), representing r pieces, where s pieces make up a whole. We write this pair of numbers using the notation r/s, and we consider two pairs r/s and m/n to be the same rational number if nr - ms = 0.
(Mathematical sophisticates would say that the rational numbers are equivalence classes of pairs (r,s) under the equivalence relation that relates (r,s) to (m,n) if and only if nr - ms = 0, and that the equivalence class of the pair (r,s) is denoted r/s).
Since not all lengths are expressible within it, the rational number system, although adequate for counting possessions, recording transactions, and counting portions of things, is inadequate for measuring lengths. A new kind of number must be constructed for that purpose. The square root of 2 is an example of this new kind of number: it measures the length of the diagonal of a square of side length 1, but it is not one of the familiar rational numbers so beloved by the Greeks.
Real numbers need to be able to describe lengths and other continuous quantities, even ones not representable by rational numbers (like the diagonal of a unit square). In short, they need to be able to describe all possible locations on the "number line". How can we construct such numbers out of rational numbers (the only kind that we so far know to exist)?
There are two ways. One is to describe a location on the number line by saying which rational numbers lie below it and which lie above it. A real number can then be defined as a partition of the rationals into two groups, satisfying certain properties (like the fact that everything in group 1 must be less than everything in group 2).
Another way (a little less elegant but more useful for writing down real numbers) is to describe a location on the number line by a sequence of rational numbers which form better and better approximations to it. One can then define a real number to be any such sequence of rational numbers.
Not all sequences converge to a single location on the number line, so we consider only Cauchy sequences. These are sequences such that, for any positive number e (no matter how small), there is some place in the sequence such that, from that point on, all terms are within e of each other. This guarantees that the sequence is forming better and better approximations to a single location on the number line.
Two sequences a(1), a(2), ... and b(1), b(2), ... may define the same location on the number line. This happens if the differences a(1)-b(1), a(2)-b(2), ... are converging to zero (meaning that, for any positive number e, all the differences from some point on are smaller than e in absolute value). In this case, we call the sequences "equivalent".
Now we define a real number to be an equivalence class of Cauchy sequences of rational numbers (that is, each Cauchy sequence defines a real number, and two Cauchy sequences define the same real number if and only if they are equivalent). This should remind you of the way rational numbers were defined as pairs of integers, with two pairs (r,s) and (m,n) defining the same rational number if and only if nr - ms = 0.
We can write real numbers in decimal notation, for example, pi = 3.1415.... What this means is we are looking at the real number defined by the following sequence of rational numbers: 3, 3.1, 3.14, 3.141, 3.1415, ... (this is a Cauchy sequence because, for any n, after the nth term all terms agree to n decimal places and hence are within 1/10^n of each other). Any decimal expression defines a Cauchy sequence of rational numbers (by truncating at successive decimal places), and hence every decimal expression defines a real number.
Another interesting property is that there are the same number of rational numbers as there are integers (even though the rational numbers contain all the integers and more besides!), but there are more real numbers than there are integers.
The well-ordering constructed in the solution to question 4 provides a way to put the positive rational numbers into 1-1 correspondence with the positive integers (which is what it means to say that there are as many positive rational numbers as positive integers): for every positive rational number r, there are only finitely many other numbers s for which N(s) <= N(r), so there are only finitely many numbers s which are smaller than r under the new ordering. This means that if you start with the smallest positive rational number, label it "1", label the next smallest "2", and so on, you will eventually reach every postive rational number r. In this way, every positive rational number r gets matched with a positive integer, leaving nothing out.
However, the real numbers can not be put into 1-1 correspondence with the integers. There's a famous argument called Cantor's diagonalization argument which proves this; we will not go any further into this here.
Note that this is the same sort of thing as a congruence statement on the integers: when "4", "5", and "2" stand for integers instead of clock-face hours, then it is no longer true that 4+5=2, but what is true is that 4+5=9 and 9 = 2 (mod 7).
Rather than work with integers and congruence relations modulo 7, it is often easier to work with a new number system (called a finite field) which has only seven numbers: 0, 1, 2, 3, 4, 5, and 6. These can be added or multiplied together, with the result always being in the same number system; for instance, 5 x 4 = 6 in this number system.
We have seen in the questions that division is well-defined in this number system, and the cancellation law holds. This is why the procedure used in solving question 7 is guaranteed to work (as long as we are working modulo a prime number).
In arithmetic modulo non-primes, the cancellation law no longer holds. For example, in arithmetic modulo 12, 3 x 4 is the product of two non-zero numbers which gives you zero. In such number systems, you cannot divide by all non-zero numbers. However, the structure of these number systems is still rich with important theoretical properties.
The definition of real numbers by Cauchy sequences relied on a notion of what the "distance" between two rational numbers r and s is. Normally, this distance is defined to be d(r,s) = |r-s|, and it is to this notion of distance we are referring when we say things like "the terms must be within e of each other". However, there are other notions of distance which satisfy all the properties a distance function needs to satisfy. These other notions of distance have a more algebraic flavour and are very important in number theory.
For instance, consider the prime number 3. For each integer n, define the weight W(n) of n to be the number of times 3 divides n. For each rational number m/n, define its weight W(m/n) to be W(m)-W(n). Now define the distance between two rational numbers r and s to be d(r,s) = 3^(-W(r-s)).
Notice that two integers are very close together if their difference is divisible by a high power of 3. This is a pretty weird distance function, but it satisfies all the properties a distance function has to satisfy, so we can define the notion of Cauchy sequences of rational numbers with respect to this distance function. This collection of Cauchy sequences forms a number system called the 3-adic numbers, in the same way that the usual Cauchy sequences of rational numbers form the real number system.
In the same way, one can define "p-adic numbers" for any prime p.
Other interesting number systems include the algebraic numbers, that is, numbers which can be described as roots of polynomial equations with rational coefficients. An example of an algebraic number is sqrt(2), a root of the equation x^2 - 2 = 0. For a while people wondered whether all real numbers might be algebraic, but the answer is no. In fact, there are many more non-algebraic real numbers than there are algebraic ones: the algebraic numbers can be put into 1-1 correspondence with the integers, but the real numbers cannot. pi and e are examples of non-algebraic real numbers.
Finally, there are the complex numbers. These can be defined as pairs of real numbers (a,b), typically written in the notation a + bi. With the appropriate definition of addition and multiplication, the pair 0+1i (usually written just as "i") squares to give -1. The fundamental importance of the complex numbers is given by
The Fundamental Theorem of Algebra. Every polynomial with coefficients in the complex numbers has a root that is a complex number.This is not true for the real numbers (for example, the equation x^2+1=0 has no real-valued roots), and it is what makes the complex numbers so useful, especially when trying to talk about geometry in algebraic terms.