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|This article/section deals with mathematical concepts appropriate for a student in mid to late high school.|
The problem of the rational numbers is that quantities that seemingly ought to exist, do not exist as rational numbers, even though the rational numbers can get arbitrarily close to what the value should be. Perhaps the simplest such number is the square root of 2. The function f(x) = x2 - 2 passes through the value zero somewhere between 1.414 and 1.415, but never achieves the value zero, when only rational numbers are considered. (See real number for an outline of the proof that the square root of 2 is irrational.)
A Dedekind cut is a subset of the rationals that satisfies:
- It contains at least one number, but not all numbers.
- If x is in the set, any y with y < x is also in the set.
- The set contains no largest member.
A little contemplation will show that a cut is generally of the form "all numbers less than X" for some rational number X. Rule number 2 is the important rule that makes this so. Intuitively, a cut is "everything to the left of [some point]". The complement of a cut is generally of the form "all numbers greater than or equal to X".
We can do arithmetic on cuts: For example, given cuts A and B, we can define "A+B" as the set of all (rational) numbers that are sums of a number in A and a number in B.
So far, it would seem that we haven't accomplished anything—every cut seems to be a rational number, and every rational number is a cut. But here is the miracle: We can define a cut as "all rational numbers that have squares less than 2". There is no rational number that gives this cut in the manner described above, but it is still a cut. We can perform cut arithmetic on it. This cut, when multiplied by itself, is the cut "all rational numbers less than 2", which is the cut associated with 2.
So we define the real numbers as the set of Dedekind cuts. The real number is defined as "that cut that consists of rational numbers that have squares less than 2".
When the real numbers are defined this way, they satisfy the intermediate value theorem. If f(x) is a continuous function that has f(a) < Q and f(b) > Q, there is a number x between a and b with f(x) = Q. Simply make the cut of those numbers with function value less than Q. That cut is the required number. (There are a few more minor details.)
The use of Dedekind cuts is one of the two famous ways of defining the real numbers, that is, completing the rationals. The other method is Cauchy sequences.