Dedekind Cut
Real analysis, which studies the theoretical foundations of the calculus of real-valued functions, depends upon an accurately defined real number concept. Such a precise definition eluded mathematicians from the time ofPythagoras (c.560-c.480BC) to the latter half of the nineteenth century. The problem was not with all real numbers, but with that subset of the reals known as the irrational numbers. Pythagoras, it is said, was so horrified by the appearance of the irrational number 2, i.e., the positive square root of 2, in his calculation of the hypotenuse of a right triangle that he swore his followers to secrecy on the matter. The Pythagoreans called such numbers "incommensurable" because they could not be obtained by multiplying two rational numbers together. A rational number can always be expressed as the quotient of two integers--that's the definition of a rational number--but these "incommensurables" cannot be expressed in this way; hence the name "irrational." Nevertheless, since the real number line is a continuum with no "gaps" or "holes," there must be a location on the real line for numbers such as 2. Another way of looking at it is that if one takes the hypotenuse of an isosceles right triangle whose sides have length 1 and places it along the real number line with one end at 0, then clearly there is a point on the line at which the other end rests. But where, exactly, does 2 reside on the real line. We can determine that it lies somewhere between the rational numbers 1.4 and 1.5. We can pin it down even more accurately between 1.41 and 1.42, or, more accurately still, between 1.414 and 1.415, and so on. Even so, with each new upper and lower bound for 2, we have still not reached the precise point on the real number line where 2 should be located. The problem is that we can approximate the positive square root of 2 by rational numbers to any desired number of decimal places.
But the square root of 2 has a non-ending and non-repeating decimal expansion; so how can we ever define the "exact" value of the positive square root of 2. EnterRichard Dedekind (1831-1916), who saw in this method of approximating irrational numbers by rational numbers a potential definition for the irrational number 2, and indeed for all irrational numbers. His idea was to define a "cut" of the rational numbers corresponding to each irrational number. Consider the case of the cut corresponding to 2. He defined this cut as the set of rational numbers which are less than 2 but which contains no largest rational number. Thus the cut contains an infinite set of rational numbers which progress to the right on the number line but which are always less than 2. Dedekind then actually defines 2 as this cut. More generally, Dedekind gives the following definition of a cut.
A set K of rational numbers is said to be a cut if (i) K contains at least one rational number, but not every rational number; (ii) if p is in K and q is a rational number less than p, then q is in K; and (iii) K contains no largest rational number.
It can be shown that each rational number is also associated with its own cut. It is then possible to define a real number as a cut, which is what Dedekind did. All the familiar properties of arithmetic, in particular, thefield properties, hold for Dedekind cuts. That is, the arithmetic of cuts obeys the same operational rules that govern the rational numbers. Because of this, Dedekind's work in this area has sometimes been called the "arithmetization" of the irrational numbers or, more broadly, the "arithmetization" of real analysis. Although Georg Cantor (1845-1918) put forth an alternative definition of the real numbers based on infinite sequences, Dedekind's characterization of real numbers as cuts of rational numbers is today the standard approach taken in most textbooks on real analysis and number theory.
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