In 1872, he published a book, Continuity and Irrational Numbers, in which he attempted to remove all ambiguities and doubts as to how irrational numbers fitted into the domain of arithmetic. Some items to be considered in this work are as follows (all numbers are shown in base ten arithmetic):

1. A rational number can be expressed in the form of a fraction a/b where a and b are integers.
2. A number which cannot be expressed as a rational fraction is an irrational number. For example,,,,. The class of real numbers is made up of rational and irrational numbers.
3. A rational number can be expressed in decimal notation and where the decimal does not terminate (end in zeroes), it repeats itself periodically. For example, 10/13 =.769230.769230.769230 and 14/11 = 1.27.27.27. An irrational number when expressed as a decimal does not terminate or exhibit the periods. It is impossible to exactly express numbers such as , or , as decimals. The value can be approximated closely but the decimal can never express the root exactly or periodically.
4. It is difficult to perform arithmetic operations with magnitudes incapable of exact expression.

   Dedekind's accomplishment was to define irrational numbers in terms of rationals. The core of his method is his concept of the "Dedekind cut". This cut is defined as a subdivision of the rational numbers into two nonempty sets satisfying the condition that any member of the first set is less than any member of the second and the first set has no largest members. This is used as a theoretical device for defining the system of real numbers.

   An irrational number is a cut separating all rational numbers into two classes, an upper and lower class (set). The , for example is an irrational number defined by a Dedekind cut dividing the set of rationals into an upper class whose members are greater than , and a lower class containing all other rational numbers.

   If the rational numbers are divided by a "cut" into two classes, A and B, then there are three possibilities:

   1. Class A has a largest element a, e.g., if A contains all rational numbers less than or equal to 1, and B contains all other rational numbers, then a=l.
      For example, Q is the set of rational numbers so:
      
   2. Class B has a smallest element b, e.g., if A contains all rational numbers less than 1, and B contains all rational numbers greater than or equal to 1, then b=l.
      
   3. Class A has no largest element and class B no smallest element, e.g., if A contains all rational numbers less than and B all rational numbers greater than . Then the cut is an irrational number separating classes A and B but belonging to neither.

   Item 3 shows that stands alone and is surrounded by the other members of the number system but fills the gap between the two sets of numbers. In a co-ordinate system where each point of a straight line corresponds to a number, the corresponds to the point whose distance is exactly units from the origin. If this point were missing then the line would be discontinuous. This proves that the irrational numbers do exist.

   To further illustrate the concept of irrational numbers consider the following example. It can be shown that every rational number has a terminating or infinite periodic decimal expansion and every terminate or infinite periodic decimal expansion represents a rational number. The is not a rational number so it cannot be represented by a terminating or infinite periodic decimal expansion. If we work out the decimal expansion of 2 by the method of finding square roots then this rule would enable us to put any given rational number into one of two classes A or B, depending on whether it is smaller or larger than . To do this we need to find the decimal expansion of the given rational number and see where it differs from the decimal expansion of . For example, suppose the given rational number is 707/500 whose decimal expansion is 1.414. Since = 1.4142135... the rational number 707/500 would be in class A since it is smaller than . This can be done for any repeating or terminating decimal.

   Cheryl Haefner

   ---

   [Ed. note] Although Dedekind's work in defining irrational numbers seems to be of relatively little consequence in the discipline of Discrete Mathematics, the notion of the Dedekind cut is an early example of a formal procedure that can be used to partition a set (with the understanding that certain "cuts", i.e. those that represent irrational numbers, do not partition the set of real numbers). As an exercise, show that the square root of two is not an irrational number in base 2 arithmetic.

   Robert H. Orr