Website owner:  James Miller

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To quote Leopold Kronecker in regard to our number system: "God made the integers. All the rest are the work of man". The number system of classical algebra is a creation of man. It is a man-made edifice devised and defined in such a way that the rules of classical algebra would hold over it. The only numbers that were not created by man are the positive integers 1, 2, 3, .... . All the rest, the positive rationals, the negative numbers, the complex numbers, etc. are all creations of man. The integers have a natural intuitive meaning. Using the integers as building blocks man invented the positive rational numbers as a means for designating measures of continuous quantities. Thus the positive rationals can be given an easy intuitive interpretation as a measure of some continuous quantity (i.e. distance, area, volume, time, etc.). The negative numbers may have intuitive interpretations in some applications (i.e. negative temperatures, debits in accounting, etc.). As far as an intuitive interpretation for the complex numbers is concerned, I am not aware of any. As far as I know they are just very abstract mathematical entities created by man, without any physical or intuitive interpretation. One can ask the question, "What is an integer? (such as the integer "5" or the integer "8") and then answer the question in about the same way as he would the question, "What is a horse?" It has a definite, concrete meaning and he knows what it is. But if he asks, "What is a negative integer? (such as the number "-5" or "-8") the answer is more difficult. A negative integer is more abstract, just a defined mathematical entity, although it may have intuitive interpretations in some instances. If one asks, "What is a complex number?" the answer is that it is just a very abstract mathematical entity with no concrete meaning or interpretation. It is an entity that mathematicans work with without knowing what it is. It is an entity defined by them but lacking an intuitive interpretation. Thus we have intuitive interpretations for some kinds of numbers but not others. In the same way, among the operations on numbers (addition, subtraction, multiplication and division), some operations have physical interpretations and some don't. The sum of two positive numbers has a natural meaning, a natural intuitive interpretation. Subtracting a smaller positive number from a larger one has an intuitive interpretation. The product of a positive number times a positive number has a intuitive interpretation. The division of a positive number by a positive number has an intuitive interpretation. When negative numbers are involved in operations there may be intuitive interpretations and there may not.


The concept of the rational numbers starts with the idea of "the n-th part of unity" (i.e. 1/n). Any positive integer "m" times the n-th part of unity gives us the positive rationals (i.e. denoting the n-th part of unity by "n*" any rational number is given by m(n*), where m and n are arbitrary positive integers).

Why are the rules of classical algebra what they are? They are what they are because of the way the number system over which they operate was defined.

Man first created the number system of classical algebra then he went on to devise other number systems for other "number-like" entities such as vectors, matrices, etc. -- setting up rules of operation for them. Thus he broadened and generalized his concept of "number" as well as his concept of "operation".

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