Hi.

That "cut-the-knot" Web site is great! Nice job!

I have a question about some terminology used in:

Fundamental Theorem of Algebra

This page presents the following definition:

"In the beginning there was counting which gave rise to the natural numbers (or integers): 1,2,3, and so on."

The next page (fundamental2.html) also includes statements consistent with this definition, such as:

"No such integer exists that when added to 20 gives 4. We say that the set of integers is not algebraicly closed."

and:

"To see how this works let's start with integers (the set N of numbers 1,2,3,...)"

This definition of "integer" is different from the one I am accustomed to. In my research on mathematical terminology consistency, I have come to the following conclusions in this area:

The "cardinal" numbers are:

The set of the cardinalities of all finite sets. {0,1,2,3,...}

The "positive integer" numbers are:

The set of the cardinalities of all non-empty finite sets. {1,2,3,...}

The "negative integer" numbers are:

The set of values for x in the solutions to all equations of the form x+n=0, where n is any "positive integer" number. {-1,-2,-3,...}

The "integer" numbers are:

The union of the set of "cardinal" numbers and the set of "negative integer" numbers.

References:
http://www.pcwebopaedia.com/integer.htm
http://whatis.com/integer.htm

The "natural" numbers are:

The same as either the "cardinal" numbers or the "positive integer" numbers, depending on the author.

The "whole" numbers are:

The same as either the "integer" numbers, the "cardinal" numbers, or the "positive integer" numbers, depending on the author.

Your comments on this significant terminology discrepancy regarding "integer" would be welcome.

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J. E. Lillge
Software Architect
Santa Clara Valley California

Copyright © 1996-2008 Alexander Bogomolny

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