Subject: Re: The terms of mathematics
Date: Wed, 04 Dec 1996 18:14:19 -0500
From: Alex Bogomolny

Dear Hans:

  1. 0 is a number with well defined properties. For example, a+0=a and a*0=0 for any number a.
  2. 0 can be used in arithmetic calculations regardless of its representation. For example, 7*(5*5-4*4-3*3)=0.
  3. 0 has a name - "zero", while an undefined term does not. Indeed, a term may be undefined in a variety of ways. If you talk of integers, for example, the 15/5 is defined while 12/5 is not. If you talk of real numbers, sqrt(9) is defined while sqrt(-9) is undefined. 1/0 is undefined unless you think of what's known as the Nonstandard Analysis. Even then the meaning of 1/0 is not quite defined.
    Thus, when I say zero, you know what I mean; whereas when you say undefined - I do not :-)

Just in case, you may note that being unknown is not the same as being undefined. When something is unknown one still assumes that it is possible to associate a value (perhaps not unique) with this something. On the other hand, you can't associate a value with an undefined term without removing the reason for its being undefined. (I.e., to remove the blemish from 12/5, one has to define and switch to rational numbers.)

Hope this helps. Please let me know if you still have doubts.

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