Science Forums 2

[CraigD]

Originally Posted by Don Blazys

Is there inconsistency, disagreement and confusion in both the "math community",and on the internet, as to what the words "proper factor" and "proper divisor" should mean?

Yes.

However, this is true of practically any words used in any discipline, if the “community” considered is sufficiently broad. “The math community” and “the internet” are a very broad communities.

Among recognized mathematicians (eg: people with PhDs in Math and related disciplines), I think there’s much inconsistency, but little disagreement or confusion about the meaning of any phrase, because mathematicians try to carefully define the words they use in a particular context, rather than relying on a pre-defined usual and traditional meanings.

Thus, while the usual meaning of “proper divisor of n” and “proper factor of n” are “an integer d such that 0 < d < n and the remainder of n ÷ d is 0” and “an integer d such that 1 < d < n and the remainder of n ÷ d is 0”, as Modest notes in post #105, and Don observes by finding a contradiction in an “Ask Dr. Math” webpage, variations in meaning are permitted and not uncommon.

The adjective “proper” usually means that the meaning of the noun following it is restricted in some special way. The Wolfram Mathworld entry for “proper” gives the terse definition “in general, the opposite of trivial.” I’d define it more liberally as “used in the way I’m using it right here”. So, when one encounters a phrase like “proper divisor”, the “proper” is a clue that one must search out a precise definition of the phrase.

Personally, I think math is most easily read when written with the smallest vocabulary. The standardization of and widespread familiarity of math readers with real and pseudo programming languages has, I think, greatly aided communication, because it’s possible to exactly define functions and their output as (usually) short programs. For example, the perfect numbers are defined by the following MUMPS program:

f  s N=N+1,S=0 x "f D=1:1:N-1 s:N#D=0 S=S+D" q:N=S

which sets the variable N to the next perfect number greater than its initial value. Because the interpreters of such program are themselves explicitly defined programs, they can’t be misinterpreted due to human subjectivity.