Re: Factors and Terms

["Leif Svalgaard" <leif@xxxxxxxx>]

From: Richard Jackson <richardjackson@richardjackson.net>

> I must be confused.
>
> Please humor me and identify the numbers (other than 1 and 7) that, when
> multiplied together, equal 7.  Since 7 is itself prime, by definition 7 has
> only two factors - 1 and itself.

it has only one factor, namely 7. If you want to call 1 a factor of 7, then
I'll argue that 7 has seven factors, viz. 7, 1, 1, 1, 1, 1, 1, because
undeniably 7 = 7*1*1*1*1*1*1.

To be precise, there is a distinction between a divisor of a number
and a factor of a number. The concept of a divisor is best illustrated
by the example of "perfect numbers". From ancient times the number
itself is not considered a (proper) divisor, but 1 is, viz. the definition
of perfect numbers:

Perfect numbers are those integers which are the sum of their positive
proper divisors. The first three perfect numbers are
6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14, and
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.

A factorization of a number by convention excludes the divisor 1,
in order to maintain the desirable property that every number larger
than 1 can be written as a unique product of prime numbers raised
to integral powers. The factorization of 6 is 6 = 2**1 * 3**1, of 28 is
28 = 2**3 * 7**1, of 7 is 7 = 7**!, etc.

So, by convention, divisors of n include 1 but exclude n, and factors
of n include n, but exclude 1. Fuzzy math does not make that sharp
distinction. One *could* qualify the words by talking about "proper"
divisors and "prime" factors, but that is rarely done.


>
> The word used was factor, not term.  Factors are things multiplied.  Terms
> are things added.  As far as I know, (assuming base 10 and Euclidian
> geometry and a flat earth and other ordinary stuff) the sum of all the
> factors of 7 is 8 and the product of all of the factors of 7 is 7.
>
> What is fuzzy here?  Perhaps I have just lost my sense of humor ... :)
>
> Richard Jackson
> mailto:richardjackson@richardjackson.net
> http://www.richardjacksonltd.com
> Voice: 1 (303) 808-8058
> Fax:   1 (303) 663-4325
>
> -|-----Original Message-----
> -|From: owner-midrange-l@midrange.com
> -|[mailto:owner-midrange-l@midrange.com]On Behalf Of Leif Svalgaard
> -|Sent: Friday, October 20, 2000 9:02 AM
> -|To: MIDRANGE-L@midrange.com
> -|Subject: Re: IBM sales/earnings...
> -|
> -|
> -|Richard is just confused. The original comment said ":sum of its
factors",
> -|and Richard was musing about "product of its factors". But in these days
> -|where we are bombarded with FUZZY MATH dealing with trillions (or
> -|was that gazillions?), who cares?
> -|
> -|----- Original Message -----
> -|From: <D.BALE@handleman.com>
> -|To: <MIDRANGE-L@midrange.com>
> -|Sent: Friday, October 20, 2000 8:16 AM
> -|Subject: RE: IBM sales/earnings...
> -|
> -|
> -|> What the HECK are you talking about???  Is this NEW MATH or what???  Or
> -|maybe
> -|> it's just that I never took a philosphy course in college?
> -|>
> -|> -----Original Message-----
> -|> 7 is most definitely the product of its factors, like every other whole
> -|> number.  However, it is a little sad - there are only two of them.
> -|>
> -|> Richard Jackson
> -|> mailto:richardjackson@richardjackson.net
> -|> http://www.richardjacksonltd.com
> -|> Voice: 1 (303) 808-8058
> -|> Fax:   1 (303) 663-4325
> -|>
> -|> -|-----Original Message-----
> -|> -|From: owner-midrange-l@midrange.com
> -|> -|[mailto:owner-midrange-l@midrange.com]On Behalf Of Joel Fritz
> -|> -|Sent: Thursday, October 19, 2000 5:36 PM
> -|> -|To: 'MIDRANGE-L@midrange.com'
> -|> -|Subject: RE: IBM sales/earnings...
> -|> -|
> -|> -|
> -|> -|Is that a reference to Chapter 7, or 7 not being the sum of
> -|its factors,
> -|> -|or...?
> -|> -|
> -|> -|> -----Original Message-----
> -|> -|> From: Don [mailto:dr2@cssas400.com]
> -|> -|> Sent: Thursday, October 19, 2000 12:34 PM
> -|> -|> To: MIDRANGE-L@midrange.com
> -|> -|> Subject: Re: IBM sales/earnings...
> -|> -|>
> -|> -|>
> -|> -|>
> -|> -|>
> -|> -|> Al,
> -|> -|>
> -|> -|> this is true, and if they keep this great management
> -|philsophy in full
> -|> -|> force, they'll get to see just how perfect the number 7
> -|really is...:)
> -|> -|>
> -|> -|> Don
> -|> -|>


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