Are Negatives Factors of Other Numbers?

Date: 07/16/2003 at 12:16:42
From: Adam
Subject: Are Negative Numbers Factors of Other Numbers

Dear Dr. Math:

Are negative numbers factors of other numbers?  In other words, is -3
a factor of 3?  

Is -3 a multiple of 3?

What is confusing about is that when examples of factors and multiples
are given, negatives are never used. I teach test taking math and need
to give students these kind of quick, utilitarian answers.

My understanding is that the answer to both questions is yes, but I
couldn't find confirmation. 

Thank you so much.

Date: 07/17/2003 at 00:45:06
From: Doctor Greenie
Subject: Re: Are Negative Numbers Factors of Other Numbers

Hello, Adam -

You will undoubtedly get different opinions from different sources, 
including the math doctors here. I will give you my opinion and 
invite other doctors to add theirs.

You should not get much difference of opinion about -3 being a 
multiple of 3.  Most if not all serious mathematicians will agree 
that any number obtained by multiplying 3 by an integer is a multiple 
of 3.

But I think you will find a difference of opinion about -3 being a 
factor of 3. In my opinion, in most contexts, factors of a number are 
only positive integers. The prime factorization of a number "n" is the 
unique set of positive prime numbers which, when multiplied together, 
yield the number n. And when we list "the factors of" a number, we 
only list positive integer divisors. The "factors" of 6 are 1, 2, 3, 
and 6; we don't include -1, -2, -3, or -6 in the list.

However, some serious mathematicians may have what they think are 
valid reasons for considering -3 to be a factor of 3....

I hope this helps.  And I hope you get more responses from other 
doctors here.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2003 at 01:59:33
From: Doctor Achilles
Subject: Re: Are Negative Numbers Factors of Other Numbers

Hi Adam,

Thanks for sending an interesting question.

Of course, this all depends on how you define "factor" and "multiple." 
I think that the answer Dr. Greenie sent you is excellent, and I found 
it informative.  I am not sure that I disagree, but I'd like to 
suggest another possible way to answer your question that uses a 
different set of definitions.

First of all, I agree with Dr. Greenie that multiples are given over 
all the integers.  That is: the multiples of a number are all the 
numbers you can get by multiplying that number by an integer.

Now, you can define factors as only positive integers. This is in many 
ways simpler than allowing negative factors.

However, another possibility is that because factors and multiples 
are related concepts, you can define them in terms of each other. So 
the factors of a number would be all the numbers for which it is a 
multiple. So since 3 is a multiple of -3, by definition -3 would be 
a factor of 3.

If you have any other thoughts or questions you'd like to share on 
this matter, please write in.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2003 at 08:46:46
From: Doctor Peterson
Subject: Re: Are Negative Numbers Factors of Other Numbers

Hi, Adam.

My opinion is like the others you have had, but with a slightly 
different twist.

I have come to see that the definition of any word, whether in 
ordinary English or in mathematics, is dependent on context. Sometimes 
we are talking only about natural numbers, and so we consider only 
positive factors (or even multiples). That is generally true when we 
are looking at primes, for example; so if you are asking about the 
meaning of "factor" _within a definition of prime numbers_, then I 
would say only positive factors are considered (though it is best if 
that is explicitly stated). There may be other contexts in which it 
makes perfectly good sense to talk about negative factors; certainly, 
for example, in factoring a polynomial we don't worry about whether 
the factors are positive.

So I have to respond with a question: what is the context of your 
question?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2003 at 10:40:49
From: Doctor Ian
Subject: Re: Are Negative Numbers Factors of Other Numbers

Hi Adam,

As Dr. Peterson points out, context is often the key to resolving
questions like this. So perhaps the place to start is with your
assertion that you teach 'test taking math'. In that context, the
'right' answer is the one that the test writers are most likely to
think is correct, regardless of what mathematicians are doing.  

I believe that most test writers would say that 'factor' means
'positive integer divisor', in which case 1 and 3 are factors of 3,
but -3 is not. However, your best bet would be to contact the creators 
of the tests that your students are preparing to take. 

Moving beyond the context of test-taking, mathematicians are often
motivated to eliminate special cases, which is how we end up with
definitions like 

  0! = 1

as well as concepts like 'degenerate' shapes (e.g., a point as a
circle with radius zero, a line segment as a rectangle with width
zero, and so on). In general, the more inclusive you can make your
categories, the more power you can get from theorems about members of
those categories. 

For example, the other day some of us at the Math Forum were trying to
come up with a sensible answer to the question: Is a parallelogram a
trapezoid?  According to the traditional definition, the answer is no:
a trapezoid has exactly one pair of parallel sides, while a
parallelogram has two pairs. 

However, some textbooks are moving toward a more inclusive definition, 
i.e., a trapezoid has _at least_ one pair of parallel sides. Under 
that definition, two pairs is at least one pair, and so a 
parallelogram is also a trapezoid, in the same way that a square is
also a rectangle. 

As far as we could tell in about a half-hour of discussion, there are
no theorems or formulas about trapezoids that become untrue when you
apply them to parallelograms. This is hardly conclusive, but there
seems to be no harm in using the more inclusive definition, and there
may be some benefit to simplifying the categorization structure for
quadrilaterals.  

So one way to approach the question is this: Are there accepted 
theorems about factors that are true if we consider factors to be
positive integers, that become untrue if we change the definition to
include negative integers? I can't think of any offhand, but I'll
leave that question for other math doctors who might be able to
provide a more comprehensive answer. 

Finally, I'll muddy the waters a little by pointing out that when we
move from arithmetic to algebra, we use the term 'factoring' to
describe operations like 

   -3x - 9x^2 = -3(x + 3x^2)              'Factor out -3'

              = -3x(1 + 3x)               'Factor out x'

So if pressed for an opinion, I'd say: As a mathematician, I'd accept
-3 as a factor of 3. But as a student taking a standardized test, I'd
say that 1 and 3 are the only factors of 3, because such tests are
ultimately about compliance, rather than correctness. 

I hope this helps!

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2003 at 16:34:43
From: Adam
Subject: Thank you (Are Negative Numbers Factors of Other Numbers)

Dear Doctors Greenie, Achilles, Peterson, and Ian:

Thank you so much for the erudite and rapid replies. I got more than I 
asked for and much to ponder. 

Dr. Peterson, the context was standardized tests. After reading all 
your replies I can say, pretty confidently, that the organizations 
that write the tests I coach would either say that factors are not 
negative or they would simply punt by not testing the issue. 

Thank you all again. 
Adam