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
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