what does it mean to factor out the greatest common factor
Having just talked nigh definition issues in geometry, I thought a recent, brusque question related to a definition would exist of interest. We know what the Greatest Mutual Divisor (GCD, also called the Greatest Common Factor, GCF, or the Highest Mutual Gene, HCF) of two numbers is; or do nosotros?
Negative GCD?
Here is the question, with an included movie:
Q) In my volume it is written that – "a gcd of 12 and 18 is half dozen. Nosotros detect that -6 also is a gcd of 12 and 18."
Gcd is greatest common divisor and I don't retrieve that -6 is greatest so is the in a higher place line wrong? I think it should say that -6 is simply a common divisor. The picture of the line in my book is given beneath.
Here is a typical definition of Greatest Common Cistron (at an elementary level), from Math Is Fun:
The greatest number that is a factor of two (or more) other numbers.
When we find all the factors of ii or more numbers, and some factors are the aforementioned ("common"), and then the largest of those common factors is the Greatest Mutual Factor.
How could -6 be "greatest" or "largest"?
How about the more than conscientious definition of Greatest Common Divisor from Wikipedia?
In mathematics, the greatest common divisor (gcd) of ii or more integers, which are not all nix, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4.
This definition requires the GCD to exist a positive integer, so this certainly doesn't apply.
An extended definition of GCD
And so what is the book talking about? I pointed out a hint from what was quoted:
I suspect that if you lot showed us the previous paragraph, that would contain the answer to your question.
They say, "… it is largest,only in the sense of (two)". Thus, they take given yous a meaning for "largest" that is not what you are expecting. If they are referring to the line I can half-see just above your clip, which begins with "(2)", that makes sense. You are expecting "largest" in the sense of "greatest integer". They are defining it, I think, such that a divisor is "largest"if any other common divisor is a divisor of this one.
In preparing this post, I searched for the relevant phrase and establish what appears to be a re-create of the textbook (either a different edition, or a related book by the aforementioned authors):
The notation says that (i) r must evenly divide both m and due north, and (2) if any other integer l divides k and n, and sol divides r. That is, any other common divisor is a divisor of r, only as I had guessed. In the example, -half-dozen is a divisor of both 12 and 18, and any other mutual divisor (1, 2, three, 6, -1, -2, -3) is a divisor of -six. So information technology is a "greatest" common divisor in this sense. And this is why the book called it a gcd (non the gcd).
I connected:
This definition is used, for example, in the first section of this page:
https://proofwiki.org/wiki/Definition:Greatest_Common_Divisor
At that place they are defining the GCD for something called an "integral domain" — that is, for any set up of "numbers" that has certain properties that are a generalization of integers, but may not even have an ordering at all, so that you tin can't define "greatest" literally. But this definitiontin be used in that context. They continue to state the usual definition for integers, and afterward discuss its equivalence.
Then your volume goes on to country that if you restrict the gcd to positive integers, it becomes unique, as you expect.
So what is happening here is that this textbook,Challenge and Thrill of Pre-College Mathematicsby 5 Krishnamurthy and C R Pranesachar, similar proofwiki, is taking the concept of the GCD and extending it to provide a definition that doesn't depend on being able to say that one number is greater than another. This extended definition turns out, equally they signal out, to yield more than than 1 GCD (a positive one and a negative one), only when applied to integers, we can adjust the definition to give the expected consequence.
In fact, did you notice the section title and last line in the larger pick I quoted above? What they are about to practise is to apply this new definition of GCD to polynomials rather than to integers — and since polynomials don't take the belongings that any polynomial is either less than, equal to, or greater than whatever other, this new definition is just what they need. So the context tells us all nosotros need to know to brand sense of what they have said.
(When I read to my children when they were young, I would often suspension to ponder with them what might be happening in the story, and 1 of them would say, "But keep reading — it volition tell you!" That is truthful of children'south books, and it is true of math — unless in fact they already told you, and you forgot.)
That was the cease of this exchange; but in preparing this post, I checked to run across if we take e'er discussed this, or whatsoever other, variant definition of the GCD, and found that we have. That commonly happens.
Starting time, the specific subject of the GCD of polynomials underlies the following page, which is at a much higher level than my target audience, then I'll let yous read information technology on your own if you choose:
What Makes Polynomials Relatively Prime?
An LCM question
But a very similar effect with regard to the LCM (Least Common Multiple) arose in the following long discussion:
Least Mutual Multiple with Nada I'm trying to find any reference about the least common multiple of two numbers when one (or both) is zero. Could y'all help me, please?
Doctor Rick answered this question using the standard definition:
No number except aught is a multiple of zip, because aught times annihilation is zip. The only multiple that, say, 0 and five have in common is 0. Thus, if the LCM of 0 and 5 exists at all, it must be 0. We do not count zero as a common multiple. If we did, then nil would be the least common multiple of whatsoever 2 numbers (unless we as well counted negative multiples, in which instance at that place would be no least common multiple of any two numbers). Either nosotros brand an exception in this instance, then that the LCM of zero and any number is zero, or nosotros make no exception, in which case the LCM of nothing and whatever number does not exist. To me it makes more sense to say that the LCM is defined but for positive numbers. See the definition of LCM here: Least Common Multiple - Eric Weisstein, Earth of Mathematics http://mathworld.wolfram.com/LeastCommonMultiple.html Information technology says, "The least common multiple of two numbers a and b is the smallest number m for which there exist POSITIVE integers n_a and n_b such that n_a*a = n_b*b = m." [Accent is mine.] If the LCM of 0 and 5 were 0, we'd have a = 0, b = v, n_a = any number, and n_b = 0 - which is not a positive integer, so information technology fails this definition. Thus, while the definition does not explicitly say that the ii numbers must be positive, this is implied by the definition. I have to ask: Why practise y'all care? Is there a context in which y'all need the LCM of nada and some other number?
"Pops", who had asked the question, and so gave his context:
I'm a computer science professor and I'1000 proposing to the students a programme to obtain the LCM of two numbers. My aim is to explicate the correct answers in all possible "legal" situations.
So what he wanted was just a conventional definition of the LCM that would include all reasonable inputs.
The extended definition of LCM
But 12 years later, a math major names Danny wrote in to object to part of what Doctor Rick said, starting with a different definition of LCM — one that is parallel to the 1 we discussed in a higher place, taking divisibility as a stand-in for "less than":
Doctor Rick states that if we allow 0 to be the LCM of any non-zero number, so 0 volition exist the LCM of all numbers. Where a|c reads "a divides c," the definition of c = LCM{a,b} is: i) a|c and b|c two) if a|d and b|d, then c|d Now suppose that c = LCM{3,2} = 0. And then let d = half dozen. We have 3|d and 2|d. Merely 0 = c does not dissever d, since no number m exists such that g|0 = 6 = d. So in fact, letting 0 be the LCM of a pair of numbers will lead to a contradiction in the usual setting. I believe the author mixed up the usual society relation in the definition of LCM, thinking that a least common multiple must exist less (in numerical value) than other mutual multiples. Danny has missed the fact that the definition Doctor Rick explicitly used does involve "the usual order relation", and that this is the usual definition exterior of abstract math. After Medico Rick pointed this out, Danny gave proficient reasons why his more sophisticated definition helps (bear with him, if y'all are not up to his level):
I believe that this definition of LCM is useful when dealing with group theory, ordered sets in general, and lattices. In particular, it remains divers in dealing with the case of infinite sets, and sets where the natural club has been changed or altered. Also the definition works in the case where LCM is required for two numbers, one of which is zero. Goose egg volition be a common multiple of all pairs of numbers, so using divisibility every bit an order relation 0 becomes the top element of an infinite ordered set similar the naturals with zero. This set then forms a complete lattice with bottom element 1 and tiptop chemical element 0. It is a lattice considering every pair of elements has a to the lowest degree upper bound, and greatest lower spring; and it is complete because the unabridged prepare has a to the lowest degree upper bound, 0, and a greatest lower bound, 1. The LCM of, for case, 0 and 3, will be 0, while the LCM of two and 3 volition be 6. I suspect that the definition you give is a useful version that works well in nearly cases; notwithstanding, perhaps the definition I gave is a fashion that the LCM tin can be more general. Could you give me an example of why your definition might be more useful?
At Doctor Rick'south request, Dr. Jacques answered, first agreeing with Danny'southward determination:
In fact, I would say that in that location is no problem in because that 0 is a common multiple of a pair of integers: after all, it is a multiple of each of them.... The point is that it is not the to the lowest degree such multiple, where "least" must be understood with respect to the fractional ordering induced by divisibility, since this is the pregnant used implicitly in "to the lowest degree mutual multiple." The problem is maybe in the "implicit" aspect. In any case, I think you know all this.
That is, to a mathematician, "least" here doesn't mean what we normally call back it ways, and that is of import.
He went on to talk about why we have two different definitions, which is something mathematicians in isolation can easily forget:
The concluding question is why we have the "simpler" definition. It is already taught in elementary school, at a time where the general definition would exist out of achieve. Even at that stage, the definition can be very useful in applied applications (like adding fractions); the same is true for greatest common divisor (GCD). For many people, that is about all the mathematics they will need. I think that, if y'all want to larn math seriously, you have first to unlearn many of the (false or incomplete) things you were taught in school, because information technology was not possible at that time to requite you strictly right and complete definitions.
Why we need two definitions
I then joined the conversation to sum up, since this thing of definitions has been of interest to me:
I'd like to tie things upward with a comment on the big film. It is quite common for a concept to start with a unproblematic idea and a "naive" definition, and later on be generalized. In this particular case, as has been mentioned, both definitions MUST go on in apply in unlike contexts, because only the naive initial definition is understandable by well-nigh people who need the concept, while only the sophisticated general definition applies to cases beyond natural numbers. Most online sources, including Wikipedia and MathWorld, give the definition applicable to natural numbers. This is the advisable definition for utilize as the Least Mutual Denominator of fractions, since denominators can't exist zero. It as well fits the name: information technology is exactly what it says, the To the lowest degree (positive integer) multiple of the given numbers. This definition is undoubtedly the source of the entire concept. Some sources give that same definition, but then add that if 1 of the numbers is aught, the LCM is 0 (with or without explaining why this extension makes sense). This is probably the answer that should have been given to the original question (from a computer science context, just looking for a reasonable value to give in this case).
We take 2 very dissimilar contexts: ordinary arithmetics with whole numbers, and abstract algebra (which is algebra, or in this case number theory, applied to all sorts of entities that may or may not be numbers). Each has its ain needs; if either forced its definitions on the other, it would fail.
This simple definition had to be extended in social club to cover other contexts, equally you noted. Every bit Physician Rick pointed out, just applying the bones definition to such cases would not work. The definition you lot are using is the issue of a search for an appropriate extension, and is based on a theorem that is true in the natural number context, and turns out to be usable as a definition in the general case. It certainly would not be appropriate to start with this every bit a definition in elementary grades, just it would be possible to brand the transition before getting to abstract algebra -- though it would never be especially helpful in understanding the concept in its everyday applications.
This is a mutual way in which definitions are extended: Theorems in ane mathematical realm become definitions or axioms in a larger realm, where the original foundations accept been left backside.
I had trouble searching for a source for your definition, since the elementary definition is overwhelmingly common. Every bit I mentioned, Wikipedia gives the uncomplicated definition, only adds https://en.wikipedia.org/wiki/Least_common_multiple Since division of integers past zip is undefined, this definition has meaning only if a and b are both different from zero. Still, some authors ascertain LCM(a,0) every bit 0 for all a, which is the result of taking the LCM to be the least upper bound in the lattice of divisibility. This terminal thought leads to your definition, which is given at the bottom of the page in defining the lattice of divisibility: The to the lowest degree common multiple tin can be defined mostly over commutative rings as follows: Permit a and b exist elements of a commutative ring R. A common multiple of a and b is an element thousand of R such that both a and b split up thousand (i.e., there exist elements ten and y of R such that ax = m and by = k). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other mutual multiple n of a and b, m divides n. In general, two elements in a commutative band can have no least common multiple or more than than one. Still, any two least common multiples of the aforementioned pair of elements are associates. In a unique factorization domain, whatsoever two elements have a least common multiple. In a principal ideal domain, the to the lowest degree common multiple of a and b can be characterised equally a generator of the intersection of the ethics generated by a and b (the intersection of a collection of ideals is always an platonic).
If y'all live in Danny's world of higher math, yous may empathise this. If not, just know that up in that location where the air is thin, the ideas of GCD and LCM change their foundations and look very unlike, but, having undergone a reconstruction, go along to exist useful in much the same ways. And the conclusion (that an LCM tin be null) filters back down to the mundane world.
And so, summing things upwards, the definition Md Rick gave is very useful in the everyday world; yours is useful in higher-level mathematics -- and both can peacefully coexist because they requite the same issue where both use. Both are "the correct definition" inside their ain context.
Source: https://www.themathdoctors.org/greatest-common-divisor-extending-the-definition/
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