1. Can all fractions be factored? (not by itself and 1 )
I try 1/3 i thought it can't be factored but was wrong.

2.

3. Could you be more specific as to what you are trying to do?

4. What does it mean to factor 1/3?

5. Originally Posted by TheObserver
What does it mean to factor 1/3?
What i mean is to find 2 or more numbers to multiply to get the product 1/3 .
like factoring: x2 - y2 with (x+y)(x-y) or 12 with 3 X4
:4/9 = 2/3 X 2/3 (what i really don't know is if fractions is considered a numbers )
:1/3=2/5 and 5/6 are factors so we can say
there will be no such thing as unfactorable fractions or decimals?

These brings me to another wonderings.. are base10 primes same as base3 primes?
Cant help to be a beginner, just want some mathematical adventures.

Thanks

6. Originally Posted by TimeSpaceLightForce
What i mean is to find 2 or more numbers to multiply to get the product 1/3 . like factoring: x2 - y2 with (x+y)(x-y) or 12 with 3 X4 :4/9 = 2/3 X 2/3 (what i really don't know is if fractions is considered a numbers ):1/3=2/5 and 5/6 are factors so we can say there will be no such thing as unfactorable fractions or decimals?
As far as I know, "prime" and "composite" numbers refer to the set of natural numbers and their factorization. So if there does exist a standard concept concerning the primality of rationals, it'd be completely new to me. Sounds interesting!

It seems that you can rewrite any real number as the product of two or more real numbers. I don't think this counts as "factoring" in the sense of prime and composite numbers since it is not performed for natural numbers over the set of naturals. So any concern of primality is probably irrelevant, but other approaches to the subject might exist, which I'm keen to see.

7. Umm, well the title does not seem accurately to reflect the question!

Lets see - consider the real numbers. This is field and is generally denoted by . And as such is closed under arithmetic multiplication, that is for any and all then . Thus any be factored as

Now suppose the integers, usually denoted by . This is not a field (if you must know it is a ring), but one defines another field whose elements are of the form for any and all . This field is in fact a subfield of the real numbers, and therefore each element must be "factorizable".

Now to the "prime" part. There is a fundamental theorem, called the prime factorization theorem that states that any can be factored as a product of a unit () and a unique set of primes.

Question for reader: since in the are subject to this theorem, does this mean that all elements in also have a unique prime factorization?

And before you smile indulgently and think "there he goes again, being his usual unlovable self" remember that mathematics is ALL ABOUT PROOF

8. Originally Posted by Guitarist
Question for reader: since in the are subject to this theorem, does this mean that all elements in also have a unique prime factorization?
Ah, that's much clearer! I think the answer is no; although I don't think I can prove it. Not rigorously anyway...

9. No, not all fractions can be factored. That is simple, factorization implies the use of normal numbers. And fractions are not part of normal numbers. They are real, just as prime numbers cannot be divided. two prime numbers cannot be factorized with normal numbers.
But sure, with real numbers anything can be factorized. except zero.

10. Originally Posted by Guitarist
Question for reader: since in the are subject to this theorem, does this mean that all elements in also have a unique prime factorization?
Er... As with most trivial subjects, this is far above me. However, I can throw in that certain algebraic properties of sets do not necessarily translate to their subsets, which may be the case here. For example, is closed under subtraction, however , despite being a subset of the reals, is not (whereby if the subtrahend is larger than the minuend, where both operands are members in the set of naturals, then the difference is negative -> therefore, N is not closed under subtraction).

11. Your bigger problem here is that is general defined as a set of equivalence classes and thus you have to worry about your representations and that your choices do not matter in your proof.

12. Here is my homework on the subject.

The factorization of integer numbers can formalized it in this way :

Let with n > 1.
Then, it exists p1 ... pn and k1 ... kn such as
Where
• are prime numbers,
• such as then
This formalization can be easily extended to rational numbers simply by having the ki exponents as signed integers and , with the classical convention
Thus, any rational number z may be written as
Where
• are prime numbers,
• such as then
It is trivial that such a factorization can always be done using the factorisation of integers.
It is easy to prove that this factorization is unique (but it takes some time to enter it with TexT).

13. As all things, one must be given to the other two in strenthing the edges to split. Thus, when the three is split, thus out of sheer force of things, the 1 missing will manefest itself as whole of the 3 thus the 3 becoming 2 then 1. That is a basic of survival becoming annihalation. Or vise versa. If you remember what I did out of myself to accomdate 2 deleted by gamma radiation due to hypnosis, all 3 were wiped out. Thus the unwanted one was taken out and the 2 strenthened themselves. To become 1 then 2 then 3 but if the third is not to be used but to separate then the whole of the is erased and started with the structure of the part. I don't know how to solve algebra because of the trauma I endured, but I can bring it out into social forms and then you can put it into mathematics.

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