That's a statement that goes quite against the claims of the article and I kinda want to ask for a citation ... But I already feel that this discussion is way over my head and I wouldn't understand any journal articles that would clear this up ...
> Mathematicians call this new system a number “ring”; they can create an infinite variety of them, depending on the new values they choose to incorporate.
Isn't this a little wrong?
As far as I remember ring is any set with multiplication, negation, and addition defined so that they satisfy a few conditions. No need for the "+ b * something" part. The usual integer numbers we use form a ring too, as well as booleans.
I might be missing something, and it's irrelevant to the main subject of the article.
"Number ring" has a specific meaning [0]. I am not sure if it is equivalent to the integers adjoined with some element though.
>In fact the usual integer numbers we use form a ring too.
Depending on definitions, the integers can be viewed as the ring of even numbers adjoined with the element 1. This does require that we do not define rings to necessarily contain 1. In my experience this definition is is becoming more of a historical footnote though.
There is also a natural generalization of adjoining multiple elements to the natural numbers (which still results in a ring) In this case, the natural numbers would just be a special case of adjoining 0 elements.
The ring of real numbers, in contrast, cannot be constructed by adjoining any finite set of elements to the integers.
The ring of integers mod n is also a ring, but does not contain the integers as a subring (and therefore cannot be thought of as the integers adjoined with any set (finite or infinite) of elements).
The polynomials with coeficients mod n form a non-finite ring which does not contain the integers.
I suspect the point that the author was attempting to make was just that Z[√5] formed a type of structure that mathematicians are familiar with.
The number system that you generate by adding sqrt(5) to the integers is a ring because it satisfies the definition. And you can create an infinite variety of rings by adding new values to the integers.
It's a ring, but not all rings have to be in that form. I can see that you think they weakly imply the latter, which really is just a bug in our (humans') natural language processing skills. :)
[+] [-] ginnungagap|8 years ago|reply
Those factors are irreducibles, but they aren't primes, which is the reason why the uniqueness of factorization fails.
A nonzero non unit element of a ring is called prime if p|ab implies p|a or p|b.
A nonzero non unit element of a ring is called irreducible if p=ab implies that a is a unit or b is a unit (invertible element).
Primes are irreducibles in an integral domain, but the converse is true in unique factorization domains and Z[√-5] is not one.
[+] [-] wodenokoto|8 years ago|reply
[+] [-] ajuc|8 years ago|reply
Isn't this a little wrong?
As far as I remember ring is any set with multiplication, negation, and addition defined so that they satisfy a few conditions. No need for the "+ b * something" part. The usual integer numbers we use form a ring too, as well as booleans.
I might be missing something, and it's irrelevant to the main subject of the article.
[+] [-] gizmo686|8 years ago|reply
>In fact the usual integer numbers we use form a ring too.
Depending on definitions, the integers can be viewed as the ring of even numbers adjoined with the element 1. This does require that we do not define rings to necessarily contain 1. In my experience this definition is is becoming more of a historical footnote though.
There is also a natural generalization of adjoining multiple elements to the natural numbers (which still results in a ring) In this case, the natural numbers would just be a special case of adjoining 0 elements.
The ring of real numbers, in contrast, cannot be constructed by adjoining any finite set of elements to the integers.
The ring of integers mod n is also a ring, but does not contain the integers as a subring (and therefore cannot be thought of as the integers adjoined with any set (finite or infinite) of elements).
The polynomials with coeficients mod n form a non-finite ring which does not contain the integers.
I suspect the point that the author was attempting to make was just that Z[√5] formed a type of structure that mathematicians are familiar with.
[0] http://mathworld.wolfram.com/NumberRing.html
[+] [-] btilly|8 years ago|reply
The number system that you generate by adding sqrt(5) to the integers is a ring because it satisfies the definition. And you can create an infinite variety of rings by adding new values to the integers.
[+] [-] infinity0|8 years ago|reply
[+] [-] kevinr|8 years ago|reply
(inb4 all the Proust fans downmod me to oblivion)
[+] [-] unknown|8 years ago|reply
[deleted]
[+] [-] Modj|8 years ago|reply
[+] [-] msds|8 years ago|reply