Talk:Rational number

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Mathematics grading: B Class High Importance  Field: Number theory
Needs a history section. All sections need expanding with prose to accomany algebra. Tompw A vital article.

Removed the following as it seems to me to be too formal and technical for an encyclopedia entry. I've preserved it here for discussion. hawthorn

Contents

[edit] Construction

Mathematically we may define them as an ordered pair of integers (a, b), with b not equal to zero. We can define addition and multiplication upon these pairs with the following rules:

(a, b) + (c, d) = (a × d + b × c, b × d)
(a, b) × (c, d) = (a × c, b × d)

To conform to our expectation that 2/4 = 1/2, we define an equivalence relation ~ upon these pairs with the following rule:

(a, b) ~ (c, d) if, and only if, a × d = b × c.

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.

We can also define a total order on Q by writing

(a, b) ≤ (c, d) if, and only if, ad ≤ bc.

Removed the following as it seems more suited to a page on p-adic numbers. Maybe someone can find it a new home. hawthorn

[edit] Other metrics

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field: let p be a prime number and for any non-zero integer a let |a|p = p-n, where pn is the highest power of p dividing a; in addition write |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p. Then dp(x, y) = |x - y|p defines a metric on Q. The metric space (Q, dp) is not complete, and its completion is given by the p-adic numbers.

It could be argued that the Egyptian fraction stuff deserves its own page. however I've left it for now. hawthorn


I'm restoring both. they're important. -- Tarquin 09:53 26 Jun 2003 (UTC)

Couldn't you have done it without throwing out the baby with the bath water! An encyclopedia entry on the rational numbers shouldn't have to start off in such an abstract and formal way- they are just fractions for goodness sake! Whay can't we say so right from the start. Even a non-mathematician can understand this concept. I'm in favour of keeping it as general as possible as long as possible. Move the formalism to the end.

I disagree that the exised stuff is all that important. The first extract is pretty much the field of fractions construction in the special case that the ring is the ring of integers, which seems like trying to sink a tack with a sledgehammer to me. The second stuff on the p-adic metric and p-adic numbers - well it just isn't what I'd expect to find on a page on the rationals is all. hawthorn

I agree that entries shouldn't start off in an abstract or formal way. Pizza Puzzle


Sorry. I was a bit hasty. You're right, we should start with a layperson-friendly overview. But after the first screen-full of text, it's fine to get technical! -- Tarquin 21:38 26 Jun 2003 (UTC)

[edit] nominator and denominator

Perhaps mentioning the formal names of nominator and denominator would be in order in this document, just to let people know how the numbers above and below dividor line are called. It is useful information especially to people that are non-native english speakers (like me).

The terminology numerator and denominator is explained in the article on vulgar fractions, which is prominently linked to from the present article. That doesn't mean the same information couldn't be repeated here, but then again the case could be made that vulgar fraction as a whole is redundant. --MarkSweep 20:01, 11 Dec 2004 (UTC)

[edit] work

this don't explain muh what i need to know!!!!!! u need more explaination to adda to it !!!!!!!!!

[edit] Request addition

I would like to request that information as to how Cantor proved the cardinality of the rationals to be \aleph_0.

Thanks in advance.Guardian of Light 23:10, 9 December 2005 (UTC)

Quick answer, form the sequence (1/1), (1/2, 2/1), (1/3, 2/2, 3/1), (1/4, 2/3, 3/2, 4/1), ... (brackets to make the ordering more understandable, remove them at the end so to speak). Now cross off all the fractions that aren't in simplest terms. You now have an ordered sequence containing all the rationals, which is what you need to prove that result. The long answer is a lot more rigorous. Confusing Manifestation 15:50, 17 February 2006 (UTC)

[edit] Incorrect characterization of the rationals

I removed the following:

As a totally ordered set, the rationals are uniquely characterized by being countable, dense (in the above sense), and having no least or greatest element

This is not correct; for instance the field Q(φ), where φ is the golden ratio, is also totally ordered, dense, countable, and has no least or greatest element. Gene Ward Smith 00:38, 14 May 2006 (UTC)

Yes, and hence Q(φ) is isomorphic as a totally ordered set to the rationals. I've restored the characterization. —Blotwell 16:40, 14 May 2006 (UTC)

[edit] Category:Set theory

Why is this there? Rational numbers are set theory only in the sense that all of mathmatics is set theory. Gene Ward Smith 00:43, 14 May 2006 (UTC)

[edit] Define Z Earlier

I think it would be a Good Thing if someone defined "Z" as a part of the definition of "Q" found in the very first section. Something on the order of "where Z is the set of integers" would be useful for those who don't immediately recognize it. I'd do it, but this isn't my field... Peter Delmonte 00:24, 25 October 2006 (UTC)

I agree, and I changed the article accordingly. -- Jitse Niesen (talk) 01:02, 25 October 2006 (UTC)

[edit] 1/49, 1001/997002999, etc

I feel a bit foolish by asking this, but, ¿aren't rational numbers suposed to have a structure like: {number}.{sequence of numbers}{period repeated infinite times}, such as 3.23777877787778777877787778...? I'm a bit confused because this numbers (1/49, 1001/997002999...) show funny series instead of a period.

Victorlj92 16:27, 25 October 2006 (UTC)

They do have a period. E.g., 1/49 has the repeating sequence 020408163265306122448979591836734693877551. --Zundark 18:12, 25 October 2006 (UTC)
And there's a nice proof for it. If you want I can place it somewhere. --CompuChip 18:33, 22 November 2006 (UTC)
And here is is:
Theorem: The decimal expansion of the real numer a is periodic iff a \in \mathbb{Q}.
Proof: Suppose that the expansion fo a is periodic, so of the form {number}.{sequence}{repeating sequence}. By multiplying with a sufficiently large power 10k we can make it so that 10ka is purely periodic, so of the form {number}.{repeating sequence}. Suppose the minimum period length is l, then there exist integers A and N such that 0 ≤ N ≤ 10l - 1 and 10ka = A + N 10-l + N 10-2l + N 10-3 l + ... -- this is the geometric series and it equals A + N/(10l - 1), so a is rational.
Now for the opposite, suppose a is rational. Note that by multiplying with 10k for suitable k, we can arrange for the denominator q of 10ka to be relatively prime with 10. Then we can write 10ka = A + x/q for some integers A and x with 0 ≤ x < q. Let r be the smallest number so that 10r \equiv 1 (mod r) and write N = x (10r - 1)/q. Then 0 ≤ N < 10r - 1 and 10ka = A + N/(10r - 1). By expanding to the geometrical series we see that 10ka has a purely periodic expansion, hence a has a periodic expansion (after a sequence of k numbers the same block repeats every r numbers). Q.E.D.
P.S. In fact you can prove that the expansion is purely periodic iff a is rational and it's denominator is relatively prime with 10. In the general case that a is rational one can say something about the (minimal) period length of the expansion, namely the order of 10 in the unit group (\mathbb{Z}/q\mathbb{Z})^* of the integers modulo q, where q is the denominator of a with all factors 2 and 5 divided out (so gcd(q, 10) = 10).
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