Talk:Rational number

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Needs a history section. All sections need expanding with prose to accomany algebra. Tompw

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	The following comments have been left for this page: Needs a history section. All sections need expanding with prose to accomany algebra. Tompw (edit)

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

1 Unique Form
2 Construction
3 Other metrics
4 nominator and denominator
5 work
6 Request addition
7 Incorrect characterization of the rationals
8 Category:Set theory
9 Define Z Earlier
10 1/49, 1001/997002999, etc
11 no simplest form for 0?
12 Recurring Decimals?
13 Should Z be used two times in the definition of Q

[edit] Unique Form

In the given form the definition of Q seems to contract the text in the introduction. In Q 1/2 and 2/4 are two distinct elements, while the text suggests that they are two representations on the same rational number and should hence be the same element in Q. A better way to define this might be two define a set like your Q (but called differently) then define equivalence classes on it with all representations of the same number and make Q the set of these classes. 212.201.44.249 16:52, 28 October 2007 (UTC)

I'm not completely sure what you mean, since your wording is a bit unclear, but I think you are misunderstanding the concept of rational numbers, mistaking one particular way of representing a rational quantity for the quantity it represents. "¹⁄₂" and "²⁄₄" are two distinct fractions associated with the same rational number, that is, with the same element from the set of rational numbers $\mathbb{Q}$ ; two different but nonetheless equivalent ways of representing a given rational number as a quotient of integers, which is just one of several possible ways of referring to a rational number. Each rational number is not the same as each fraction; each rational number, as the formal construction says, is an equivalence class of fractions, that is, it is the quantity underlying a certain set of equivalent fractions. For example, the rational number represented in decimal as "0.5" is the same quantity, the same unique rational number as the one represented by any of the fractions in the equivalence class {¹⁄₂, ²⁄₄, ³⁄₆, ⁴⁄₈...}. That is, each rational number corresponds to one class of equivalent fractions, not to one individual fraction. Each rational number has an infinite number of possible representations as fractions, but only one (or at most two) possible representations as a positional string of digits in a given radix (e.g., in decimal, said number can be represented either as "0.500000..." or as "0.499999..."; and the rational number corresponding to the fractions ¹⁄₃, ²⁄₆, etc. has the unique decimal positional representation 0.333333...). A completely unique way of representing the absolute value of each non-zero rational number is possible in an analogous way to how the natural numbers can be uniquely represented by their prime factorization, using a extended form of prime factorizations where the exponents are not limited to $\mathbb{N}$ but to $\mathbb{Z}$ . For example, the unique prime factorization of the rational number representable in decimal form as "0.500000..." or as "0.499999...", and in fraction form as "¹⁄₂", "²⁄₄", "³⁄₆", ..., is 2⁻¹; that of "1.5", "1.49", "³⁄₂", "⁶⁄₄", etc., is 2⁻¹×3; and so on. That is, the numeric value of every non-zero rational number can be uniquely identified by an ordered set of signed integers, representing the exponents of successive primes in its unique prime factorization:

{0, 0, 0, 0, 0, ...} = 1 = ¹⁄₁ = ²⁄₂ = ³⁄₃ = ... etc.

{−1, 0, 0, 0, 0, ...} = 0.5_dec = 0.49_dec = 0.1_bin = 0.8_hex = 0.6_doz = ¹⁄₂ = ²⁄₄ = ³⁄₆ = ... etc.

{1, 0, 0, 0, 0, ...} = 2 = ²⁄₁ = ⁴⁄₂ = ⁶⁄₃ = ... etc.

{0, −1, 0, 0, 0, ...} = 0.3_dec = 0.01_bin = 0.5_hex = 0.4_doz = ¹⁄₃ = ²⁄₆ = ³⁄₉ = ... etc.

{0, 1, 0, 0, 0, ...} = 3 = ³⁄₁ = ⁶⁄₂ = ⁹⁄₃ = ... etc.

{1, −1, 0, 0, 0, ...} = 0.6_dec = 0.10_bin = 0.A_hex = 0.8_doz = ²⁄₃ = ⁴⁄₆ = ⁶⁄₉ = ... etc.

{−1, 1, 0, 0, 0, ...} = 1.5_dec = 1.49_dec = 1.1_bin = 1.8_hex = 1.6_doz = 1¹⁄₂ = ³⁄₂ = ⁶⁄₄ = ⁹⁄₆ = ... etc.

{−2, 0, 0, 0, 0, ...} = 0.25_dec = 0.249_dec = 0.01_bin = 0.4_hex = 0.3_doz = ¹⁄₄ = ²⁄₈ = ³⁄₁₂ = ... etc.

{2, 0, 0, 0, 0, ...} = 4 = ⁴⁄₁ = ⁸⁄₂ = ¹²⁄₃ = ... etc.

{−2, 1, 0, 0, 0, ...} = 0.75_dec = 0.749_dec = 0.11_bin = 0.C_hex = 0.9_doz = ³⁄₄ = ⁶⁄₈ = ⁹⁄₁₂ = ... etc.

{2, −1, 0, 0, 0, ...} = 1.3_dec = 1.01_bin = 1.5_hex = 1.4_doz = 1¹⁄₃ = ⁴⁄₃ = ⁸⁄₆ = ¹²⁄₉ = ... etc.

{0, 0, −1, 0, 0, ...} = 0.2_dec = 0.19_dec = 0.0011_bin = 0.3_hex = 0.2497_doz = ¹⁄₅ = ²⁄₁₀ = ³⁄₁₅ = ... etc.

{0, 0, 1, 0, 0, ...} = 5 = ⁵⁄₁ = ¹⁰⁄₂ = ¹⁵⁄₃ = ... etc.

{1, 0, −1, 0, 0, ...} = 0.4_dec = 0.39_dec = 0.0110_bin = 0.6_hex = 0.4972_doz = ²⁄₅ = ⁴⁄₁₀ = ⁶⁄₁₅ = ... etc.

{−1, 0, 1, 0, 0, ...} = 2.5_dec = 2.49_dec = 10.1_bin = 2.8_hex = 2.6_doz = 2¹⁄₂ = ⁵⁄₂ = ¹⁰⁄₄ = ¹⁵⁄₆ = ... etc.

{0, 1, −1, 0, 0, ...} = 0.6_dec = 0.59_dec = 0.1001_bin = 0.9_hex = 0.7249_doz = ³⁄₅ = ⁶⁄₁₀ = ⁹⁄₁₅ = ... etc.

{0, −1, 1, 0, 0, ...} = 1.6_dec = 1.10_bin = 2.A_hex = 2.8_doz = 1²⁄₃ = ⁵⁄₃ = ¹⁰⁄₆ = ¹⁵⁄₉ = ... etc.

{2, 0, −1, 0, 0, ...} = 0.8_dec = 0.79_dec = 0.1100_bin = 0.C_hex = 0.9724_doz = ⁴⁄₅ = ⁸⁄₁₀ = ¹²⁄₁₅ = ... etc.

{−2, 0, 1, 0, 0, ...} = 1.25_dec = 1.249_dec = 1.01_bin = 1.4_hex = 1.3_doz = 1¹⁄₄ = ⁵⁄₄ = ¹⁰⁄₈ = ¹⁵⁄₁₂ = ... etc.

{−1, −1, 0, 0, 0, ...} = 0.16_dec = 0.0010_bin = 0.2A_hex = 0.2_doz = ¹⁄₆ = ²⁄₁₂ = ³⁄₁₈ = ... etc.

{1, 1, 0, 0, 0, ...} = 6 = ⁶⁄₁ = ¹²⁄₂ = ¹⁸⁄₃ = ... etc.

{−1, −1, 1, 0, 0, ...} = 0.83_dec = 0.1101_bin = 0.D5_hex = 0.A_doz = ⁵⁄₆ = ¹⁰⁄₁₂ = ¹⁵⁄₁₈ = ... etc.

{1, 1, −1, 0, 0, ...} = 1.2_dec = 1.19_dec = 1.01_bin = 1.3_hex = 1.2497_doz = 1¹⁄₅ = ⁶⁄₅ = ¹²⁄₁₀ = ¹⁸⁄₁₅ = ... etc.

{0, 0, 0, −1, 0, ...} = 0.142857_dec = 0.001_bin = 0.249_hex = 0.186A35_doz = ¹⁄₇ = ²⁄₁₄ = ³⁄₂₁ = ... etc.

{0, 0, 0, 1, 0, ...} = 7 = ⁷⁄₁ = ¹⁴⁄₂ = ²¹⁄₃ = ... etc.

And so on. Add a way to represent the sign and there you have a unique way to identify each non-zero element of $\mathbb{Q}$ , without needing to mess with esoteric stuff like equivalence classes that only unnecessarily complicates things (although the apparently simple introduction of rational numbers as "a number which can be expressed as a ratio of two integers" is probably the main cause of misunderstanding, because inadvertent people may take it to mean that "a rational number is the same as a fraction", which is not). Uaxuctum 04:41, 30 October 2007 (UTC)

I am not completely sure what you mean, since your wording is very verbose :). In any case the article defines a set Q which basically contains all pairs in integers and calls that the rational numbers. As I said and you seem to agree this is bad, since this Q includes a separate element for each representation of a rational number. Hence Q should be called the set of fractions. This leaves us with the problem of how to define Q correctly, and I believe introducing equivalence of fractions that represent the same rational number and making these equivalance classes elements of Q is by far the easiest way to do it. That is probably also why it is used in the Formal definition section of the article, but unfortunately the introduction clearly says something different and hence we have a contradiction. 212.201.44.249 13:13, 30 October 2007 (UTC)

[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 d_p(x, y) = |x - y|_p defines a metric on Q. The metric space (Q, d_p) 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)

under this  order, isn't the rationals not dense? is it possible for a set to be dense depending on the order? 66.160.66.86 (talk) 16:19, 22 January 2008 (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 10^k we can make it so that 10^ka 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 ≤ 10^l - 1 and 10^ka = A + N 10^-l + N 10^-2l + N 10^{-3 l} + ... -- this is the geometric series and it equals A + N/(10^l - 1), so a is rational.

Now for the opposite, suppose a is rational. Note that by multiplying with 10^k for suitable k, we can arrange for the denominator q of 10^ka to be relatively prime with 10. Then we can write 10^ka = A + x/q for some integers A and x with 0 ≤ x < q. Let r be the smallest number so that 10^r $\equiv$ 1 (mod r) and write N = x (10^r - 1)/q. Then 0 ≤ N < 10^r - 1 and 10^ka = A + N/(10^r - 1). By expanding to the geometrical series we see that 10^ka 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).

[edit] no simplest form for 0?

In the second paragraph of this article it says that every non-zero rational number can be written in exactly one simplest form in which the numerator and denominator have no common divisor except 1 and the denominator is positive. Why does a rational number have to be non-zero to be written in this form? If 0 is written as 0/1, then the only common divisor of the numerator and denominator is 1, and the denominator is positive, right?

[edit] Recurring Decimals?

Does this article mention anywhere that recurring decimals are rational? I looked pretty hard, but couldn't see it. I know that to someone who has alot of mathematical expierience it would be relatively obvious, but from reading the page on '0.999... = 1' arguments it seems that it is unclear to the majority that recurring decimals are rational. --Cmdr Clarke (talk) 20:32, 18 December 2007 (UTC)

[edit] Should Z be used two times in the definition of Q

In this article Q is defined as:

$\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\}$

But I would say it would be sufficiently to define it as:

$\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{N} \right\}$

I think it's sufficient to have a nominator which can be positive and negative and a denominator which can only be positive. Does anyone disagree? Snailwalker | talk 14:45, 7 March 2008 (UTC)

You would still need the $n \ne 0$ , as N is often defined to include 0. But I don't think it's a good idea anyway: the point is that Q is the field of fractions of Z. N doesn't come into it. --Zundark (talk) 18:57, 7 March 2008 (UTC)

Nah the definition of $\mathbb{N}$ is the set of nonzero positive integers so $n\neq 0$ is not needed. I just think the second definition is more beautiful and more strict. Snailwalker | talk 11:24, 8 March 2008 (UTC)

That's a definition of N, but it's not the definition. I still don't see why you want to introduce N here, but I agree with your point that there's no need to use Z twice: $\mathbb{Q} = \left\{\frac{m}{n} \,\Big|\, m,n \in \mathbb{Z} \text{ and } n \ne 0 \right\}$ . --Zundark (talk) 12:02, 8 March 2008 (UTC)

Hmm it just seems redundant to use Z twice since all rational numbers can be reached by using N and Z. But I can follow your argument about field of fractions. Snailwalker | talk 14:22, 8 March 2008 (UTC)