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alternative proof that $\sqrt{2}$ is irrational

Major Section: 
Reference
Type of Math Object: 
Proof

Mathematics Subject Classification

11J72 no label found12E05 no label found11J82 no label found13A05 no label found

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Hi, are there more people thinking that non-real complex numbers (in Continental Europe: imaginary numbers) are irrational -- because they are not rational? I think such a terminology is exceptional.
Jussi

An irrational number is any real number that is not a rational number. This definition can be extended to complex numbers, but I think there's no need to do this.

But if you have to do this, then I suppose that irrational number is any complex number that is not rational. That's how the definition works:

set of irrational numbers is a complement (in any field of characteristic 0) of set of rational numbers.

joking

Jussi, I just noticed that PM currently has a definition that agrees with your usage, not mine:

"An irrational number is a real number which cannot be represented as a ratio of two integers."

I find this definition ***highly*** unusual, as every basic algebra textbook that I have looked at classifies nonreal numbers as irrational. I am extremely curious as to why akrowne chose to define irrational numbers this way. I will file a correction to "irrational" in hopes that, even if akrowne chooses not to change the PM definition, he at least mentions the alternative definition.

I am sorry that I did not notice this earlier.

Warren

Pi (approximately 3.14159...) is irrational but still real, right? What is i * pi? Irrational and imaginary? What about pi + i? Irrational and complex?

I've never seen a definition that addresses non-real numbers as either rational or irrational. I think the definition as written is just fine, and standard.

Are there any other entries in PM that rely on the proposed broader definition (or that are incorrect because of this narrower definition)?

Roger

By the way, the Springer on line encyclopedia is curiously indeterminate about this. They say that an irrational is a "number that is not a rational number" but go on to discuss it in terms of a geometric length incommensurate with 1. All the examples are on the real line.

Mathworld simply does not say, but again all the examples are on the real line.

> By the way, the Springer on line encyclopedia is curiously
> indeterminate about this. They say that an irrational is a
> "number that is not a rational number" but go on to discuss
> it in terms of a geometric length incommensurate with 1. All
> the examples are on the real line.

I think I might replace the word "curiously" with "horribly" there. In some sense, there's no such thing as a number. One would be hard-pressed to find a common definition which included all objects that we've come to know as numbers.

In the particular case, I think it's fairly well agreed upon that without any preceding text to the contrary, an irrational number is a real number that is not rational. If pressed, I would call e*i complex irrational.

Cam

> But if you have to do this, then I suppose that irrational
> number is any complex number that is not rational. That's
> how the definition works:
>
> set of irrational numbers is a complement (in any field of
> characteristic 0) of set of rational numbers.

I agree with the first (unquoted) part of your post, but I think this last definition is unsatisfactory -- What are the rational numbers in an arbitrary field of characteristic zero (i.e. what are the integers)?

Cam

> Mathworld simply does not say, but again all the examples are on the real line.

The second sentence of the Mathworld article:
"Irrational numbers have decimal expansions that neither terminate nor become periodic."

Such numbers must be real.

> I've never seen a definition that addresses non-real numbers as either rational or irrational.

I have to say the same as Roger -- during about 47 years I have worked with higher mathematics, I have never seen or heared that some non-real complex number could be called irrational (such numbers are called in Europe imaginary).

Historically, it's clear that irrational and rational numbers are only real (cf. e.g. http://members.aol.com/jeff570/r.html). In Wikipedia (http://en.wikipedia.org/wiki/Irrational_number), irrational is real.

> Are there any other entries in PM that rely on the proposed broader definition (or that are incorrect because of this narrower definition)?

Many entries narrower, e.g. http://planetmath.org/encyclopedia/TheoryOfRationalAndIrrationalNumbers....

Jussi

mathcam asks: I agree with the first (unquoted) part of your post, but I think this last definition is unsatisfactory -- What are the rational numbers in an arbitrary field of characteristic zero (i.e. what are the integers)?

Let K be a field of characteristic zero. Then there is a "one" element in K, call it 1, and n times 1 = 1+...+1 (n times) is not zero for any natural n (because of the assumption on the characteristic). The numbers n x 1 are your integers. Also, since K is a field, every n x 1 (not zero) has an inverse which we call 1/n. Thus, there is a copy of Q (or isomorphic to Q) inside K.

Alvaro

I will definitely edit my entry according to the PM convention, as most everyone who has contributed to this thread seem to support it; however, I still find it very odd to use the convention that "irrational" and "not rational" do not mean the same thing.

Warren

Leaving aside issues of what is correct or standard terminology,
I like to use the word "rational" to describe complex numbers whose
real and imaginary parts are rational numbers. Likewise, I would
call elements of algebras such as quaternions, polynomials, and
matrices rational when they have all rational coefficients. The
unifying theme here is that the rational elements form a subalgebra
which is obtained by starting with the generators of the algebra and
applying the basic operations of addition, subtraction, multiplication, and division a finite number of times (i.e. excluding
infinite sums and such operations).

A lot of the issue here has to do with the nature of this website, in
particular the fact tht it is being written as a collection of more
or less independant entries. For instance, if I was writing a book
or even an article in which polynomials whose coefficients were
rational numbers or complex numbers whose real and imaginary parts
were rational numbers played an important role, I would simply
define the term "rational polynomial" or "rational complex number"
suitably and not consider the matter further because this use of the
terminology would not conflict with standard usage but be an
extension of the term "rational number" which would be of use in the
context of my work. However, here, the usual situation is for a
trerm to be defined in one entry and used in a differrent entry
which makes for a tricky situation when dealing with non-stnadard
extensions of standard terminology.

Jussi (pahio) wrote:

> non-real complex numbers (in Continental Europe: imaginary numbers)

I consider these two phrases to be synonyms also, so it's not just continental Europe. On the other hand, I know some people who use "imaginary number" to refer to what I would call a "purely imaginary number"; i.e., with real part equal to 0 (excluding the number 0 itself because it is not imaginary!).

Warren

I certainly concede that every field contains a subring naturally isomorphic to the integers, but for example, in the characteristic 0 field of the p-adic rationals, this definition does not coincide with the p-adic integers. In C_p, the complex p-adics, a realm in which we would actually like to make a distinction between rational and irrational, the distinction is even more exacerbated.

Cam

(that is, every characteristic zero field contains a subring...)

> I will definitely edit my entry according to the PM
> convention, as most everyone who has contributed to this
> thread seem to support it; however, I still find it very odd
> to use the convention that "irrational" and "not rational"
> do not mean the same thing.
>
> Warren

I guess I'd still argue they do mean the same thing. The subtlety is that defining the complement of a set is a relative, and not absolute, notion -- It requires the knowledge of a bigger set. I don't think either of us would argue that a banana is an irrational number simply because it's not a rational number.

So the distinction between the two approaches is only in answer the question "If it's not in the rationals, where is it?" The convention that we're espousing is that we take the complement of the rationals inside the real numbers, whereas you take the complement inside the (perfectly valid, but less common) complex numbers.

Cam

> What is i * pi? Irrational and imaginary?
> What about pi + i? Irrational and complex?

Both are "non-real" complex numbers (or in Continental Europe, imaginary numbers, since their imaginary parts \pi and 1 are distinct from 0). Why would we have some need to call them irrarional?

> I certainly concede that every field contains a subring naturally
> isomorphic to the integers, but for example, in the characteristic 0
> field of the p-adic rationals, this definition does not coincide with
> the p-adic integers. In C_p, the complex p-adics, a realm in which we
> would actually like to make a distinction between rational and
> irrational, the distinction is even more exacerbated.

Every field has a simple field as subfield, i.e. smallest (inclusion) subfield of F. If a characteristic of a field is p>0, then this subfield is isomorphic to Z_{p} (notice that p is prime). If a characteristic is 0, then this subfield is isomorphic to Q.

So if you consider any field F of characteristic 0, then you can assume that Q is a subfield of F. There's even more - there's no other subfield of F isomorphic to Q (since every subfield of F contains Q and Q is simple). So then definiton makes sense:

Irrational element (yeah... this is my definition :) ) in char 0 field F is any element in F\Q.

Irrational number is irrational element in R = real numbers.

Also C_{p} is isomorphic (as fields) with C = complex numbers. Of course Q_{p} can be embedded in C_{p}, so Q_{p} is nothing else then subfield of C = complex numbers. So I don't understand where's the problem?

Of course all of this is as a bit strange. I don't know whether there's use of irrational elements (instead of numbers).

joking

> If pressed, I would call e*i complex irrational.

Who may press you, Cam? Also e is a complex number and irrational. But if one wants to characterise the number ie, isn't it simply a transcendental number which happens not to be real?
Jussi

> I still find it very odd to use the
> convention that "irrational" and
> "not rational" do not mean the same thing.

Dear Warren, "Irrational" (from Latin "irrationalis" containing the negative prefix "in" and the adjective "rationalis" 'rational, reasonable') in general means 'not rational', but when speaking of numbers the established meaning is obviously 'non-rational real'. And I think it would give us no benefit, if we would call "irrational" the non-rational elements of any extension field of Q (e.g. of Q(i), R(X), C[[X]]).
Jussi

I favour such an extension of the word "rational" -- it is useful.

Hi Warren, I think this meaning of the terms "imaginary" and "purely imaginary" is very rational and practical but quite rare in America! Historically, "imaginary number" has of course meant 'non-real number', i.e. 'non-real complex number'.
Jussi

P. S. -- BTW, I and some other people can say that also 0 is purely imaginary (but not imaginary!). What could prevent it? Then all purely imaginary numbers form an additive group and (R-module).

Interesting that C has a subfield isomorphic to Q_p. However, Q_p is not isomorphic to R (cf. http://planetmath.org/encyclopedia/NonIsomorphicCompletionsOfMathbbQ.html). That subfield contains imaginary numbers, I suppose.

Jussi

> P. S. -- BTW, I and some other people can say that also 0 is purely imaginary (but not imaginary!). What could prevent it? Then all purely imaginary numbers form an additive group and (R-module).

From a mathematician's viewpoint, I agree that it definitely makes sense to consider 0 as purely imaginary, but I can attest to the fact that a person unfamiliar with mathematics is easily confused by this notion: I have had students before that told me that 0 was not a real number! I am almost certain that their logic runs as follows: If an object is "adverb adjective" where "adverb" does not have the meaning of "not", then that object must also be "adjective". For this reason, when I teach students complex numbers, 0 is not purely imaginary so that students do not think that 0 is imaginary.

In mathematics, there are quite a few definitions that do not follow the above "adverb adjective" logic. For example, if I remember correctly, a topological space can be path connected or simply connected but not connected. On the other hand, it is quite unlikely that I would teach topology! Also, these sorts of definitions are quite rare at the pre-college and early college levels.

> For example, if I remember correctly, a topological space can be path
> connected or simply connected but not connected.

Not quite true. :) Every simply connected space is path connected (definiton). Also every path connected space is connected (this is because interval [0,1] is connected). There are spaces that are connected, but are not path connected (for example countable set with cofinite topology).

> If an object is "adverb adjective" where "adverb" does not have the
> meaning of "not", then that object must also be "adjective".

Hmmm... I can't find any counterexample (of course this doesn't meen that there are no counterexamples).

joking

mathcam said: --I certainly concede that every field contains a subring naturally isomorphic to the integers, but for example, in the characteristic 0 field of the p-adic rationals, this definition does not coincide with the p-adic integers. In C_p, the complex p-adics, a realm in which we would actually like to make a distinction between rational and irrational, the distinction is even more exacerbated.--

Definitely. The issue here is that the term rational has several meanings. The term rational (as in rationals and integers) has a different meaning than rational (as in rational and irrational). The "integers" of the "rationals" Q_p are the elements of Z_p, but Z_p, for certain p, contains elements of C (complex numbers) which are definitely not rational (as in "irrational"). For example, Z_5 contains i=sqrt(-1), or Z_7 contains sqrt(2). My point then is that even though we do want to call Z_p the "integers" of Q_p, the elements of Z_p are not necessarily what we want to call "rational" (not irrational) inside C_p.

Alvaro

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