Is i irrational? (was Re: Versionincrementaphobia)
The Ghost In The Machine
<jw2...@bris.ac.uk>
wrote
on Tue, 07 Mar 2006 20:34:56 GMT
<Q7mPf.94623$Q22....@fe1.news.blueyonder.co.uk>:
> Ian Pilcher wrote:
>> James Westby wrote:
>>
>>>What makes you think that i is rational?
>>
>>
>> I'll bite. What do you put after the decimal point?
>>
>
> I don't know, but I imagine that I couldn't write it in closed form. For
> that matter what do you pu in front?
>
> I always assumed that i would be irrational, but maybe it is neither
> irrational nor rational.
An interesting philosophical question. If a number is
rational it can be expressed as p/q, where p and q are
integers, q != 0. Therefore i is not rational.
However, it's not clear whether i is irrational or not,
since irrational also means "not rational", but I for one
would think that an irrational number is real, and i is
not real.
Also, there are at least two different ways of defining
a real number (Dedekind cuts and Cauchy sequences), and
neither can be used for i, since i is not part of the
total real ordering, nor can it be the limit of any sequence of
rationals, even allowing for silly claims such as -1 = 1+2+4+8+... .
(Briefly: if x = 1+2+4+..., then 2*x=2+4+8+... = x-1;
therefore 2*x = x-1 or x = -1, despite all partial sums
of the series being positive. But it's not i.)
On the flip side, though,
http://mathworld.wolfram.com/IrrationalNumber.html simply
defines an irrational number as any number which cannot
be a quotient p/q of two integers, but the definition is a
bit sloppy since it implies irrational numbers have decimal
expansions (though it has real and imaginary parts, which can).
This appears to be a "definitional bug".
It gets bizarre though, as i is an algebraic integer,
and a unit of the algebraic number field. (There are a
lot of units in that field, as opposed to the two units +1
and -1 in the rational field.)
So now we have a non-rational algebraic unit. At this
point it's probably best to head out for a coffee or
tea break as one's brain is probably screaming for aspirin
at this point. :-)
Followups to a slightly more logical discussion area. :-)
>
>
> James
--
#191, ewi...@earthlink.net
It's still legal to go .sigless.
The World Wide Wade
The Ghost In The Machine <ew...@sirius.tg00suus7038.net> wrote:
> An interesting philosophical question.
No it's not.
mathman
Gottfried Helms
rational :: ratio of two integers , like 2/3
ir-rational :: no two integers can be found, like with sqrt(2)
(latin "in-" (="not") preceding a word starting with "r..."
is conventionally changed into "ir-", see also "ir-regular",
"ir-relevant" and so on, similarly like "il-" "literate"
instead of "in-" "literate,
but *not*, for instance, "is-stance" instead of "in-stance"
or "is-stead" instead of "in-stead" ;-) ,
so using "ir-rational" meaning "not-a-ratio of integers"
seems not to be too irrational, IMHO)
I think, it is without contradictions to
accept integer multiples of i for complex integers,
so complex rationals and complex irrationals may be
defined analoguously.
Gottfried
LuckyOne
Ken Pledger
The Ghost In The Machine <ew...@sirius.tg00suus7038.net> wrote:
> ....
> However, it's not clear whether i is irrational or not,
> since irrational also means "not rational", but I for one
> would think that an irrational number is real, and i is
> not real....
Asking whether the number i is rational or irrational is like
asking whether I (a New Zealander) am a Democrat or a Republican. I'm
neither, being right outside the U.S.A. and its political system. The
number i is neither rational nor irrational, being right outside the
real field.
Ken Pledger.
Stephen Montgomery-Smith
You mean it's not interesting or not philosophical?
(I would completely agree that it is not a philosophy question, but
would only 99% agree that it is not interesting.)
James Burns
And then there's Gelfond's Theorem: a^b is transcendental if
a and b are algebraic numbers with a neither 0 nor 1 and with
b irrational. In this context i counts as irrational. (For
example, i^i is transcendental).
The question of whether, no, /when/ i is irrational would be,
I think, a question of the philosophy of language, not
the philosophy of mathematics. Whether i should be taken to
be irrational depends on what irrational means within the current
context. What that meaning is /should/ be made clear by whoever
is expounding, but it might take a bit of thought on the part
of the listener. I think it might be impossible to be ambiguous
if one expressed oneself in "pure math", but that might be
more a devout wish on my part.
The philosophy of language would be handy in exploring the notion
of a definition of rational number /implicitly/ expressed in,
for example, my natural language expression of Gelfond's Theorem,
above.
Jim Burns
The World Wide Wade
<1141774851.9...@j33g2000cwa.googlegroups.com>,
"LuckyOne" <gwl...@nukove.com> wrote:
> Hey, Wade, what's your beef? Lighten up!
I wrote that it wasn't an interesting philosophical question. It
isn't. In fact it's not even a philosophical question. It is a
question, however, which is a shame because we were close to a
hat trick.
Minus XVII
it is a well-known fact (from alternating series?) that
x = -1 = ...0000. - ...0001. = ...9999., so that
"the second root of minus one equals the second root of infinity."
so, the question is, is infinity irrational by Gelfond?
> > (Briefly: if x = 1+2+4+..., then 2*x=2+4+8+... = x-1;
> And then there's Gelfond's Theorem: a^b is transcendental if
> a and b are algebraic numbers with a neither 0 nor 1 and with
> b irrational. In this context i counts as irrational. (For
> example, i^i is transcendental).
thus:
dog; you forgot tripolar ones....
"Time is the only dimension!"
>Other types of coordinates, like polar or spherical and Bucky's are
>alternate ways of designating the first 3 dimensions in relation to a
>set point and a set of axes. As I understand it (albeit in a limited
>way), Bucky's "Natures Coordinate" system is based on the way energy
>events at the smallest level aggregate to fill all-space in stable
>configurations with the tetrahedron being the basic structural quanta.
--Welcome to the googolplex; you can login any time,
like, you, but.... http://tarpley.net/bush23.htm
Robert Low
> In comp.lang.java.advocacy, James Westby
>>irrational nor rational.
> An interesting philosophical question. If a number is
> rational it can be expressed as p/q, where p and q are
> integers, q != 0. Therefore i is not rational.
Too much confusion about what one means by 'number'.
A *real* number is rational if it's p/q where p is
an integer and q is a (positive) natural. Usually,
'real' is taken for granted when one discusses whether
a number is rational or not.
If you no longer insist that your numbers be real,
then I guess you could consider (eg) a commutative ring
with unity, which sits inside a field, and then
talk about elements of that field being rational
if they're in the field of quotients of your original
ring.
But I don't think it's particularly philosophically
interesting. YMMV.
G. A. Edgar
irrational. And, of course, i is also not rational.
i is an algebraic integer, but not a rational integer...
In some p-adic fields, there are solutions to x^2=-1. When such a
solution x is expanded in base p, we see that the expansion is not
eventually periodic, since the eventually periodic expansions
correspond exactly to the rational numbers... In this sense, i is
irrational.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Pubkeybreaker
irrational. And, of course, i is also not rational. "
It is also irrational when viewed by the Gelfond-Schneider Theorem,
since e^(i pi) is rational.
Dave Rusin
G. A. Edgar <ed...@math.ohio-state.edu.invalid> wrote:
>In some p-adic fields, there are solutions to x^2=-1. When such a
>solution x is expanded in base p, we see that the expansion is not
>eventually periodic, since the eventually periodic expansions
>correspond exactly to the rational numbers... In this sense, i is
>irrational.
Well, all right then, what about fields like F_5, where x^2 = -1
if x=2 or if x=3 ? Certainly 2=2/1 and 3=3/1 are rational,
so in this sense i is rational!
That leaves only the question of whether i = 2 or i = 3 ...
dave
Oliver Wong
"The Ghost In The Machine" <ew...@sirius.tg00suus7038.net> wrote in message
news:nsa1e3-...@sirius.tg00suus7038.net...
> In comp.lang.java.advocacy, James Westby
> <jw2...@bris.ac.uk>
> wrote
> on Tue, 07 Mar 2006 20:34:56 GMT
> <Q7mPf.94623$Q22....@fe1.news.blueyonder.co.uk>:
>> Ian Pilcher wrote:
>>> James Westby wrote:
>>>
>>>>What makes you think that i is rational?
>>>
>>>
>>> I'll bite. What do you put after the decimal point?
>>>
>>
>> I don't know, but I imagine that I couldn't write it in closed form. For
>> that matter what do you pu in front?
>>
>> I always assumed that i would be irrational, but maybe it is neither
>> irrational nor rational.
>
> An interesting philosophical question. If a number is
> rational it can be expressed as p/q, where p and q are
> integers, q != 0. Therefore i is not rational.
Good point, which raises this one:
Real is to Integer as Imaginary is to ____ ?
If we had an answer to that, we could perhaps define "imaginary
rationals", and then rename "rationals" to "real rationals", and thus create
a new definition for "rationals" which is the union of "imaginary rationals"
and "real rationals".
>
> However, it's not clear whether i is irrational or not,
> since irrational also means "not rational", but I for one
> would think that an irrational number is real, and i is
> not real.
I too was under the impression that the set of (real) irrationals is a
subset of the set of reals. If this impression is true, and since i is not a
real number, it cannot possibly be a (real) irrational number.
- Oliver
thefre...@hotmail.com
Thank you very mutch.
Bill Dubuque
>
> so i is not irrational. And, of course, i is also not rational.
Many respected authors use irrational to mean "not rational",
in domains larger than just the reals (esp. complex numbers).
E.g. if you search books.google.com for "irrational algebraic"
you'll find such usage by many eminent mathematicians: e.g.
Conway, Gelfond, Manin, Ribenboim, Shafarevich, Waldschmidt
(esp. in diophantine approximation, e.g. Thue-Siegel-Roth
theorem, Gelfond-Schneider theorem, etc).
--Bill Dubuque
Gerry Myerson
"Oliver Wong" <ow...@castortech.com> wrote:
> Real is to Integer as Imaginary is to ____ ?
The Gaussian integers (the numbers a + bi, where a and b are integers)
play a role in the complex numbers similar to that of the integers
in the reals.
> I too was under the impression that the set of (real) irrationals is a
> subset of the set of reals. If this impression is true, and since i is not a
> real number, it cannot possibly be a (real) irrational number.
We're arguing about definitions. Definitions are a matter of
convenience, not of right and wrong. As long as you make your
definitions clear at the start, there's no problem.
With the usual definition of the rationals, i is (of course) not
a rational number. Whether that makes it an irrational number
depends on what is convenient for whatever you're doing. When I
teach Dirichlet's Theorem on Diophantine Approximation, I am
careful to say explicitly that it applies to all *real* irrationals,
since I don't want someone saying that i is a counterexample.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
G. A. Edgar
"In Mathematics, it's not quite true that what is not rational is
irrational."
http://www.cut-the-knot.org/do_you_know/numbers.shtml
G. A. Edgar
http://www.thefreedictionary.com/irrational+number
irrational number
n.
Any real number that cannot be expressed as a ratio between two
integers.
Minus XVII
imaginary integers have rations, just like real ones;
ratios of gaussian integers are also known as "bilinear fractions,"
a la Mobuis.
thus:
I didn't know that he had *any* other unknown claim!... but,
I don't think that he stated a different proof for n=3;
n=4 is obviously a special case, that his method did not apply to, and
possibly it only applied to "noncongruences modulo Fermat curves
(curves given by integer exponents in the F"L"T equation)....
that was an example of how much further Fermat went
in analytic geometry, then Descartes, although
he restricted himself to the plane. anyway,
my conjecture has long-been, that the "last" theorem is actually
an early dyscovery of his, a method that he used for many problems....
the most important unsolved problem of his, of course, is that
of characterizing the Fermat primes (he retracted the claim
that they were all prime, in a letter to Frenecle (sp.?)).
> Fermat never _publicly_ asserted FLT, you know. But he did assert the
> cases of cubes and fourth powers, both of which were proved long ago. I
> think he realized he was wrong about the general case.
>
> But what about these two claims by Fermat:
> -- Every natural number is a sum of three triangular numbers
> 0,1,3,6,10,15...
> -- The (misnamed) Pell equation
> xx = Dyy + 1
> has a solution with x>1, provided that D>0 and D is not a square.
> Both have been proved, but the first is pretty deep, while for the
> second, Fermat claimed to have an attractive proof by descent, and no
> proof by descent is known today. In these two cases I think it is
> possible that the old chap knew something that we don't.
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The Ghost In The Machine
Oh, I'll agree with that. :-) I for one think that the question is a
bit like asking what sign a complex number is. :-)
--
#191, ewi...@earthlink.net
Windows Vista. Because everyone wants a really slick-looking 8-sided wheel.
The Ghost In The Machine
> The World Wide Wade wrote:
>> In article <nsa1e3-...@sirius.tg00suus7038.net>,
>> The Ghost In The Machine <ew...@sirius.tg00suus7038.net> wrote:
>>
>>
>>>An interesting philosophical question.
>>
>>
>> No it's not.
>
> You mean it's not interesting or not philosophical?
Maybe it's not a question! ;-)
>
> (I would completely agree that it is not a philosophy question, but would
> only 99% agree that it is not interesting.)
I'd have to look, but science was at one point called "natural
philosophy"; I'm not sure what mathematics evolved out of, and it's
probably outside this newsgroup to speculate. :-)