Is i irrational? (was Re: Versionincrementaphobia)  
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The Ghost In The Machine
PostPosted: 2006-3-8 6:00:00 Top
java-programmer, Is i irrational? (was Re: Versionincrementaphobia) In comp.lang.java.advocacy, James Westby
<email***@***.com>
wrote
on Tue, 07 Mar 2006 20:34:56 GMT
<Q7mPf.94623$email***@***.com>:
> 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


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