Math question: How to decide about rational and irrational numbers

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mcc
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Math question: How to decide about rational and irrational numbers

Post by mcc » Sat Mar 03, 2018 2:48 pm

Hi,

WARNING!
I am no math guru...by far not. The opposite is true...

As far as I had understood...

An integer number is a number, which has only '0's as fractional part...which is identical to "no fractional part at all".
A rational number is number, which can be expressed as a quotient from two integer numbers.
An irrational number is a number which is no integer number and no rational number and no complex number.

There are rational numbers, which fractional part stops at some point. For example 1 4 / 0.25
There are rational and irrational numbers, which fractional part doesn't stop but is repetive. For example 1 3 / 0.3333333333.....
Every fractional part, which is non-repetive and does not stop is part of a irrational number.

But:
Is there any mathematical proof, that the fractional part of a rational number is repetive or none-repetive or whether it stops or not?
Is this property an effect of the underlaying number system (decimal, etc) only?

For example:
How do we know, that PI is irrational...that is: The fractional part non-repetive?
It may be that afte the 9e9999999999999999999999999999999999999999999999999999999999999999999999 digit the fractional
part DOES repeat itsself....
(pi isn't a good example...but...)

I fear, that a brute-force attack to this problem with my DM42 will only empty the batteries... ;)

Thanks a lot for any enlightment! :)

Cheers
Meino
DM 42 - SN: 00373, Firmware v.:3.9.1 / 3.9.1 as compiled by SwissMicros

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Walter
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Re: Math question: How to decide about rational and irrational numbers

Post by Walter » Sat Mar 03, 2018 5:16 pm

Look, Meino, I'm no mathematician either. But I know generations of mathematicians have investigated pi. IIRC, some millions of digits of pi are known today. No repetitive sequence was detected so far. Thus, the hypothesis that pi is irrational can't be rejected.

OTOH, all rational number feature either a limited number of digits (like 1/8 = 0.125) or a repetitive sequence (like 1/11 = 0.090909..., 1/7 = 0.142857142857...).

HTH

EDIT: Reading Wikipedia about rational and irrational numbers is recommended as well. It may enlighten you ... ;)
Last edited by Walter on Sat Mar 03, 2018 5:45 pm, edited 1 time in total.
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keithdalby
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Re: Math question: How to decide about rational and irrational numbers

Post by keithdalby » Sat Mar 03, 2018 5:29 pm

I find it interesting that pi can take different values (effectively, but not really. I'm talking about the Ricci scalar if my memory serves me correctly) in noneuclidean geometry.

mcc
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Re: Math question: How to decide about rational and irrational numbers

Post by mcc » Sat Mar 03, 2018 5:56 pm

Hi,

as said, PI was only an example and it was a bad one (as mentioned).
I reached for a more general and abstract answer.
I read wikipedia in beforehand and what I understood dont gives me a clue.
So I returned to here...
OTOH, all rational number feature either a limited number of digits (like 1/8 = 0.125) or a repetitive sequence (like 1/11 = 0.090909..., 1/7 = 0.142857142857...).
Proof?

Cheers
Meino
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Walter
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Re: Math question: How to decide about rational and irrational numbers

Post by Walter » Sat Mar 03, 2018 6:05 pm

mcc wrote:
Sat Mar 03, 2018 5:56 pm
OTOH, all rational number feature either a limited number of digits (like 1/8 = 0.125) or a repetitive sequence (like 1/11 = 0.090909..., 1/7 = 0.142857142857...).
Proof?
Hmmh, did nobody teach you saying "please" and "thanks"? :shock: Then teach yourself, please!
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Thomas Okken
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Re: Math question: How to decide about rational and irrational numbers

Post by Thomas Okken » Sat Mar 03, 2018 6:13 pm

Pi is irrational; that question is settled.

N.B. The proof by Lindemann, that pi is not just irrational but also transcendental, settled the ancient question whether or not it is possible to square the circle. (It is not.)

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Re: Math question: How to decide about rational and irrational numbers

Post by Thomas Okken » Sat Mar 03, 2018 8:22 pm

mcc wrote:
Sat Mar 03, 2018 5:56 pm
OTOH, all rational number feature either a limited number of digits (like 1/8 = 0.125) or a repetitive sequence (like 1/11 = 0.090909..., 1/7 = 0.142857142857...).
Proof?
How does long division work?

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ijabbott
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Re: Math question: How to decide about rational and irrational numbers

Post by ijabbott » Sat Mar 03, 2018 8:24 pm

mcc wrote:
Sat Mar 03, 2018 5:56 pm
OTOH, all rational number feature either a limited number of digits (like 1/8 = 0.125) or a repetitive sequence (like 1/11 = 0.090909..., 1/7 = 0.142857142857...).
Proof?
Informally, after the initial division of a/b (where a>=0 and b>=1), there will be some remainder r_0. In the next step of long division, you "bring down the zero" and calculate r_0*10/b to yield some digit d_1 (the first digit after the decimal point) and some remainder r_1. Continue the long division by calculating r_1*10/b to yield some digit d_2 and some remainder r_2. Keep going to generate new digits and new remainders. In general, r_(k)/b will yield a digit d_(k+1) and a remainder r_(k+1). Note that each digit d_(k+1) is a function of r_k and that each remainder r_(k+1) is a function of r_k. In other words, each distinct remainder produces a particular digit and a particular remainder at the next step. Now the fun part is that there are only b possible remainders, so after at most b steps you will generate a remainder that has been seen previously. At that point, the sequence of generated digits and remainders from the previous occurrence of the repeated remainder up to and including the step that generated the repeated remainder will repeat ad infinitum. Note that a terminating decimal just means that it ends with the digit 0 repeated ad infinitum. In fact, a remainder 0 will always generate a digit 0 and a remainder 0 at the next step.

(Actually, the above works in any base, with the multiplier "10" representing that base.)

mcc
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Re: Math question: How to decide about rational and irrational numbers

Post by mcc » Sat Mar 03, 2018 11:23 pm

Hi ijabbott,

Yeah! Thanks a lot! That was, what I was looking for!

Another, but I think similiar thing is also in my mind (and I have not looked up in wikipedia this specific thing
due to lack of words to name that problem. I looked up this one https://de.wikipedia.org/wiki/Grenzwert_(Folge) and
this one https://de.wikibooks.org/wiki/Mathe_f%C ... _Beispiele (sorry...dont found the english equivalent
again for the lack of words.).

When I remember correctly the sum of all 1/n for n: 1->oo is oo (infinum).

On the other side, I think, that the sum of all 1/(10^n) for n: 1->oo is 0.1111111... .

Is there a sequence of the form "sum of all 1/f(n), which somewhere "in between" the both above,
for which it is not possible to decide, whether it goes "ad infinum" or has a certain limit?
I dont mean a sequence, for which it is not yet possible to have an answer (like certain conjectures, which
are not proofen yet)...I mean a sequence, for which is proofen, that it is not possible to calculate/proof, that
it has a certain limit.

Or is this whole "logic" completly nuts?

Cheers!
Meino
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ijabbott
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Re: Math question: How to decide about rational and irrational numbers

Post by ijabbott » Sun Mar 04, 2018 8:42 am

Any series that does not converge to a limit is called a divergent series, but there are some alternating, divergent series with partial sums that eventually reach a point where successive partial sums are bounded between two values. The simplest example of such a series is Σ(-1)^n.

That's probably not what you're asking though. I think you are asking whether it is possible to construct a series where its convergence/divergence is undecidable. I don't know, but that might be the case due to Gödel's incompleteness theorems.

It may also be possible to prove that a series has a limit even if no one has determined the limit yet. Perhaps someone proved that the sum of inverse squares had a limit before Euler proved that the limit was (π^2)/6.

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