0.9 repeater = 1

[bjornart]

I will delay posting the answer to my trick question in case others wish to weigh in with an opinion.

Me, me, I want to weigh in. I think it's irrational. I couldn't prove it beyond saying rational numbers are have repeating or terminating decimals though.

 

[whitefork]

Perhaps it's rational under base 2? Hmmmm. No time to work it out now, so I'll take that wild guess.

Very nice thread here.

 

[bjornart]

I would call it rational. The number is the sum of the series

10^-n, where n = 1 to inifinity. The sum of rational numbers is rational.

I didn't notice until Kullervo pointed it out, but your series is wrong, the real one is:

<the sum> n = 1 to infinity
of

10 to the power of

<the sum> i = 0 to n-1
of
-(1 + i)

And now my head hurts.
Someone please post a simpler way of defining that.

 

[bjornart]

Perhaps it's rational under base 2? Hmmmm. No time to work it out now, so I'll take that wild guess.

Very nice thread here.

My WAG would be that irrational numbers are irrational in all natural bases.

 

[whitefork]

And now my head hurts.

Stay away from the akvavit.

 

[slimshady2357]

For example, here's a number that is easier to describe as process, even though the number itself is not a process:

.101001000100001000001...

I assume you can guess the process (or algorithm) that describes this number. Each group of consecutive zero digits is one larger than the previous group.

Trick question - is the above number rational or irrational?

Hmmmmm at first glance I thought that was Liouville's number (or one of t hem ) 0.10100100000010000000000000000000000001... where each group of zeros is 1!, 2!, 3!, 4!.... long.

And so of course it's transcendental (and therefore irrational)

But now I see it's different..... and I'm wondering if the series doesn't converge....

Dammit, it sure looks irrational! But you say it's a trick.... hmmmmmmmmmmmmmm....

1/10 + 1/1,000 + 1/1,000,000 +1/10,000,000,000...

Hell, it's been too long, what's the word xoup?

Adam

 

[Tez]

I believe this number is actually transcendental, hence irrational, but I can't prove it.

Actually the proof is quite easy. It makes use of a powerful theorem (of Liouville I think?) which basically says this:
If a number is "well approximated" (in a certain precise sense) by rationals, it is transcendental!

e.g. Pi is well approximated by 22/7, 343/113 etc etc. Sqrt(2) cannot be so well approximated.

This is the main way that any number has been proven transcendental. Somewhat counterintuitive...

 

[hammegk]

Tez, you must be more careful with what you say. That post broke the board for hours, transcendentally and irrationally.

 

[whitefork]

Actually the proof is quite easy. It makes use of a powerful theorem....

yabbot, how easy is the theorem to prove?

 

[ceptimus]

Actually the proof is quite easy. It makes use of a powerful theorem (of Liouville I think?) which basically says this:
If a number is "well approximated" (in a certain precise sense) by rationals, it is transcendental!

e.g. Pi is well approximated by 22/7, 343/113 etc etc. Sqrt(2) cannot be so well approximated.

This is the main way that any number has been proven transcendental. Somewhat counterintuitive...

This can't be right can it? And the PI approximation is 355/113, not 343/113 (Pi = 3.0354 )

I'll be back with some approximations for Sqrt(2)

ceptimus.

 

[slimshady2357]

Actually the proof is quite easy. It makes use of a powerful theorem (of Liouville I think?) which basically says this:
If a number is "well approximated" (in a certain precise sense) by rationals, it is transcendental!

e.g. Pi is well approximated by 22/7, 343/113 etc etc. Sqrt(2) cannot be so well approximated.

This is the main way that any number has been proven transcendental. Somewhat counterintuitive...

Are you sure about that? I'm pretty sure the way most numbers are proven to be transcendental is to assume they are algebraic and then derive a contradiction. Are the two methods somehow related?

Oh and as I mentioned already, yes, his name was Liouville

Adam

 

[patnray]

.1010010001…= 10^-1 + 10^-3 + 10^-6 + 10^-10 + …

The series 1, 3, 6, 10, …, n is the sum of the integers from 1 to n, which is given by n * (n + 1)/2

Thus the decimal is the sum of 10^-(n* (n + 1)/2) for n =1 to infinity.

Since this decimal is the sum rational numbers, it is itself rational.

BTW, pi is not the SUM of a series of rational numbers. It is the LIMIT of the sum of a series of rational numbers.

The proof that the sum of integers from 1 to n = n * (n + 1)/2 is one of my favorite proofs because it is so clever. It is somewhat similar to Brown’s method for proving 0.999… = 1 (with some rearrangement).

 

[whitefork]

Hold on there a minute.

e is the (limit of the) sum of 1 + 1/1! + 1/2! + 1/3! ....1/n! as n increases without bound, all of which are rational

Xouper's number is (the limit of) 10^-1 + 10^-3 + 10^-6 + 10^-10 + …all of which are rational.


How do these differ so e is irrational and x's number is not?

 

[patnray]

A number is rational if it is the SUM of rational numbers. In your series expansion for e, e is the LIMIT of the sum of the series. Not the same thing. The series approaches e, as close as you want to get, but it never EQUALS e...

 

[rwald]

The sum of a finite number of rational numbers is rational.
The sum of an infinite number of rational number may or may not be rational.
Xouper's number is the sum of an infinite number of rational numbers; therefore, this fact cannot be used to determine the number's rationality.

The number neither repeats nor terminates. Therefore, it's irrational. It's that simple.

 

[ceptimus]

OK I said I'd be back with some Sqrt(2) approximations. Assuming my C compiler and brain are both working correctly (big assumption there) then these are the best approximations for PI and Sqrt(2) for denominators up to 10,000

6/ 2 = 3.000000 (error = 4.507034%)
13/ 4 = 3.250000 (error = 3.450713%)
16/ 5 = 3.200000 (error = 1.859164%)
19/ 6 = 3.166667 (error = 0.798131%)
22/ 7 = 3.142857 (error = 0.040250%)
179/ 57 = 3.140351 (error = 0.039527%)
201/ 64 = 3.140625 (error = 0.030801%)
223/ 71 = 3.140845 (error = 0.023796%)
245/ 78 = 3.141026 (error = 0.018049%)
267/ 85 = 3.141176 (error = 0.013248%)
289/ 92 = 3.141304 (error = 0.009177%)
311/ 99 = 3.141414 (error = 0.005682%)
333/ 106 = 3.141509 (error = 0.002649%)
355/ 113 = 3.141593 (error = 0.000008%)

3/ 2 = 1.500000 (error = 6.066017%)
4/ 3 = 1.333333 (error = 5.719096%)
7/ 5 = 1.400000 (error = 1.005051%)
17/ 12 = 1.416667 (error = 0.173461%)
24/ 17 = 1.411765 (error = 0.173160%)
41/ 29 = 1.413793 (error = 0.029731%)
99/ 70 = 1.414286 (error = 0.005102%)
140/ 99 = 1.414141 (error = 0.005102%)
239/ 169 = 1.414201 (error = 0.000875%)
577/ 408 = 1.414216 (error = 0.000150%)
816/ 577 = 1.414211 (error = 0.000150%)
1393/ 985 = 1.414213 (error = 0.000026%)
3363/2378 = 1.414214 (error = 0.000004%)
4756/3363 = 1.414213 (error = 0.000004%)
8119/5741 = 1.414214 (error = 0.000001%)

I don't know if that proves anything, other than how strangely good the 355/113 approximation is - but I think that's just lucky.

ceptimus.

 

[slimshady2357]

The number neither repeats nor terminates. Therefore, it's irrational. It's that simple.

That's what I thought (and others too). And yet, xouper says it's a trick....

What is the trick if it's that simple?

What do you think the trick is rwald?

And xouper, tell us already, what's the trick?

(ya, that's right, I'm impatient )

Adam

 

[rwald]

Well, I didn't notice the "trick" part earlier, so I'll have to think about it now...

Maybe something to do with the fact that the infinitith repetition of the pattern is actually a repeating 000...? So it ends up being something like 0.101001000100001...000..., which as a whole is eventually a repeating decimal? But I don't know if a number which is only a repeating decimal after an infinite number of digits would count as a rational number.

Is there some way to determine the rationality of a number based on the sum used to compute it? Something that would distinguish between 1/2 + 1/4 + 1/8 + 1/16 ... and 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 ... ) ?

 

[daver]

OK I said I'd be back with some Sqrt(2) approximations. Assuming my C compiler and brain are both working correctly (big assumption there) then these are the best approximations for PI and Sqrt(2) for denominators up to 10,000
ceptimus.

You might want to look into continued fractions. They're pretty fun, and could have saved you some work.

 

[xouper]

Regarding .101001000100001000001...

A rational number is often described as having a pattern that repeats. The trick part of my question is that while there is indeed a repeating "pattern" to the decimal expansion, is this pattern of the kind that qualifies a number as rational? The answer is no.

<div style="border: solid black 1px; margin: 0 32px 0 0; padding: 24px; background-color: #fff8f0;">Review - If we let R represent a finite sequence of digits, then a rational number is of the form .XRRR... where R repeats forever and X represents an optional finite sequence of digits. R can also be a single zero digit. R is called the repetend.</div>

Answer: .101001000100001000001... is irrational. It neither repeats nor terminates.

I'll try to address all the other questions raised, but if I overlooked any, please ask again.

Kullervo: Perhaps it's rational under base 2?

No. As bjornart wagged, if a number is rational in one natural base, then it is rational in all other natural bases. This is easy to see if you observe that an integer in one base is always an integer in any other base. Thus, given any two natural bases b and c (greater than 2), and any rational number n/m in base b, we can always find a rational number p/q in base c:

(n/m)_{base b} = (p/q)_{base c}

This is just another way of saying that changing bases does not change whether the number is rational or irrational.

patnray: .1010010001... = 10^-1 + 10^-3 + 10^-6 + 10^-10 + ...

The series 1, 3, 6, 10, ..., n is the sum of the integers from 1 to n, which is given by n * (n + 1)/2

Thus the decimal is the sum of 10^-(n* (n + 1)/2) for n =1 to infinity.

Correct.

patnray: Since this decimal is the sum rational numbers, it is itself rational.

As rwald observed, the sum of a finite sequence of rational numbers is always rational, but the sum of an infinite sequence of rational numbers is not necessarily rational. In other words, we can't use this as a test for rationality of an infinite series.

patnray: BTW, pi is not the SUM of a series of rational numbers. It is the LIMIT of the sum of a series of rational numbers.

Perhaps there is some confusion about jargon here. It is not incorrect to say that Pi is the sum of an infinite sequence of rational numbers. This sum is defined in terms of a limit (as you observed), but it is still a sum of the terms.

http://mathworld.wolfram.com/Sequence.html
http://mathworld.wolfram.com/Series.html

Consider also that Pi = 3 + 1/10 + 4/100 + 1/1000 + 5/10000 + 9/100000 + ... which is the sum of an infinite sequence of rational numbers.

patnray: The series approaches e, as close as you want to get, but it never EQUALS e...

Except that the infinite series does exactly equal e, in the same manner that the infinite series .999... exactly equals one.

slimshady2357: I'm wondering if the series doesn't converge

It converges. The formal test is simple. For a geometric series expressed as the sum (n=1 to infinity) of r^n, it is convergent if |r|<1. Thus a series with r=1/10 is convergent. The number I gave is less than that, so it too is convergent. But the formal test is not necessary if you notice that the number I gave is less than 1/9.

slimshady2357: at first glance I thought that was Liouville's number

I assume you are referring to Liouville's Constant, which as you observed, is slightly different than the number I gave, although its construction is similar. There is a whole class of Liouville Numbers. For bonus points, is the number I gave a Liouville Number? If it is, then it is transcendental, as Tez observed.

Tez: Actually the proof is quite easy. It makes use of a powerful theorem (of Liouville I think?) which basically says this: If a number is "well approximated" (in a certain precise sense) by rationals, it is transcendental!

More on that here:
http://mathworld.wolfram.com/TranscendentalNumber.html