0.9_ = 1?

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Well, there aren't many instances where the intuition of most people with little enough education to not know when they're wrong is so spectacularly in error.

As for Zeno's paradox, it turns up occasionally and you get some interesting responses (I've noticed a very popular response among those with no calculus experience is to say it proves space must be discrete), but most people intuit that motion is in fact possible so they aren't likely to perseverate on a fallacious idea.

what I had meant is that if it had already been proven, then what is the controversy?

I am aware of base b expansion. It is in Rosen's Number Theory text from my Number Theory and cryptography class. While, the base b expansion shows that a number is rational iff its base b expansion is periodic or terminates. The central question I want to know is if .9 repeating is technically rational. If it is not then I do not think that .9 repeating = 1. Because that would violate the rationality of 1 and in consequence would seem to violate the central existence of rational numbers (and whole numbers and integers, etc. ) Now, .9 repeating obviously does not terminate. Does it technically qualify as periodic? (according to the standard base b definition of periodic).

A base b expansion (.c1c2c3....)b is called periodic if there are positive integers N and k such that Cn+k=Cn for n >(equal to) N.

My text states that every real number has a unique decimal expansion excluding the possibility that the expansion has a tail end that consists entirely of 9s. in that case .9 repeating is irrational. and 1 is rational. So as far as I took 5 minutes to peek at it... it doesn't look like .9 repeating equals 1.

Now it is possible that .9 repeating does equal 1, or that .9 repeating is a special case number (transcendental, etc.) that is not defined by a fraction, but rather only algebraically.

It is most likely that the system of math while consistent, is not complete. And problems like these will always exist.

I think more that not so much our system of mathematics is here not complete than our system or representing numbers has a weak point. Decimal are designed to represent rational numbers - for nothing more. 0.9_ is by a certain formality able to get written down, but does not really make sense. Therefore the statement 0.9_ as the formal result of 0.1_ * 9 and there fore 1/9 * 9 to be counted as 1 makes sense.

I think there are gaps in the numbers system, like the way it says that if X=0.9_ then 10X-X=9 and they forget that there is one less 9 on the end of 9.9_ (when you times something by 10 all the digits shift to left, leaving a gap on the right) and the way that it assumes that 1/infinity is 0 even though 0*infinity isn't 1.

It is less a problem of our system of numbers, but of our representation of our system of numbers. If take the one abstract step further, you will see that it isn't a problem of the number system at all.

Taken as a topological space (actually a metric space) the real number system is locally compact. Not a gap in sight.

ruveyn

Human perseveration in the face of deductive proof; that is the conundrum here.


The rationals are defined as the field of quotients over the integers. You're working backwards; we ought to look rather at to what the series representation of .9... converges, which is quite clearly 1. Even from the definition you give, it is quite clear that .999... is periodic with a period of one digit. For all natural numbers n,m c_n=c_(n+m)=9.


The reason that a base 10 expansion has uniqueness excluding cases in which the tail consists of 9s is solely because if you include tails of 9s then every rational number gets two expansions as much as .9... and 1 are the two expansions of 1. See again the problem quoted from my analysis book on the ternary expansion of a number.

Let's suppose .999... is irrational. Then it must be the limit of a Cauchy sequence of rational numbers (by the usual definition by which the rationals are completed). This cauchy sequence is the sequence of partial sums of 9(Σ1/10^k-1). This sequence clearly converges though in the rationals to 1. hence we have a contradiction, and thus it is rational.


While there are things that are uncertain in math, this is not one of them. This is a false problem predicated on a misunderstanding of mere notation.

Non terminating decimal expansions are limits of series; that .9... = 1 follows immediately from the definition of .9...

decimals do not represent just rational numbers. they represent real numbers being either rational or irrational.

I don't just have copies of your books lying around so its not like I can flip to the page and follow. But I am not to stuck on the idea that it is impossible for .9 repeating to equal 1. I am ok with it if it is. when I had looked up base b expansion I had first thought that .9 repeating was rational but they added the additional part about the tail of 9s and I have been out of math long enough for that to make only partial sense. It is less that I refuse to agree and more that I needed it explained in a better way. 1/3 *3 doesn't really work for me. But I found a decent explanation and a decent proof. so it is ok. Consider me convinced (as much as you will convince me of anything)

I looked it up online and I concur that

The number 1 can be represented exactly by itself. It can also be regarded as the limit of the sequence

0.9 0.99 0.99 0.999 0.9999

where the q-th term of the series is given by

1 - 1
-----
10^q

where q is infinite (and I presume an integer >= 0...)

I would agree that 1 has two decimal expansions, itself exactly and the sequence that represents .9 repeating.

http://whyslopes.com/Number_Theory/Deci ... ation.html

I would also concur that every non-zero, terminating decimal has a twin with trailing 9s and therefore such expansions are non-unique.

Decimals can't represent an irrational number - e.g.:

(2)^(1/2) = 1.41421356 ...
e = 2.7182818 ...
ln (3) = 1.098612289 ...

whilst rational numbers are well to represent:

6/(-5) = -1.2
3.2 * 4.8 = 15.36
1/3 = 0.3_

Irrational numbers - all nonterminating, nonrepeating decimals (numbers that can't be represented as a ratio of integers)

http://books.google.com/books?id=qtuySd ... ZqD-Sw_YbI

0.9_ is a repeating decimal, and 1 isn't. So if 0.9_ is equal to 1 then you can say that 1 is a repeating decimal and it isn't, which is the same as saying x equals 1 and x does not equal 1, which is impossible.

A decimal isn't a kind of number but rather a way to express a number. And that expression need not be unique. Another example is series. You can write a general expression for an infinite series summing up to pi (in fact there are many different ways to do this), or you could just write "pi". In fact, you can think of decimals as a form of series, i.e. 1.23=1*10^0+2*10^-1+3*10^-2.

I suspect you two mean different things by "represent". You seem to be saying represent as in "give an impression of", while by represent Dussel seems to mean "fully describe".

I think 0.9_ is in a sense useful because one can immediately deduce a series 9*10^-n from n=1 to infinity which gives 1. In contrast, I don't think it is immediately obvious from 3.14159... how Pi can be summed up, though this has eventually been done in many different ways.

lol?

A) whatever gave you the impression that 1.0_ isn't a repeating decimal?
B) whatever gave you the impression that numbers are defined by their representations? (instead of the other way around)
C) whatever gave you the impression that each representation of number is unique?
D) Why would you cite Uncyclopedia as a reference, unless you were a troll? (in deed or intent, it doesn't really matter)



All you are trying to do is disprove the above... which is silly, because it is an identity.

There are an infinite number of ways to represent 1 (or any other number).



Do you deny that that that is 1?



What about that?



Or that?

Yes. It is pi squared over 6.

ruveyn