.9 repeating = 1? Is our numerical system flawed?

Originally posted by Xar Ke Zeth
Good point, it'd just be neglecting the problem. But maybe that problem only exists with our decimal system?

Yeah, so? It works. Better than ignoring a problem than to have a fixable problem.

Dunno, I was just throwing around ideas willy nilly, hoping to come up with something that could possibly remedy the problem.

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The problem remains in binary as well. [base 2]

0.111111111. . . = 1.0

1/2 + 1/4 + 1/8 + 1/16 + 1/32 . . . = 1

I am guessing this a point of contention in any number system where the point is followed by the highest value symbol is repeated infinitely.
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We all know the real line is composed of rational and irrational numbers. The infinite sum thing is a convention Newton created to get around the somewhat irrational nature of theoretical mathematics. In reality, most functions have no real solution, or an infinite number of solutions. In order to be useful, a function must conform, be rational, and have a finite number of real solutions. Those are the exceptions.

I Think. Don't flame me too bad on this one, Am.

Originally posted by Icarus Rising
We all know the real line is composed of rational and irrational numbers. The infinite sum thing is a convention Newton created to get around the somewhat irrational nature of theoretical mathematics. In reality, most functions have no real solution, or an infinite number of solutions.

Wow, this thread was very informative.
Icarus, Agreed. We also invented the Sq root symbol to get around the "irrational" nature of mathematics as well. I skipped a couple of pages so I dont know if anyone mentioned this.
We still dont know the exact value of sqrt(2). We know its close to 1.4142.
The good ol' "speechless" number.

Someone asked how .9~ can be represented as a fraction. I can only do it using infinities

((10^infinity)-1/10^infinity)....again we're stuck and we have to use limiting thms.

gasp!

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The problem remains in binary as well. [base 2]

0.111111111. . . = 1.0

Please people, do research before you throw out what you think is true. Binary is a fixed place dual number system. 0.11111 IS NOT binary. It has no decimal points. For the love of all that is Holy!

Binary is cool because, the system, not its implementation, is in the integer number system, so you can never have things like .99999 or the like.

stop bad science!

ktprktpr! You have revived me, I had a heart attack there.

Please, people, read the post above this. Please.

IR: Sorry. Without calculus, we would honestly never move because of what I mentioned earlier, Xeno's Paradox. You need to have infinite sums converge, and they do. You're right about most functions, however. Most functions aren't even solvable with current methods, and many just can't be described.

I_S_I_S: We invented square roots for irrationals? We invented square roots because they're the opposite of squaring. sqrt(2) is irrational, yes, because of the number 2 and the nature of the square root function. Oh well, sucks. Thank god for rounding.

And, uh, .9 can be expressed as a fraction. Easily. 9/10.

Originally posted by Amorymeltzer
And, uh, .9 can be expressed as a fraction. Easily. 9/10.

Uh, the fraction was to represent .999999999999999999999999~.

I was just playing around with my graphs and realized that (x-1)/x converges to 1(limit) but really keeps going to .9999999~.
Hey I was bored Ok?

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Excuse me dear,
BUT

decimal :
. . . 10^2, 10^1, 10^0, . 10^-1, 10^-2 . . .
. . decimal point . . . . . . ^

binary:
. . . 2^2, 2^1, 2^0, . 2^-1, 2^-2 . . .
. . binary point . . . . ^

you do understand how a floating point number works don't you?

scitec.uwichill.edu.bb...
zeus.cs.pacificu.edu...
www.asic-world.com...
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there are certain numbers, that when converted from decimal to binary, will produce a never ending sequence of 1s or 0s.

you could of course represent .99~ as the sum of a sequence.
i.e. Sum of 9/(10^n) from n=1 --> infinity...
i don't know if that clears anything up.

Well, y'all finally pulled a longtime lurker out of hiding with this one. This is my first post.

I aced real analysis during my math studies, but its been a while, so sorry if I make a mistake.

.999... = 1 by definition. An infinite decimal sequence is defined to represent the its limit. You can think of .999... as the sequence 9/10, 99/100, 999/1000..., which has a limit of 1. This isn't really a flaw in the real number set. You just end up with a dual representation of every finite decimal sequence. If you were so inclined you could simply disallow sequences ending in repeat nines. In my studies in mathematics though, I have come to believe the idea of infinite decimal expansions is flawed. I'll have to give the fast explanation, but I'll elaborate more if y'all are interested.

Someone metioned countable infinities versus uncountable infinities. I think it was Cantor that thought this up, and it did make him a little nutty by the way. A set can be finite, countably infinite, or uncountably infinite. The integers are a countable set. Any set that has a bijection with the integers is countable. That means you could make an infinitely long list of the elements of the set assigning each one an integer, or natural number. It turns out the reals are uncountable. Cantor came up with a brand new method of proof for it, diagnolization. Its slightly easier to prove that (0,1) is uncountable. (0,1) can be put into a bijection with the reals, so they have the same cardinality.

Suppose the all the numbers in (0,1) could be put into bijection with the natural numbers, the consider the list:

.a_0,a_1,a_2,...
.b_0,b_1,b_2,...
.c_0,c_1,c_2,...
.
.
.

now, create a number by going down the list, if a_0,b_1,c_2 and for each one adding a 5, unless that digit is 5, and then add a 6, Hence, this new number is different from each number in the list, so (0,1) is uncountable.

This is a really interesting fact. Notice that if you tried the above proof with binary, it wouldn't work, because the list could arranged to have all 0s on the diagonal and to make your number different it would be .111..., which didn't need to be in the list to begin with since .111 = 1 in binary.

Another interesting fact is that in a real analysis book, the only time decimal expansions are used as a representation of the reals is for the above proof.

The above was pretty much fact (except for some sloppiness), the following is sort of my opinion on math philosophy:

I think it is a flaw to include all possible infinite decimal expansions in the real numbers, because some of them cannot be "chosen". A number isn't much use to anybody if you can't represent it in some sort of finite form. Don't confuse expressable in finite form with having a finite decimal expression. 1/3 is a perfectly valid representation of a number, even though it has an ifinite decimal representation.

A guy named Alan Turing is the father of computer science (I'm sure many of you here know this, but humor me). He came up with the idea of another kind of number: a "computable" number. A computable number is any number that can be the output of a computer program in some sort of decimal form, given that the program was given an infinite time to run. For instance, Pi is computable, because we can write a program that will keep spitting out digits of Pi as long as the computer runs. This program could be taken as a finite representation of Pi. It turns out there are only countably many computable numbers.

This leads me to wonder what the heck the rest of the numbers are. There are uncountably many real numbers that cannot be expressed by a computer program. These numbers don't seem useful at all. If there exists no process for creating the number, how would you express it in any finite form? The problem seems to lie in an over generous definition of convergence. It makes sense to allow .9999... to equal 1, since 1 is the limit of .9999, but real analysis allows any decimal expansion to be valid, because it defines convergence as Cauchy (sp?) convergence, where sequences are allowed to converge without converging to anything. It suffices to simply be able to create an infinite sequences of monotically decreasing bounds on the original sequence. By allowing any infinite decimal expansion into the real numbers, we have let in uncoutably many wacky numbers that aren't any good to anybody.

Wow, that was a long post. Hope I don't get in trouble for it. There's information available on the net about all this and I would love to discuss it further if anybody is interested.

Originally posted by cmdrpaddy
there are certain numbers, that when converted from decimal to binary, will produce a never ending sequence of 1s or 0s.
(...)

That is right. It all depends on the base you choose.

1/3 is in decimal infinite to write, but in base 3 (amongst others) is totally finite. One third in base 3 is 0.1 ...

In the same way, 1/7 (decimal) is 0.1 in base 7, 0.2 in base 14, 1/19 (decimal) is precisely 0.1 in base 19, so on...

pingo, I'm not sure but I think what you were referring to as infinite uncountable sets and infinite countable sets would be related to dense and non-dense sets. A dense sets is simply a set in which between any two numbers you can find another number, i.e. between 1 and 2 you can find 1.5 and between 1.5 and 2 you can find 1.75 etc. obviously a non-dense set would one that there is no numbers between two successive numbers (eg the set of integers).
I don't think there is nothing wrong with non computable numbers, if we ignored them or excluded them we wouldn't have a complete number system and its not like they are doing any harm by being there!

Originally posted by cmdrpaddy
(...)
A dense sets is simply a set in which between any two numbers you can find another number, i.e. between 1 and 2 you can find 1.5 and between 1.5 and 2 you can find 1.75 etc. obviously a non-dense set would one that there is no numbers between two successive numbers (eg the set of integers).
(...)

I don't think the questioon really is there... As asked, the question relates to a dense set of numbers. On the other hand, I talked above of infinites, which is obviously related to the question, and, for instance, the natural numbers set is not dense.

The infinite is in this way something that depends on our own comprehension of numbers, and I think it is so far (nearly) uncomprehendable. It is just too big or small, but our mind is not ready to go that far yet.

GAH 4 pages??

0.9 Recurring is not equal to 1 no matter how hard you try.

0.9 Recurring tends to 1.


0.9 recurring = x and 1x=1 are in confliction - its a stupid proof which should not take 4 pages to solve.

[edit on 30-3-2005 by Vanguard]

Originally posted by Vanguard
GAH 4 pages??

0.9 Recurring is not equal to 1 no matter how hard you try.

0.9 Recurring tends to 1.


0.9 recurring = x and 1x=1 are in confliction - its a stupid proof which should not take 4 pages to solve.

[edit on 30-3-2005 by Vanguard]

Except for the fact that they are equivalent

Originally posted by utrex

Originally posted by Vanguard
GAH 4 pages??

0.9 Recurring is not equal to 1 no matter how hard you try.

0.9 Recurring tends to 1.


0.9 recurring = x and 1x=1 are in confliction - its a stupid proof which should not take 4 pages to solve.

[edit on 30-3-2005 by Vanguard]

Except for the fact that they are equivalent

They are nearly the same by an infinitely immeasurable amount but not equal.

How many ways can i state this??

[edit on 30-3-2005 by Vanguard]

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One way of thinking about it is, to whatever level of precision you want to choose 0.9999... will always round to 1.0

begs the question: "When is an irrational number 'rational enough'?"

When is some number very close to 3.0 accepted as 3.0?

3.0~= 3.000000000.....000002891038574...
or
3.0~= 2.999999999......999996582556571...

Quantum implies a discreet Universe at the very smallest quantities. But when dealing with real quantities they are almost certainly not precise and essentially an irrational quantity.
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This is maths - not the real world

Originally posted by Vanguard

They are nearly the same by an infinitely immeasurable amount but not equal.

How many ways can i state this??

[edit on 30-3-2005 by Vanguard]

It doesn't really matter how many ways you state it: it's not true. There have been perfectly acceptable proofs in this thread showing their equivalence.

p.s. I think the word you are looking for is "infinitesimal."