Poll: Does 0.99 Recurring = 1

Does 0.99 Recurring = 1

  • Yes

    Votes: 225 42.5%

  • No

    Votes: 304 57.5%
  • Total voters
    529
Status
Not open for further replies.
let a = b

a² = ab

Multiply both sides by a

a² + a² - 2ab = ab + a² - 2ab

Add (a² - 2ab) to both sides

2(a² - ab) = a² - ab

Factor the left, and collect like terms on the right

2 = 1

Of course, everything is fine until the last step. As both sides are zero (a-b=0), it is simply declaring 2x0=0.

:)
 
Originally posted by w11tho
let a = b

a² = ab

Multiply both sides by a

a² + a² - 2ab = ab + a² - 2ab

Add (a² - 2ab) to both sides

2(a² - ab) = a² - ab

Factor the left, and collect like terms on the right

2 = 1

Of course, everything is fine until the last step. As both sides are zero (a-b=0), it is simply declaring 2x0=0.

:)
Click to expand...

Yes, which means that it's not really a proof. :p
 
If this is a strictly mathematical question, then 0.9r does indeed = 1, but this is as much convention as anything else.

In my (philosophical) opinion the confusion is caused like this:

There are two points that exist an undetermined finite space apart from each other.

How much pure "space" is in between these points?

You would try to work this out by finding out how many times you can divide up a space (or just 'space'), and when you get to the smallest amount, you count how many "atoms" of space there are in between the two points.

You then realize that you can divide space an apparently infinite number of times

You then also realize that even an infinite amount of "space" would have no value (distance in this case) because the idea of having "units" or "atoms" of space is a non-concept

It is a mis-use of reason to attempt to find atoms of space because space is a concept required to have the concept of objects in the first place.

We can have units to measure distance between objects because the units are relative to other objects which exist in space, but you cannot measure space itself.

I believe this is the basis of the age-old confusion.

[/philosophical mode]
 
Last edited:
Originally posted by Karaboudjan
(Currently at P2 A Level knowledge) in algebra that says that the x couldn't be divided by something. IIRC there wasn't a x =//= 0
Click to expand...
In the early derivation of The Field of Reals (ie the real number line) its built into the "system" that you cannot divide by zero, even if that zero is "cleverly" disguised as (a-b) or (x-y) or even x. You can divide it by something really small, "dx", but you cannot divide by zero.

And before someone posts "What about (sin x)/x" or any other removable singularity, do a Taylor series expansion and see what you get ;)

Put it like this, if there existed such a simple to show (and correct) paradox in mathematics that even those who have no mathematical ability could understand it, maths wouldn't have lasted as long as it has. There isn't even a paradox or contradiction anywhere in it so far.
 
If you haven't seen a Taylor series, this might not make much sense, but I'll put it up anyway. Here is another (incorrect) proof of something - this time Log(2)=0. It goes like this:

Doing the Taylor expansion of Log(1+x) and setting x=1 gives:

Log(2) = 1 - 1/2 + 1/3 - 1/4 + 1/5 - ......

And we we rearrange it as follows:

= (1 + 1/3 + 1/5 + 1/7 + ....) - (1/2 + 1/4 + 1/6 + 1/8 + ....)
= (1 + 1/3 + 1/5 + 1/7 + ....) + (1/2 + 1/4 + 1/6 + 1/8 + ....) - 2(1/2 + 1/4 + 1/6 + 1/8 + ....)
= (1 + 1/2 + 1/3 + 1/4 + ....) - (1 + 1/2 + 1/3 + 1/4 + ....)
= 0

:eek: :confused:

However, there is a subtle reson for this. It's because you can't arbitrarily rearrange the terms in a sum unless it is absolutely convergent. The sum for Log(2) is only conditionally convergent. This has all been proved many moons ago in Real Analysis.

:)
 
Originally posted by AlphaNumeric

Put it like this, if there existed such a simple to show (and correct) paradox in mathematics that even those who have no mathematical ability could understand it, maths wouldn't have lasted as long as it has. There isn't even a paradox or contradiction anywhere in it so far.
Click to expand...

Well, godel created a mathematical paradox, so that's a little inaccurate. But if I misunderstand you, then feel free to shoot me. :)

oh, and like alpha says, maths isn't defined by the rules of algebra, it's the other way round. Algebra is a language, and still has to obey the laws of maths. Just because you can write x/0, doesn't mean you can do it.
 
Originally posted by growse
Well, godel created a mathematical paradox, so that's a little inaccurate. But if I misunderstand you, then feel free to shoot me. :)
Click to expand...
A lot of Godels work is very mind bending, and IIRC he spent the vast majority of his time on such things as paradoxes. His idea was more about the general flaws in pretty much communication and description, be it English, French or Maths. "I am lying" is a paradox of language and logic.

Suffice to say, there doesn't exist a thing where someone says "Oh, ignore that, or it proves everything wrong" in maths. It shows that there exists problems towhich a solution cannot be found (plenty of things in maths without an analytic solution). It shows we cannot do everything, but then we knew that ;)
 
Last edited:
Originally posted by AlphaNumeric
A lot of Godels work is very mind bending, and IIRC he spent the vast majority of his time on such things as paradoxes. His idea was more about the general flaws in pretty much communication and description, be it English, French or Maths. "I am lying" is a paradox of language and logic.

Suffice to say, there doesn't exist a thing where someone says "Oh, ignore that, or it proves everything wrong" in maths. It shows that there exists problems towhich a solution cannot be found (plenty of things in maths without an analytic solution). It shows we cannot do everything, but then we knew that ;)
Click to expand...

Ah, yes, but I think it's important to say that it goes a bit further. Godel managed to show that there is such a thing as an unprovable truth in maths - a statement that is true, but not provable. But yes, we already knew that :)
 
Originally posted by growse
Ah, yes, but I think it's important to say that it goes a bit further. Godel managed to show that there is such a thing as an unprovable truth in maths - a statement that is true, but not provable. But yes, we already knew that :)
Click to expand...
This is true - but that doesn't means there are holes in maths. I think the jist of the theorem is that within ANY closed form logical system, it is possible to create (true) theorems, which cannot be proven.

This isn't to say maths is wrong, it just means that it can't do EVERYTHING! I just wanted to clarify this, to those who were saying "Maths is wrong, and I have proved it because obviously 1 != 0.999.....".

:)
 
Isn't that pretty much a given within any axiomatic system? (at Growse)

Actually, scratch that, I'm sure I just sweeped over probably enough logical reasoning to choke a large animal of somekind. I'd expect such a statement to be true, though couldn't prove it (somewhat ironic isn't it :p).

Godel worked on areas that I think people like Harley have been talking about in this thread. There are times when you need to look at axiomatic systems as a whole, and question general trends, possibilities and ideas. These processes can produce some very interesting ideas, discussions and results, and from what I've gathered from reading, Godel thought in a way that few people do, and had genius to help it along. Probably like a mixture of Prof Gowers and Prof Korner (though non-cambridge people, and even non-mathmos might not know them) and possibly applified in ability (scary thought that). I'd be ecstatic if I had 1/10 the ability of such people.......

Suffice to say that 0.9r = 1 is a provable truth, so The Incompleteness Theorem doesn't apply in this case ;)
 
Last edited:
In reference to this equation:

x = 0.9r

(times 10)

10x=9.9r

- x

9x = 9

x= 9/9

x=1

One of my fellow maths students is trying to disprove it using:
x = 0.9r

(times 1.5)

1.5x = 1.49r

-x

0.5x = 0.49r

x = .49r/0.5

x = 0.9r

and not 1
Click to expand...

Somebody please help me put him right
icon14.gif
 
haha, he given up:p

Geometric Progession (for 0.9r)
i.e
0.9 + 0.09 + 0.009 ....

The work out sum to infinity, we use formula a / (1-r)
where a= first term
and r=common difference


sooo...

0.9 / (1- 0.1)

= 0.9 / 0.9

= 1
 
Originally posted by hendrix
In reference to this equation:
One of my fellow maths students is trying to disprove it using:
x = 0.9r

(times 1.5)

1.5x = 1.49r

-x

0.5x = 0.49r

x = .49r/0.5

x = 0.9r

and not 1

Somebody please help me put him right
icon14.gif
Click to expand...

I'm not a real mathsy person but when you subtract x, you have subtracted 1 and not 0.9r which you have stated = 0.9r in the first line?

I may be wrong!!
 
Status
Not open for further replies.
Back
Top Bottom