0.9999....(recurring) = 1?

Maelwys · 2007-03-13 03:27:23

Anthony.R.Brown wrote:

Quote: "
So where does the 1 go in the second number? If the 0s are infinite/recurring, always the same, with no after, then there's nowhere to put the 1. "
A.R.B
The 1 does not go anywhere!
with 0.9999..... even though we can see many .9's we are in fact always looking at one .9
with 0.0001..... even though we can see many .0's we are in fact always looking at one .1
because .1 is the actual Infinite Difference! from the Start!
There has to be a Value between 0.9 and 0.1 and that is 0.000...the Decimal Shift!
The Infinite Difference does not Grow! is stays the same and is shifted!

Aha! This is what I've been looking for, the basic place where we're apparently having trouble speaking the same language... and it seems to have been right at the basic definition of an infinitely recurring decimal, which is obviously why we think so differently on the equality (or lackthereof) of 0.999...

Lets take it back one step further then, and avoid 0.999... for now, since by my understanding of things, it's a special case (not in yours obviously, but by picking a different infinitely recurring decimal for now, we can avoid that incongruity).

So, instead we'll deal with 0.888... as I understand what you're saying, when I look at 0.888... I'm actually looking at 0.8? And the difference between 1 and 0.888... is 0.2? Is that right?

luca-deltodesco · 2007-03-13 04:09:54

anthony, what is your first language (please say its not english)

Anthony.R.Brown · 2007-03-13 04:10:09

To Maelwys
Quote:

"So, instead we'll deal with 0.888... as I understand what you're saying, when I look at 0.888... I'm actually looking at 0.8? And the difference between 1 and 0.888... is 0.2? Is that right?"

A.R.B agreed! let's look at the above! p.s I will only be online for the next 45 minutes! incase I cut off the conversation!

Anthony.R.Brown · 2007-03-13 04:16:50

To

luca-deltodesco

Quote: " anthony, what is your first language (please say its not english) "

A.R.B

I'm good at Sign language! wish I could show you!

Last edited by Anthony.R.Brown (2007-03-13 04:17:36)

luca-deltodesco · 2007-03-13 04:19:50

what is your first language? IS it english? If not, perhaps we might be able to communicate better in whatever your first language is if we know it.

Anthony.R.Brown · 2007-03-13 04:21:18

To

luca-deltodesco

I doubt if WE! could with you!

Maelwys · 2007-03-13 04:50:11

Anthony.R.Brown wrote:

To Maelwys
Quote:
"So, instead we'll deal with 0.888... as I understand what you're saying, when I look at 0.888... I'm actually looking at 0.8? And the difference between 1 and 0.888... is 0.2? Is that right?"
A.R.B agreed! let's look at the above! p.s I will only be online for the next 45 minutes! incase I cut off the conversation!

Hey, I think we might be getting somewhere here!

Now here's my next question... if 0.888... is really 0.8, then what about the thing you said earlier, that we have to remember where the numbers come from.
To get 0.888... I divide 8 / 9
To get 0.8 I divide 8 / 10
But surely 8/9 = 8/10 can't be true, is it? So how is that 0.888... could be equal to 0.8?

Anthony.R.Brown · 2007-03-13 04:53:42

To Maelwys

A.R.B

as I said I will have to go for now! I will answer! 14/03

Sekky · 2007-03-13 09:31:40

Anthony.R.Brown wrote:

To Sekky
Is the above ENGLISH!!

Your attitude absolutely stinks, you're going to end up failing at everything you do if you won't admit your mistakes, especially when the counter-proof is staring you right in the face. Maybe in a thousand years when you finally grow up you'll be able to discuss some vaguely convtroversial maths, this is trivially obvious to anyone with a basic understanding of calculus.

krassi_holmz · 2007-03-13 09:36:14

To : Anthony
Note: very funny way of posting
It could be better if the posts were true.

Anthony.R.Brown · 2007-03-14 23:35:09

To Maelwys

First of all Congratulations! With your persistence's in trying to understand the problems! Associated with Infinite/Recurring 0.9 Everyone will all Benefit from it!
I have not been able to explain fully! What I'm trying to say, because we have only been concentrating on Infinite/Recurring 0.9 as if it is unique in the world of Infinite/Recurring Number/Values.
I have always known this not to be true! All Infinite/Recurring Number/Values behave in the same way according to the type!! They belong to,there are Normal Number/Values and Infinite/Recurring Number/Values and Infinite/Recurring Difference Number/Values.

Example Normal 0.1 = 0.1

Example Infinite/Recurring 0.1 = 0.111...

Example Infinite/Recurring Difference 0.1 = 0.001...

With the three examples above! They can all equal the Value 1

Normal 0.1 + Normal 0.9 = ( 0.1 + 0.9 ) = 1

Infinite/Recurring 0.1 + Infinite/Recurring 0.9 = ( 0.111... + 0.999... ) = 1

With the second example above! Because the 0.1 and 0.9 are Infinite/Recurring the same Start Values are being repeated over and over again!and actually equal ( 0.1 + 0.9 ) and not ( 0.111... + 0.999... ) as if the Values are getting larger which of course would be 1.111...etc.

Infinite/Recurring Difference 0.1 + Infinite/Recurring 0.9 = ( 0.001... + 0.999... ) = 1

With the third example above! Because the 0.1 is the Infinite/Recurring Difference we Know that the Value has come from ( 1 - Infinite/Recurring 0.9 ) there will always be a .1 Difference the .00 is the Decimal Point Shift!

The above Shows how there can be Three Different Types of 0.1 and why its Important to know which one is being used in any Calculations!

---------------------------------------------------------------------------------------------------------------

To Maelwys Examples for 0.8

Again you are Comparing Normal Number Calculations with Infinite/Recurring Number Values!

8 / 10 = 0.8

8 / 9 = 0.888...

For 0.9 The Normal Calculations above give Different Results! And also for some other 0.1 to 0.9 Values!

9 / 12 = 0.75

9 / 11 = 0.8181818...

9 / 10 = 0.9

9 / 9 = 1

9 / 8 = 1.125

So the above shows using the Same Normal Calculations! There is no Infinite/Recurring 0.9
What we need is a more Precise way for Calculating Any Infinite/Recurring Number Values!

The best way I know is Multiplying the Original Start Value! By Infinite 1.111...
This will then give a True Infinite/Recurring Number Value! For Any 0.1 to 0.9 Value.

Now going back to your original question how can 8 / 10 be equal to 8 / 9 the Answer is they are not! Concerning Normal Number Calculations! Even though they both have a 0.8
Firstly the 0.8 for 8 / 10 is 0.8 and not the Start! For 8 / 9 it does have a 0.8 at the Start! But as I have pointed out,it is not possible with some other 0.1 to 0.9 Infinite/Recurring Number Values!
Calculated in the way you have put forward!

Ricky · 2007-05-18 14:00:26

I'm really hoping not to revive this topic, but I came across this today. My Real Analysis book asks the following question:

Consider real numbers between 0 and 1 with decimal digits of only 4 and 7. Is this set dense in the real numbers?

After a bit of playing around, I found the numbers:

0.477777777...
0.744444444...

It should be apparent that there is no number of this set between these two numbers. But this got me thinking: what if I did this with 0's and 1's:

0.011111111...
0.100000000...

Now remember that any decimal real number we can write as a binary real number. So since the real numbers are dense in the reals, these numbers should be dense. But given the above example, they are not. That is, unless you accept that:

0.011111111... = 0.10000000...

So either reals aren't dense, or the above two numbers are equal.

Anthony.R.Brown · 2007-05-19 01:27:32

To Ricky

Quote: " 0.011111111... = 0.10000000...

So either reals aren't dense, or the above two numbers are equal. "

A.R.B

So are you saying the two Numbers above are the Same?

Ricky · 2007-05-19 05:51:46

In binary, yes.

Anthony.R.Brown · 2007-05-19 23:54:16

To Ricky

Quote:" In binary, yes. "

A.R.B

In that case! it has nothing to do with this Post??

Ricky · 2007-05-20 02:17:10

There, you are wrong. This shows that:

And similar results can be obtained for infinite decimals in any base.

George,Y · 2007-05-20 03:36:45

Simple, Ricky, don't know why you have gotten stucked.

The relation between 0.0111... and 0.1 in binary is just that between 0.999... and 1 in ten system.

The "infinite" sum concept may help you understand it.

Ricky · 2007-05-20 11:31:38

Simple, Ricky, don't know why you have gotten stucked.
The relation between 0.0111... and 0.1 in binary is just that between 0.999... and 1 in ten system.

Yes, that was the point of my post. I'm not sure how you think I've gotten "stucked" (not making fun of grammar, I honestly don't know what you mean).

George,Y · 2007-05-20 21:40:51

Uh, got it. I think the misunderstanding came from the either one conclusion-I thought you began to doubt whether reals are dense if they are not equal.

Anthony.R.Brown · 2007-05-21 00:29:22

To Ricky

Quote: Post 00.17.10 " There, you are wrong. This shows that: "

\sum_{i=2}^\infty \frac{1}{2^i} = \frac{1}{2}

A.R.B

for the above you are using Fractions again!! The Infinite/Recurring 0.9 problem uses Normal Decimal Calculations!

Nothing changes the fact that Infinite 1.111.... x 0.9 will Never equal 1

A.R.B

Last edited by Anthony.R.Brown (2007-05-21 00:30:05)

Sekky · 2007-05-21 03:13:53

Anthony.R.Brown wrote:

Nothing changes the fact that Infinite 1.111.... x 0.9 will Never equal 1

Of course not, apart from the fact that it clearly does to anybody with a shred of sense and you've seen it proved a million times over, but other than that, of course not.

Ricky · 2007-05-21 07:35:22

I thought you began to doubt whether reals are dense if they are not equal.

Ah, I see. What I'm saying George is that if you don't accept that 0.0111... = 0.100..., then you must conclude that the real numbers are not dense. Similarly, if you don't accept that 0.999... = 1, then you must conclude that the reals are not dense.

Anthony.R.Brown · 2007-05-21 22:57:47

Quote:" Of course not, apart from the fact that it clearly does to anybody with a shred of sense and you've seen it proved a million times over, but other than that, of course not. "

A.R.B

all we need to see is the Math on how from 0.999...(.9) to (1) is done!
it can only be done by + (.1) which as we all know! is no longer Infinite/Recurring!!..................

Sekky · 2007-05-22 05:12:51

Anthony.R.Brown wrote:

all we need to see is the Math on how from 0.999...(.9) to (1) is done!
it can only be done by + (.1) which as we all know! is no longer Infinite/Recurring!!..................

No.

Hey Anthony, can I see your degree?

George,Y · 2007-05-22 11:47:53

Sekky wrote:

Hey Anthony, can I see your degree?

This is a typical elitism

Math Is Fun Forum

#626 2007-03-13 03:27:23

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#634 2007-03-13 09:31:40

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#644 2007-05-20 21:40:51

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#645 2007-05-21 00:29:22

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#646 2007-05-21 03:13:53

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