0.9999....(recurring) = 1?

George,Y · 2007-06-09 03:11:23

Ricky wrote:

Compared to older models of integers, rationals, or finite digits, the one of infinite digits is really logically weak. If you truely want to use it mikau, you have to tolerate contradictions.
You have never shown a contradiction, the only thing you have shown is "I don't like the standard definition of real numbers."

well, well, well, I really doubt your ability to comprehend or your memory.

0.999... been composed of values like 0.9, 0.09, ... from left to right, and infinitesimals from right to left. One sum, two natures of elements. Isn't this enough??

Identity · 2007-06-09 04:58:48

Anthony, why can't you directly challenge the proof offered in post #25? What's so hard to accept about that? If you are right, then algebra is wrong, and I don't think algebra is wrong. Could you please state what you think is wrong about the proof?

Ricky · 2007-06-09 08:34:34

0.999... been composed of values like 0.9, 0.09, ... from left to right, and infinitesimals from right to left. One sum, two natures of elements. Isn't this enough??

That doesn't even make sense. The real numbers don't consist of infinitesimals, let alone the rational numbers. And what do you mean one sum? What do you mean two natures?

Forget it George. You don't even accept the existence of irrational numbers, the construction of the real numbers from the rationals, or how we define decimal expansion of a real number. It is therefore pointless to even try to talk about 0.999...

You don't see a definition for what it is: a definition. We are just trying to state something explicitly so that everyone is talking about the same thing, for example with decimal expansion of real number. A definition can't be wrong, George, because that's all it is, a definition. It doesn't mean you can't do things differently, but if you're going to talk about decimal expansion of real numbers you should go off of the standard definition of decimal expansion of real numbers. You refuse to, for whatever reason.

Ricky · 2007-06-09 10:59:46

I did reading from a book I bought a while back, From Numbers To Analysis. Anyone who is interested in seeing how numbers are constructed, what it actually means to be a number, how to prove that 1 + n = n + 1 for all integers n, and the like, I strongly recommend it. I personally find it extremely fascinating.

I stumbled across the following remark:

6.4.4 Remark. Note that an expression of hte type
is not a decimal

I must say, I was a bit shocked at first. But then I continued reading:

The reason is that the number represented by this decimal is the same as by the decimal

I flipped back a few pages for their definition of a decimal:

6.4.1 Definition. A sequence a_0, a_1, a_2, ... of integers is called a decimal if for every i ≥ 1, 0 ≤ a_i ≤ 9 and there does not exist any N in the naturals such that a_n = 9 for all n > N.

So their definition is carefully crafted around not allowing the existence of 0.999... At first, I didn't like this definition. Though it had no effect on our ability to represent real numbers since 0.999... = 1, it seemed restrictive. But there are a few interesting things this does do. First, there is now a 1-1 correspondence between real numbers are decimal representations. Furthermore, two real numbers are equal if and only if their decimal representations are equal.

And interesting definition, with many what I would call positive side effects. I don't believe this is standard though, it seems as if most other mathematicians accept the definition which allows 0.999 to be a decimal.

George,Y · 2007-06-10 01:14:00

Okay, I've got it. Reals are your last shelter. I will refute all of these stuffs soon (including Cantor Set), in paper ( to save my effort in a way).

But now I insist:
As long as it's within the conventional rationals range (only integers, rationals, Cauchy limit defination), the logic problem of 0.999... does holds.

In a way you are right: I am not most mathematicians- I am special.

Anthony.R.Brown · 2007-06-10 01:33:14

To Identity

Quote: "Anthony, why can't you directly challenge the proof offered in post #25? What's so hard to accept about that? If you are right, then algebra is wrong, and I don't think algebra is wrong. Could you please state what you think is wrong about the proof?

A.R.B

First of all you need to read all of my Post #773 to Know why the so called Proof below Post #25 is being Calculated in the Wrong way!

#25

Basic proof for those of you who are confused:

x = 0.999...
10x = 9.999...
9x = 10x - x = 9
9x ÷ 9 = x
9 ÷ 9 = 1

∴ x = 1
∴ 0.999... = 1

A.R.B

The above is doing Multiple Calculations! and not the Single Calculation that is to Calculate Infinite/Recurring 0.9
By doing Multiple Calculations Any Problem can be Continuously changed! to a point where the original Problem has been lost! and Proved? in any way the person wants it to be Proved?
To show this I will take the so called end of the above so called Proof #25 and continue it in my favour!

∴ x = 1
∴ 0.999... = 1 " A.R.B and now again ( 1 / 0.9 ) x ( 0.9 ) = 0.999...

The above shows why any Math problem must be solved according to the original ( Start ) of the problem and not extended beyond all recognition! and made into another problem!

Last edited by Anthony.R.Brown (2007-06-10 01:38:49)

Ricky · 2007-06-10 04:06:55

In a way you are right: I am not most mathematicians- I am special.

To think outside the box, you must first know what lies within.

As long as it's within the conventional rationals range (only integers, rationals, Cauchy limit defination), the logic problem of 0.999... does holds.

No, not at all. There is no logic problem with 0.999... since this is a repeating decimal it is therefore rational and equal to 1.

George,Y · 2007-06-11 02:43:08

You're gonna stay in the box and play with the rules

and I'm gonna see if I can add some new paper waste to this world or not, dude

Anthony.R.Brown · 2007-06-11 22:44:41

Quote:" No, not at all. There is no logic problem with 0.999... since this is a repeating decimal it is therefore rational and equal to 1. "

A.R.B

1 Is not an Infinite/Recurring Number/Value! 1 has no ( repeating decimals! )

Infinite/Recurring 0.9 Is just one of Many Number/Values that are Infinitely < 1

Last edited by Anthony.R.Brown (2007-06-12 02:17:50)

Maelwys · 2007-06-12 03:06:33

Some questions about 0.999..., if 0.999... is less than 1.

1/ What is the result of 1 minus 0.999...?
2/ What is the result of 1 divided by 0.999...?
3/ What is the result of 2 divided by 0.999...?
4/ What is the average of 1 and 0.999...?
5/ What is the result of 0.999... multiplied by 0.999...?
6/ What is the result of 0.999... multiplied by 2?

If you believe that 0.999... < 1, please feel free to answer each of these questions above to help the rest of us understand this relationship. Thank you.

Anthony.R.Brown · 2007-06-13 00:23:36

To Maelwys

Quote:" Some questions about 0.999..., if 0.999... is less than 1.

1/ What is the result of 1 minus 0.999...?
2/ What is the result of 1 divided by 0.999...?
3/ What is the result of 2 divided by 0.999...?
4/ What is the average of 1 and 0.999...?
5/ What is the result of 0.999... multiplied by 0.999...?
6/ What is the result of 0.999... multiplied by 2?

A.R.B

1/ 1 - 0.999... = 0.001...
2/ " You can't divide an Infinite Number into 1 Because no one Knows How long an Infinite Number is "
3/ " You can't divide an Infinite Number into 2 Because no one Knows How long an Infinite Number is "
4/ " The Average of 1 is 1 " " The Average of 0.999... is 0.9 "
5/ " You can't multiply an Infinite Number Because no one Knows How long an Infinite Number is "
6/ " You can't multiply an Infinite Number Because no one Knows How long an Infinite Number is "

Last edited by Anthony.R.Brown (2007-06-13 00:35:45)

Maelwys · 2007-06-13 00:37:13

Anthony.R.Brown wrote:

2/ What is the result of 1 divided by 0.999...?
2/ " You can't divide an Infinite Number into 1 Because no one Knows How long an Infinite Number is "
3/ What is the result of 2 divided by 0.999...?
3/ " You can't divide an Infinite Number into 2 Because no one Knows How long an Infinite Number is "
5/ What is the result of 0.999... multiplied by 0.999...?
5/ " You can't multiply an Infinite Number Because no one Knows How long an Infinite Number is "
6/ What is the result of 0.999... multiplied by 2?
6/ " You can't multiply an Infinite Number Because no one Knows How long an Infinite Number is "

Those all sound like cop-out answers to me. So 0.111... x 2 isn't 0.222...?

Anthony.R.Brown wrote:

1/ What is the result of 1 minus 0.999...?
1/ 1 - 0.999... = 0.001...

For now I'll ignore that your ... doesn't make sense in that context (since it is generally accepted to represent continuing the same pattern for infinity, and your pattern seems to imply that the number is either 0.001001001001... or at least 0.00111111....

Anthony.R.Brown wrote:

4/ What is the average of 1 and 0.999...?
4/ " The Average of 1 is 1 " " The Average of 0.9 is 0.9 "

I think you misunderstood the question. You can't have an average for each number seperately (well, okay you can... but it's stupidly obvious), I'm asking what the average of the two numbers is. The average of 1 and 3 is 2. The average of 1 and 2 is 1.5. What is the average of 1 and 0.999...? (basically, what is the number halfway between them)

Anthony.R.Brown · 2007-06-13 01:00:38

To Maelwys

1 Is not an Infinite/Recurring Number/Value! 1 has no ( repeating decimals! )

Infinite/Recurring 0.9 Is just one of Many Number/Values that are Infinitely < 1

Maelwys · 2007-06-13 01:10:12

Anthony.R.Brown wrote:

To Maelwys
1 Is not an Infinite/Recurring Number/Value! 1 has no ( repeating decimals! )
Infinite/Recurring 0.9 Is just one of Many Number/Values that are Infinitely < 1

You've repeated that anthem several dozen times already on this thread. Repeating it again, without any new context, doesn't add anything to your arguments. Can you please explain which of my above questions is this meant to address? And how it addresses the specific question? Thanks.

Anthony.R.Brown · 2007-06-13 01:14:33

To Maelwys

OK! you tell me 1 / Infinite/Recurring 0.9

Maelwys · 2007-06-13 01:18:48

Anthony.R.Brown wrote:

To Maelwys
OK! you tell me 1 / Infinite/Recurring 0.9

Okay. Using my understanding of the value of 0.999...,
1 / 0.999... = 1, because 0.999... = 1, and anything divided by 1 equals itself.

But since you believe that 0.999... < 1, obviously this solution doesn't work for you, which is why I'm curious what you believe the answer would be to a question like that.

Anthony.R.Brown · 2007-06-13 01:22:28

To Maelys

Quote:" Okay. Using my understanding of the value of 0.999...,
1 / 0.999... = 1, because 0.999... = 1, and anything divided by 1 equals itself.

A.R.B

How can a Number/Value ( 1 ) which is not Infinite/Recurring = a Number/Value that is (0.999...)

Maelwys · 2007-06-13 02:14:13

Anthony.R.Brown wrote:

To Maelys
Quote:" Okay. Using my understanding of the value of 0.999...,
1 / 0.999... = 1, because 0.999... = 1, and anything divided by 1 equals itself.
A.R.B
How can a Number/Value ( 1 ) which is not Infinite/Recurring = a Number/Value that is (0.999...)

You're avoiding my question again. For the moment I'm not arguing that 1 and 0.999... do or don't equal each other. I'm just asking, based on your understand that 0.999... < 1, what is the result of 1 / 0.999...
It should be a fairly straightforward question, no?

Last edited by Maelwys (2007-06-13 02:14:36)

Anthony.R.Brown · 2007-06-13 23:31:05

To Maelwys

Quote:" You're avoiding my question again. For the moment I'm not arguing that 1 and 0.999... do or don't equal each other. I'm just asking, based on your understand that 0.999... < 1, what is the result of 1 / 0.999...
It should be a fairly straightforward question, no? "

A.R.B

To be able to Divide a Number/Value into another Number/Value we have to know how long! or how many Decimal points both Number/Values have!

Example 1/0.9 = 1.111... Because 0.9 is only one decimal place! (going back to my original point about only being able to solve the Infinite/Recurring 0.9 Problem from the Sart! ) we can find the True calculation Answer!

In the case of trying to Calculate 1/Infinite 0.9 there is no way of knowing how many Decimal places long! Infinite 0.9 is! but because it is an Infinite repetition of the Start Value! we know that however long Infinite 0.9 is! it must always be < 1 the actual Value can't be wrote down! just given as 0.(n) < 1

Maelwys · 2007-06-14 00:28:49

Alright, if you'll stick by your cop-out answer, that's fine for now. But it still doesn't answer:
4/ What is the average of 1 and 0.999...?

Anthony.R.Brown · 2007-06-14 01:06:48

To Maelwys

Quote:" Alright, if you'll stick by your cop-out answer, that's fine for now. But it still doesn't answer:
4/ What is the average of 1 and 0.999...? "

A.R.B

If you can Show how to Average an Unknown Number/Value Concerning Length and/or Decimal Places! then you will be a Genius!! and Far better than any Math person Ever!

luca-deltodesco · 2007-06-14 01:13:32

hahahaha.

ok then, the average of 0.333... and 0.333... is *hold breath* 0.333....
the average of 0.333... and 0.999.... is *hold breath* 0.666....
the average of 0.111... and 0.777..., is *hold breath* 0.444....
the average of 0.12(12)... and 0.24(24)... is *hold breath* 0.18(18)...
the average of 0 and 0.111... is *hold breath* 0.0555...

Last edited by luca-deltodesco (2007-06-14 01:16:51)

Anthony.R.Brown · 2007-06-14 01:23:36

If you can Show how to Average an Unknown Number/Value Concerning Length and/or Decimal Places!

So How long/Decimal Places does 0.333... have ?

Because 0.3 is one Decimal Place!

luca-deltodesco · 2007-06-14 01:25:28

you don't need to know.

also, think about this:

0.3 is the same as 0.3000...., which just so happens to be in the same format as 0.0555... or 0.333...

Last edited by luca-deltodesco (2007-06-14 01:26:11)

Anthony.R.Brown · 2007-06-14 01:31:22

1 / 0.3 = 0.333...

1 / 0.33 = 0.030...

And not 0.3 is the same as 0.3000....,

Math Is Fun Forum

#776 2007-06-09 03:11:23

Re: 0.9999....(recurring) = 1?

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