Is 1 = 0.9999......

[MadRat]

Originally posted by: TuxDave
For instance, I'm ASSUMING that pi has an infinite number of digits after it to represent the value. (I'm not sure, but it's the best I can do). If a digit can exist at the infinite place, then there does exist a last number to pi right? Does that make sense? Does it make sense to claim that an infinitely long stairway has a top? If you think so, then this becomes a philosophy of math debate.
-Dave-

Pi is 22/7ths, correct? Then the fraction of 22/7ths is best defined in a base 7 numbering system, not a base ten.

Infinite is undefined, just as a variable of x can be undefined to solve for other variables. With all undefined variables we can use its meaning to exclude values, but we can never assign a value to it. Nor can we use a limit to define an undefined value, which is why we use limits to draw a line definitive as to what it is NOT. The math geniuses around here do not want to stick to the strict meaning of limits.

 

[bigredguy]

If .999...=1
then couldn't 1=1.00..001?

 

[RossGr]

Madrat, I addressed your issue about the nonsensical "infinite postion". As soon as you postion a digit in a real number it occupies a fixed decimal place therefore it is NOT at infinity. Sorry this is simply how real numbers are constructed.

 

[MadRat]

Originally posted by: Dufusyte
I reply, no, 1/3 does not equal 0.3333...

In truth, 1/3 can only be expressed as a fraction. It *cannot* be expressed as a decimal. If you try to express it as a decimal, you run into an interminal process, and the numerical value of the interminal process will lie between the point where you leave off the process and 1/3, but it will never reach 1/3. So it is erroneous to say that 1/3 = 0.33333..., and hence the rest of the proof fails as well.

As someone slyly pointed out in the last debate, 1/3 is equal to .1 in base 3 numbers. The only confusion is when we stick to base 10 numbers.

 

[MadRat]

Originally posted by: RossGr
Madrat, I addressed your issue about the nonsensical "infinite postion". As soon as you postion a digit in a real number it occupies a fixed decimal place therefore it is NOT at infinity. Sorry this is simply how real numbers are constructed.

You did no such thing. You pitched a fit and waved credentials around is all, but never did you produce an argument.

Face it, your proof is flawed from points of view outside your own.

 

[RossGr]

Pi is 22/7ths, correct? Then the fraction of 22/7ths is best defined in a base 7 numbering system, not a base ten.

No, Pi is approximatly 22/7s .

To the best of knowledge there is no number base in which pi is rational.

Infinity certinaly is defined, that is how we are able to work with it consistenty. There are rules, as long as you follow the rules the results are consistent and meaningful. You break the rules and you get nonsence.

1.000...001 >1 Since in this context the ellipsis indicates a finite number of missing digits, If you terminated a string of digits it is not infinite.

 

[MadRat]

Originally posted by: bigredguy
If .999...=1
then couldn't 1=1.00..001?

These math geniuses deny the existence of any number ending with .000...1 because they short-sidedly deny the existence of any value at the infinite position other than 0.

 

[RossGr]

Would you quit worring about your blatent ignorance and read the post.

 

[MadRat]

Originally posted by: RossGr
No, Pi is approximatly 22/7s .

Okay, RossGr, to make my point I want to compute Pi on a supercomputer. What is the equation?

 

[MadRat]

Originally posted by: RossGr
Would you quit worring about your blatent ignorance and read the post.

I've pointed out your logic flaw. Blatant ignorance? LOL, I told you before this behavior just proves more so how is weak your stand.

 

[silverpig]

Originally posted by: MadRat

Originally posted by: RossGr
No, Pi is approximatly 22/7s .

Okay, RossGr, to make my point I want to compute Pi on a supercomputer. What is the equation?

There are a number of expressions which converge to Pi.

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

is probably the most common one.

 

[silverpig]

Originally posted by: Dufusyte
1/3 does NOT equal 0.3333333...

because you cannot write an infinite string of 3's.

A "repeating" number is not a number at all. It is a process, indeed, it is a process without termination.

Therefore "0.999999..." is, likewise, not a number, but rather a process without termination.

Notice that we have to write dots at the end of it to symbolize the fact that it is a non-terminating process. Similarly, if we write it with a superscript Line over the 9, the line indicates that it is a non-terminating process, and not a rightful number.

So "0.99999..." does not equal ANYTHING, since it is not even a fixed number itself.

It is a process whose numerical value lies between the point where you leave off the process, and 1. For example, if you write it as "0.9...", then its value lies between 0.9 and 1. If you write it as "0.99999..." then its value lies between 0.99999 and 1. Please note: the value NEVER reaches 1, no matter how long you extend the process.

Therefore, to the people who say:

1/3 = 0.3333...

I reply, no, 1/3 does not equal 0.3333...

In truth, 1/3 can only be expressed as a fraction. It *cannot* be expressed as a decimal. If you try to express it as a decimal, you run into an interminal process, and the numerical value of the interminal process will lie between the point where you leave off the process and 1/3, but it will never reach 1/3. So it is erroneous to say that 1/3 = 0.33333..., and hence the rest of the proof fails as well.

Hence, the "proofs" advanced in prior posts are shown to be in error.

Quod est demonstratum.

0.333... is not the representation of writing a 0 then a . then a 3, then another 3, and then another... It is the representation of a decmial point followed by an infinite number of 3s. The 3s are already there. 1/3 DOES in fact equal 0.333... because 0.333... is 0.3 with an infinite number of threes.

It is not a progression either.

But, for argument's sake, let's grant that you are right. You still have not disproven any of the proofs in this thread. The 0.333... * 3 is not a formal proof. Find me a number between 0.999... and 1.

 

[silverpig]

Originally posted by: MadRat

Originally posted by: bigredguy
If .999...=1
then couldn't 1=1.00..001?

These math geniuses deny the existence of any number ending with .000...1 because they short-sidedly deny the existence of any value at the infinite position other than 0.

You cannot have an "infinite position."

Please, put these in order from largest to smallest (feel free to throw out any you say don't exist, but please explain why they don't)

a) 0.000...0001
b) 0.000...01
c) 0.000000....00....0000...1
d) 0.0000...00....000000...000...001...00001
e) 0.000...0000001

 

[ElFenix]

Originally posted by: MadRat

Originally posted by: bigredguy
If .999...=1
then couldn't 1=1.00..001?

These math geniuses deny the existence of any number ending with .000...1 because they short-sidedly deny the existence of any value at the infinite position other than 0.

thats not the infinith position! thats a finite position!

 

[MadRat]

Originally posted by: silverpig
But, for argument's sake, let's grant that you are right. You still have not disproven any of the proofs in this thread. The 0.333... * 3 is not a formal proof. Find me a number between 0.999... and 1.

For every formal rule their is a formal exception.

Its really simple to refute his rule in this case. Since infinite is an indefinite value then you cannot halve it. Since you cannot halve it then it does not mean it has no position, rather simply that the number does NOT equal something that fits in a finite or definite form. Can you halve a null value? Why or why not? Null and infinite values are undefinitive and therefore you cannot express them with finite or definite value, making them exceptional to your rule.

 

[MadRat]

Originally posted by: ElFenix

Originally posted by: MadRat

Originally posted by: bigredguy
If .999...=1
then couldn't 1=1.00..001?

These math geniuses deny the existence of any number ending with .000...1 because they short-sidedly deny the existence of any value at the infinite position other than 0.

thats not the infinith position! thats a finite position!

Again, you confuse a position with a value. The position can equal any number. In no way does the position affect the value,

 

[RossGr]

Just what is a Real number?
A Real number consists of an integer part and a fractional part, this thread is only disscussing the fractional part so I will not address the integer part.

The fractional part of a real number is defined by an infinite sum of the terms D(n) B^-n. Where B is the base and D(n) is an integer such that 0< D(n) < B-1. A couple of examples
if B = 2, D(n) is one of the set {0,1}
if B = 16 the set of integers is {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
if B= 10 the integers are {0,1,2,3,4,5,6,7,8,9}

Notice in each set the largest integer is B-1.

The sum looks like this

D(1) B^-1 + D(2) B^-2 + D(3)B^-3 + ... + D(n) B^-n + .... This sum continues on for all integers but every term corresponds to an integer and is labled by that integer. Since there is no integer called infinity there is no digit at some place called infinity, Again every digit is a number from the set of integers for the selected base corresponds to some integer N which specifies its location in the sum. If you do not specify N they you are not talking a real number. I can specify D(n) for any N in the sum represented by .999.... , D(n) = 9 for all n.

As long as the ellipsis occurs at the end of a number it represents an infinite repetition of a pattern, if it appears in the middle of a number it represents the omission of a finite number of digits.

To say that a "number" is a Real number implies that it can be represented as an infinite sum, the set of rational numbers coresponds to real numbers where we can specifiy every D(n). so for 1/3, D(n) = 3 for all n, this is an explicit definition. I cannot do that for Sqrt(2) or Pi they are not rational numbers they are irrational.

Since this topic is about rational numbers, I do not understand why Pi is even brought up, but the fact is there are algorithms which can compute any given digit of Pi, sorry I cannot post it herel. It is on the web, do a search.

 

[Sahakiel]

Ok, first a bit of background...

1. I only scanned the first and last pages, so if I repeated or missed something, my apologies.
2. I'm an engineer, so, er.. I might be biased.... uh..yeah..

My reasoning:

Mathematics is mathematics. You can only apply the rules of mathematics to a mathematical problem. Using generalized logic is also not applicable. Why, because generalized logic does not recognized the foundations of mathematics; for example, the definitions of operators.
Logic is applicable to mathematics only when taking into account the rules upon which mathematics is based. I can always post the proposition "1 + 1 = 3" and not only claim it's true, but also prove it.
How? One way is by defining my own definition to the operator '+'. Another way would be to define the number sequence as 1,3,4,2,11,13,... As long as we're working with my rules, it is valid and true. If, however, you do what most people do instinctively and apply the rules of mathematics, it's obviously false.

Basically, my basis for largely ignoring MadRat's posts (sorry) is because each post seems to be an attempt at applying common sense or philosophy to mathematics.
I mean no offense, but I fail to see how providing philosophical concepts or implying common sense and instinct proves mathematics. A truth table can always provide a contradiction to a logical argument, but it can not provide a proof. At least, not in digital logic design.

Therefore, philosophy has no place in mathematical proof. It may provide inspiration and it may lead to new definition, but it does not govern. This is similar to modern thereotical physics. Traditionally, physics has paved the road to new ideas and technology while mathematics has followed to provide a numerical basis. An example would be Newtonian physics.
Nowadays, with so much development in "strange" territory from Cosmology to Quantum Mechanics, mathematics has begun to lead the way more and more simply because instinct fails. New discoveries are occuring not because a physicist had a sudden insight, but because the mathematical equations lead to, simplify, resolve themselves, or whatever, into seemingly strange and nonsensical forms that imply new discoveries and are only proven years later when the technology catches up.

In essence, 0.9999... = 1 not because it's easy to understand, or because common sense tells us it is (which is a bad way to prove things, btw), but because the rules of mathematics laid the foundation for the data and the process of analysing the data and the logical conclusion following the rules results in the proposition being true.
You can argue until you're blue in the face that 0.999... != 1 simply because it would be implying that sitting in a chair with one's butt glued firmly to the seat results in one being the chair. One can pull out as many degrees and accredidations as there are in the world to cement the right to provide such a proof. However, doing so is completely irrelevant and is akin to refereeing a U.S. football game with the rules from British football.

One last thing:
I don't truly 100% believe in the mathematics, but I do believe it is proven correctly. If not, well, egg on my face..
On the other hand, few can accept the time-dilation effects of General Relativity. That concept powers your light bulb, by the way. (Electro-magnetic waves, aka. light, is a relativistic phenomenon)

 

[Dufusyte]

RossGr:
10^100 is a legitimate fixed number because it *can* be written out, if you take enough time to do it.

0.99999... is not a fixed number, because it is impossible to write it out, regardless of how much time and space you have for writing.

That is why 0.9999... is not a fixed number, but 10^100 is.

0.9999... is a conceptual range of values.

---

SilverPig:
If 0.9999... were a legitimate number with a fixed value, then I could find you a number between it and 1. The problem, my friend, is that 0.99999... is not a fixed number. I consider that to be the fault of the 0.9999... Indeed, it is not a number at all: it is bogus, and should be thrown out. If you want to write 1/3 as a decimal, then write it out, and call me when you have finished writing it, and they I will continue the discussion.

I have no respect for bogus, impostor, "pseudo-numbers" that have "etc" written on their tails, as in 0.9999etc.

---

The base 3 thing is interesting, but does not prove the point. I can believe that .1 + .2 = 1 in base 3, just like I can believe that 1/3 + 2/3 = 1 in fractions. But that still does not prove that 1/3 = 0.3333...

Any numberal followed by "..." should not dare to call itself a number. If I have to make up a term, I will call a "repeating number" an "incomplete number". Another name for it could be a "non-fixed number." This is because a repeating number does not have a definite value. That is where the error enters.

 

[Haircut]

Dufusyte, I think you should email all the math professors you can find as if all numbers that have non terminating digits are not fixed then you have just wiped out all irrational numbers and most rational ones.

 

[silverpig]

Originally posted by: Dufusyte
RossGr:
10^100 is a legitimate fixed number because it *can* be written out, if you take enough time to do it.

0.99999... is not a fixed number, because it is impossible to write it out, regardless of how much time and space you have for writing.

That is why 0.9999... is not a fixed number, but 10^100 is.

0.9999... is a conceptual range of values.

---

SilverPig:
If 0.9999... were a legitimate number with a fixed value, then I could find you a number between it and 1. The problem, my friend, is that 0.99999... is not a fixed number. I consider that to be the fault of the 0.9999... Indeed, it is not a number at all: it is bogus, and should be thrown out. If you want to write 1/3 as a decimal, then write it out, and call me when you have finished writing it, and they I will continue the discussion.

I have no respect for bogus, impostor, "pseudo-numbers" that have "etc" written on their tails, as in 0.9999etc.

---

The base 3 thing is interesting, but does not prove the point. I can believe that .1 + .2 = 1 in base 3, just like I can believe that 1/3 + 2/3 = 1 in fractions. But that still does not prove that 1/3 = 0.3333...

Any numberal followed by "..." should not dare to call itself a number. If I have to make up a term, I will call a "repeating number" an "incomplete number". Another name for it could be a "non-fixed number." This is because a repeating number does not have a definite value. That is where the error enters.

So you're saying Pi and sqrt(2) are not legitimate numbers? Neither are "fixed", you cannot write either out, so they are bogus?

 

[Dufusyte]

The only bogus aspect is trying to write them in decimal form.

I have no problem with a pretty fraction, such as 1/3.

But there is a problem when somebody tries to write it as a decimal, since it can't be done (never ends): 0.33333...

Similarly, pi is an entirely legitimate geometric reality, namely the circumference of a circle with a diameter of 1. However there is a definite problem with attempting to write this value in decimal format, since it cannot be done (interminable).

The decimal format cannot express certain values, and when people use "incomplete decimals" and "interminal decimals" in their equations, they introduce errors. Since the decimal format does not express the value precisely, the equations that use them also have fuzzy erroneous results.

Rule: if you can't write it like a normal number (without an elipsis, etc), then don't use it, or else it introduces a tiny error.

- That is, if you are concerned about tiny errors. If you are doing engineering where you don't care about tiny errors, then by all means use 3.1415926. But if you are talking Math Theory, where there is no allowance for errors, then you gots to be using fixed numbers, and no incomplete decimals please. Otherwise people start running into silliness like "show me a number between 0.999... and 1". Doh; if you're doing a proof, then you have no business letting an incomplete decimal onto the page in the first place. Only exact numbers, please.

 

[Haircut]

OK, I'll try and address your post point by point

1) The only bogus aspect is trying to write them in decimal form.
Yes, you can't write a non-terminating number completely in decimal form, that is why we have the notation that we do. 0.9 with a dot above the 9 signifies infinitely repeating 9s, thus we don't need to write it out in full.

2) I have no problem with a pretty fraction, such as 1/3.
Good, neither do I.

3) But there is a problem when somebody tries to write it as a decimal, since it can't be done (never ends): 0.33333...
That's why we have the notation above.

4) Similarly, pi is an entirely legitimate geometric reality, namely the circumference of a circle with a diameter of 1. However there is a definite problem with attempting to write this value in decimal format, since it cannot be done (interminable).
Agreed.

5) The decimal format cannot express certain values, and when people use "incomplete decimals" and "interminal decimals" in their equations, they introduce errors. Since the decimal format does not express the value precisely, the equations that use them also have fuzzy erroneous results.
There are some values that we can't write out in full using decimal notation, this does not make them any different to other numbers.

6) Rule: if you can't write it like a normal number (without an elipsis, etc), then don't use it, or else it introduces a tiny error.
WTF, now this I have a problem with. So, now you're saying I can't do any math with any irrational numbers or rational numbers with repeating decimals.
If we use an incomplete decimal expansion then it will introduce an error. If I say the sqrt(2) is 1.414, then 1.414*1.414 = 1.999396.
There is nothing wrong with doing the calculation simply using sqrt(2) though, I don't have to use the decimal form.

7) - That is, if you are concerned about tiny errors. If you are doing engineering where you don't care about tiny errors, then by all means use 3.1415926. But if you are talking Math Theory, where there is no allowance for errors, then you gots to be using fixed numbers, and no incomplete decimals please. Otherwise people start running into silliness like "show me a number between 0.999... and 1". Doh; if you're doing a proof, then you have no business letting an incomplete decimal onto the page in the first place. Only exact numbers, please.
I am concerned about tiny error, that's why I choose to use pi = pi, not pi = 3.1415926.
I can still do the math, I just use a symbol to represent the number so I don't have to use an incomplete (and inaccurate) decimal expansion.

 

[TuxDave]

Originally posted by: MadRat

Originally posted by: TuxDave
For instance, I'm ASSUMING that pi has an infinite number of digits after it to represent the value. (I'm not sure, but it's the best I can do). If a digit can exist at the infinite place, then there does exist a last number to pi right? Does that make sense? Does it make sense to claim that an infinitely long stairway has a top? If you think so, then this becomes a philosophy of math debate.
-Dave-

Pi is 22/7ths, correct? Then the fraction of 22/7ths is best defined in a base 7 numbering system, not a base ten.

Infinite is undefined, just as a variable of x can be undefined to solve for other variables. With all undefined variables we can use its meaning to exclude values, but we can never assign a value to it. Nor can we use a limit to define an undefined value, which is why we use limits to draw a line definitive as to what it is NOT. The math geniuses around here do not want to stick to the strict meaning of limits.

22/7 is a quicky approximation of pi, just plug it into a calculator and you'll see that they differ after 4-5 digits. I doubt there's a base system to which you can write pi to.

Yes, I agree that limit can tell you what it approaches. And people are taught that if, for instance, as x approaches infinity, the function approaches 2, so we can GUESS that @ x=infinity, the function equals 2. So for in this case, I'm not really saying limits, I'm saying, if you write 0.999... and I say for every 9 you write, there's a 9 after that forever and ever, then whatever value I've been trying to describe will equal 1. It's not what you finished writing down, but what I'm describing.


-Dave-

 

[Dufusyte]

There are some values that we can't write out in full using decimal notation, this does not make them any different to other numbers.

Yes it does make them different:

- Interminal decimals cannot be written without resorting to suspicious looking superscripts and/or elipses

- Interminal decimals do not have a fixed value.

hint: if it had a fixed value you would be able to write it without resorting to suspicious notation.

Sometime notation is shorthand to save time, such as 10^100. But there are other types of notation that are actually quite insidious: they do not merely save time, but rather they mask an impossibility. Writing 0.9999... is *not* a time saver, as though the person would be able to write out the exact number if he had enough time and space. Rather, the elipsis insidiously masks the fact that 0.9999... is not a fixed value at all, which can never be written out. It is like a little shape-shifting critter that has snuck his way onto the page, and we should banish it forthwith.