0.999... = 1

[SuSa]

Link to original post: [drupal=3743]0.999... = 1[/drupal]



I've seen this debate on many a forum. Let's see how educated SWF is about it. I want this in blogs instead of the debate hall because of what this really turns into.....anyways let's begin.

First I will use a mathematical proof to prove that 0.999... (where 9 is repeated into infinity) is equivalent to 1.

If 1/3 = 0.333...
And 0.333... * 3 = 0.999...
And 1/3 * 3 = 1
Then 0.999... = 1

Now here is where lies the debate...

If 1/3 is not actually equivalent to 0.333... it is approximate. What is it approximately?

Else 1/3 is equivalent to 0.333... and cannot ever be exactly 1/3rd of an object. All three pieces would have to be finite, and one piece must be larger. Example:

0.333...
0.333...
0.333...4 (infinitely as small as the previous two, but ending in 4)




What do you think? I hold the personal opinion that you cannot actually divide 1 by any prime number that ends in a repeating decimal. One piece would always have to be just that much larger. Therefore 1/3 and other such division by prime numbers only serve a "practical" purpose. Such as I can say I can take 6' and divide it into 3 equal groups of 2'.

The counter argument to this is the following:
It is impossible to show 1/3 in a decimal form. We simply do not have a way to write it. It exists, however we approximate to 0.333...

 

[A Pimp Named Slickback]

it's possible of course

If you think of it theoretically, if you can divide 12 by 4, then you can divide 12 by 3. There's no reason why you wouldn't be able to divide a number by 3 and obtain an answer.

But physically measuring EXACTLY 1/3 of something is just as hard as it would be to measure EXACTLY 0.444... or 1/pi of something.

But just because it's hard doesn't mean it isn't possible.

 

[SuSa]

Divide a circle into 3 even parts. Exactly 1/3rd.

It will only give the illusion of being complete. The incomplete part is likely unnoticeable at all, but existent. It's just infinitely small.

I am speaking not of 12, not of 9, not of 6. Not even of 3. I am speaking of 1. 1 divided by 3. No other number, no other amount. Just 1 divided by 3.


An example given to me was:
If I have a log, I can measure it and cut it into exactly 3 pieces.

At that point the log is very likely larger than any measurement of "1". It's probably 4' or even 40". It's most definitely not 1.

1 foot is not 1. It is 12 inches.

 

[A Pimp Named Slickback]

I'm not sure what you mean.

But you seem to think there's something special about the fraction 1/3 for some peculiar reason.

Why would it be more difficult to cut something into 3 than into say, 5?

0.333... is simply a notation that is convenient. It is still an exact number, no different than say, 0.2, which is actually just 0.200000....

Dividing a circle into exact 3rds (120 degrees each) is no more difficult than dividing into 4ths (72 degrees each).

 

[Teran]

There's a reason I dropped maths after 16 years of age.

Makes me go -__-

 

[Jonas]

Link to original post: [drupal=3743]0.999... = 1[/drupal]
What do you think? I hold the personal opinion that you cannot actually divide 1 by any prime number that ends in a repeating decimal. One piece would always have to be just that much larger. Therefore 1/3 and other such division by prime numbers only serve a "practical" purpose. Such as I can say I can take 6' and divide it into 3 equal groups of 2'.

We're talking about math here, not personal opinion. In math, you can divide any number with any positive or negative number (ruling out 0), so of course you can divide 1/3.

Actually, you might as well say that Pi doesn't exist, because it has an infinite number of decimals, and they're not just the same decimal being repeated. This is even harder to wrap your head around because you cannot write it down on a piece of paper (or anywhere, for that matter) and because it's an irrational number (which means it cannot be expressed as a fraction with integers).

 

[SuSa]

I'm not sure what you mean.

But you seem to think there's something special about the fraction 1/3 for some peculiar reason.

Why would it be more difficult to cut something into 3 than into say, 5?

0.333... is simply a notation that is convenient. It is still an exact number, no different than say, 0.2, which is actually just 0.200000....

Dividing a circle into exact 3rds (120 degrees each) is no more difficult than dividing into 4ths (72 degrees each).

Go me for forgetting that a circle is 360 degrees.

The difference between 1/5 and 1/3 is that 5 is 0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1

1/3 is 0.333... + 0.333... + 0.333... = 1, when in reality it is equivalent to 0.999...

An infinitely small difference, is still a difference.

We're talking about math here, not personal opinion. In math, you can divide any number with any positive or negative number (ruling out 0), so of course you can divide 1/3.

Actually, you might as well say that Pi doesn't exist, because it has an infinite number of decimals, and they're not just the same decimal being repeated. This is even harder to wrap your head around because you cannot write it down on a piece of paper (or anywhere, for that matter) and because it's an irrational number (which means it cannot be expressed as a fraction with integers).

Math can be interpreted in different ways. Believe it or not there are actually math philosophies.

Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered

Logicist definitions try to reduce mathematics to logic, especially deductive logic, or set theory, for example:

* "all propositions that can be deduced from Zermelo–Fraenkel set theory"

Intuitionist definitions regard mental activity as the essence of mathematics:

* "mental activity which consists in carrying out, one after the other, those mental constructions which are inductive and effective," meaning that by combining fundamental ideas, one constructs a definite result.

Formalist definitions deny both physical and mental meaning to mathematics, making the symbols and rules themselves the object of study:

* "the manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules"

I personally like Darwin's quote.

"A mathematician is a blind man in a dark room looking for a black cat which isn't there." Charles Darwin

Also Pi, while irrational, may at some point repeat. We just haven't gotten that far.

A non-repeating number ending in the decimal 0.999... does not suddenly add 0.000...1 (0 repeated for as infinitely long as 0.999, but ending in a 1)

 

[Jonas]

An infinite row of numbers ENDING with 1? There's something you don't understand. If something is infinite, it has no end. Thus, an argument like "1-0.99999.... = 0.00000...1" does not make any sense because 0.99999 and 0.00000... represent infinite rows of numbers.

 

[SuSa]

That's where you see the problem.

When does 0.999... get the extra push to become 1?

Oh yeah.
Never.

I was merely pointing that out.

 

[Seikend]

Else 1/3 is equivalent to 0.333... and cannot ever be exactly 1/3rd of an object.

What does this line even mean?

If 1/3=0.333.... , 0.333..... doesn't equal 1/3?

 

[SuSa]

0.333... + 0.333... + 0.333... = 0.999... it's missing something to become 1 whole. Therefore forcing it to be divided by fourths, not thirds, or otherwise one measurement of one third isn't actually one third - but ever so slightly more.

What I was trying to get at ^^

It was just really, really poorly phrased. I wrote this blog at like 1am. <_<

 

[Seikend]

0.333... + 0.333... + 0.333... = 0.999... it's missing something to become 1 whole. Therefore forcing it to be divided by fourths, not thirds, or otherwise one measurement of one third isn't actually one third - but ever so slightly more.

What I was trying to get at ^^

It was just really, really poorly phrased. I wrote this blog at like 1am. <_<

But you literally just above that posted a mathematical proof that 0.99....=1 ...

I am genuinely confused, what you're saying makes no sense. Are you saying that the proof is wrong?

Also: http://en.wikipedia.org/wiki/Proof_that_π_is_irrational

 

[SuSa]

The debate is whether or not the proof is correct. Or if our use of fractions with repeating decimals is illogical.

I'm challenging that 1/3 = 0.333... is interchangeable or even correct at all! As the only way to actually divide something by 3 is to have it's quantity be larger than 1 would imply.

EG:
I said 1 circle.
But the only way to split that 1 circle is because it's composed of 360 degrees.

Therefore 1 circle is not 1 circle. It's 360 degrees. 360 != 1, therefore my example was horrible.

 

[Deleted member]

By definition:
Two Real numbers are different if there is another number in between them.

Algebraic proof
x = 0.9...
10x = 9.9...
10x - x = 9
9x = 9
x = 1

since x = 0.9...
0.9... = 1

analytical proof
link

/thread

PS the 0.3.. proof is incorrect as it is circular.

 

[SuSa]

I failed to link to this article didn't I?

Anyways, my bad on the 1/3rd thing. Not a proof but it still works out. (Again, I'm tired... -_- about to go to bed)

Quote from above article:

For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a supremum definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division. Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.

Supremum

EDIT:

I'm confused now. Although I guess the difference in ...999 and 0.999... is important to note.
P-adic number system
If x = ...999 then 10x = ...990, so 10x = x − 9, hence x = −1

 

[A Pimp Named Slickback]

0.999... EQUALS 1

there is no difference

ever learn calculus and limits? if yes, then think of it this way. As the number of 9's following the decimal point approaches infinity, the value approaches 1. At infinite 9s (which is what 0.999... is defined as) it equals 1.

There is no difference between the two. When you say that 0.333... + 0.333... + 0.333... = 0.999..., you forget that 0.999... = 1. It has been mathematically proved to be such. Mathematics aren't always straight forward and may sometimes be counterintuitive, but a proof is a proof.

Kinda like how 9^0 = 1

and irrational numbers are DEFINED to never repeat. otherwise they could be written as a fraction and void their status as an irrational number.

 

[Fried Ice Cream]

I always wondered if anything would actually come from this debate. Ever.

 

[SuSa]

As the number of 9's following the decimal point approaches infinity,

1) Infinity cannot be approached. That is a flaw in comprehension of what infinity is...
2) Since infinity cannot be approached (it is forever) - there will always be a difference between finite and infinite.

This difference means that the negligible difference between 0.999... and 1, while smaller than the human mind can comprehend - still exists.

The mathematical proof is assuming one can approach infinity. One cannot approach forever or eternity. As neither are true measurements, but the lack thereof. You cannot actually measure something that goes on for infinity - because you would never stop.

If anything this proves are mathematical system false for ever assuming one can approach the infinite.



Rant:

Spoiler

The difference becomes negligible. It does not become non-existent. If anything, this is a misconception of what "infinity" truly means. It's a mathematical twist that "negligible" is equivalent to "non-existent".

A difference so small you cannot even comprehend it doesn't make this difference not exist. It's merely outside of human comprehension.

Just like you cannot actually imagine anything for infinity. You simply cannot comprehend this, but you understand that infinite goes on forever. However you can never rationalize "forever" and at some point cease to care. This is the flaw in this proof.

While the equation works out - wouldn't that prove that math is actually wrong and cannot work like this? However - in order to further comprehend subjects dealing with the possibility of an infinite - we have twisted what infinity is in order to fit our mathematical system?

At no point does 9 cease to be 9 and become 0. The difference between 0.999... and 1 may be so negligible to our comprehension that we ignore the practically-non-existent difference - but that is not to say that this difference does not even exist!

I see it as a flaw of our system. A proof in a flawed system is meaningless.

 

[Seikend]

I'm confused now. Although I guess the difference in ...999 and 0.999... is important to note.
P-adic number system
If x = ...999 then 10x = ...990, so 10x = x − 9, hence x = −1

Someone will probably provide a better answer before I do but ah well.

...999 is infinity.
And infinity is difficult, wibbly wobbly, timey wimey stuff. Infinity isn't a real number, it's a concept, and a lot of what governs normal numbers doesn't apply to infinity. If you don't know maths well, it's not wise to try and play around with it.

I'll be honest here, I might have this wrong. It's been a while since I did maths and I wasn't exactly an expert. But to compare infinity in size as ...999 and ...990 makes no sense mathematically.

Infinity's magnitude is purely determined by it's frequency.

Let's say what are the odds that you jump at some point tommorow.

There is literally an infinite number of things you could do tommorow, right? Every second can be split up into milliseconds, and nanoseconds and so on. You will never do things the exact same way, at the exact same time.

There's also an infinite number of things you could do tommorow that include jumping. You could jump at 9 oclock or 10 oclock. Or halfway between them. Or halfway between 9 and this time. Or halfway between 9 and /this/ time. etc.

But if we said that all infinities are the same, then your chance of jumping tommorow is

1/1 * 100= 100%. But we know that's not right. You are not guaranteed to jump tommorow. Then surely, not all infinities are equal.

That's because there's a lot more occurences of you not jumping throughout the day, than you jumping, Even though it's the same period of time. It's more /frequent/ that you don't jump than you do. This makes the infinite options you have of not jumping larger than the infinite options you have of jumping.

I hope this is at least very vaguely understandable. If not I'll give a proper explanation. aaand, it might be wrong. If so, I hope someone corrects me.


If no one addressed the P -adic number system at some point I'll do that too.

 

[Deleted member]

1) Infinity cannot be approached. That is a flaw in comprehension of what infinity is...
2) Since infinity cannot be approached (it is forever) - there will always be a difference between finite and infinite.

This difference means that the negligible difference between 0.999... and 1, while smaller than the human mind can comprehend - still exists.

The mathematical proof is assuming one can approach infinity. One cannot approach forever or eternity. As neither are true measurements, but the lack thereof. You cannot actually measure something that goes on for infinity - because you would never stop.

If anything this proves are mathematical system false for ever assuming one can approach the infinite.



Rant:

Spoiler

The difference becomes negligible. It does not become non-existent. If anything, this is a misconception of what "infinity" truly means. It's a mathematical twist that "negligible" is equivalent to "non-existent".

A difference so small you cannot even comprehend it doesn't make this difference not exist. It's merely outside of human comprehension.

Just like you cannot actually imagine anything for infinity. You simply cannot comprehend this, but you understand that infinite goes on forever. However you can never rationalize "forever" and at some point cease to care. This is the flaw in this proof.

While the equation works out - wouldn't that prove that math is actually wrong and cannot work like this? However - in order to further comprehend subjects dealing with the possibility of an infinite - we have twisted what infinity is in order to fit our mathematical system?

At no point does 9 cease to be 9 and become 0. The difference between 0.999... and 1 may be so negligible to our comprehension that we ignore the practically-non-existent difference - but that is not to say that this difference does not even exist!

I see it as a flaw of our system. A proof in a flawed system is meaningless.

I just gave THREE proves that are all correct. what are you even ranting on about?

 

[Seikend]

The difference becomes negligible. It does not become non-existent. If anything, this is a misconception of what "infinity" truly means. It's a mathematical twist that "negligible" is equivalent to "non-existent".

But it is non existent, that's the entire point. Paprika posted the following:

By definition:
Two Real numbers are different if there is another number in between them.

Algebraic proof
x = 0.9...
10x = 9.9...
10x - x = 9
9x = 9
x = 1

At no point does it make a comparison between an irrational (infinite) and rational(finite) number, so you can't use that point against it.

 

[A Pimp Named Slickback]

Infinity is a concept yes.

But then tell me, what IS 0.333 repeating? How do you define it without using the concept of infinity?

How can you say that 0.333repeating + 0.333... + 0.333... is 0.999...? Where is your proof that this is so? You can't just say that "oh just add the threes." Despite what you think, this is a HUGE leap in logic. How do you know beyond a doubt that there will no number than 9 in the answer? You can't prove it, you just make that assumption. Unless you add all those threes together, you cannot provide a definite proof for "0.333... + 0.333... + 0.333... is 0.999..."

And you CAN approach infinity. Anybody who has learned calculus and limits will tell you that infinity, although cannot be reached, can be approached. Infinity has some very strange properties that I'm not sure you understand.

Just out of curiously SuSa, what would you say the function "f(x) = 1/(x)" approaches as x increases infinitely? Think about what happens to f(x) as x increases, and then when x finally hits that abstract Infinity.

 

[Deleted member]

But it is non existent, that's the entire point. Paprika posted the following:



At no point does it make a comparison between an irrational (infinite) and rational(finite) number, so you can't use that point against it.

are you referring to the first or second thing I said?

 

[Seikend]

are you referring to the first or second thing I said?

The second thing. Sorry, I initially read them as the same point. Just realised the definition is a proof in itself.

EDIT:

Also, this whole thing reminds me of this.

 

[Deleted member]

I don't see why it is necessary to make a "comparison" between rational and irrational numbers. 0.3.. and 0.9... are both rational (1/3 and 1 respectively, proof for 1/3 can be performed the same as for 1), and I'm also not entirely sure what "against it" I should have used it?

 

[SuSa]

I just gave THREE proves that are all correct. what are you even ranting on about?

And a mathematical proof based on a system that allows infinity to be approachable, this is flawed, shows what exactly? That the proofs are right or that the mathematical system is wrong?

Because the latter is very possible, as the mathematical system we have developed has been created by humans.

Infinity is a concept yes.

But then tell me, what IS 0.333 repeating? How do you define it without using the concept of infinity?

How can you say that 0.333repeating + 0.333... + 0.333... is 0.999...? Where is your proof that this is so? You can't just say that "oh just add the threes." Despite what you think, this is a HUGE leap in logic. How do you know beyond a doubt that there will no number than 9 in the answer? You can't prove it, you just make that assumption. Unless you add all those threes together, you cannot provide a definite proof for "0.333... + 0.333... + 0.333... is 0.999..."

And you CAN approach infinity. Anybody who has learned calculus and limits will tell you that infinity, although cannot be reached, can be approached. Infinity has some very strange properties that I'm not sure you understand.

Just out of curiously SuSa, what would you say the function "f(x) = 1/(x)" approaches as x increases infinitely? Think about what happens to f(x) as x increases, and then when x finally hits that abstract Infinity.

That, again, is a misconception of what infinity is. It is a concept that attempts to work with human comprehension, but utterly fails when you try to do it.

You can approach the number 10, if you count upwards in intervals of 1 starting at 0. Most people would agree this is "Counting to 10". Your goal is stationary, and thus approachable.

When you view the concept of infinite - your goal is ever moving and it does this at a rate which cannot be defined. Even assuming infinity was somehow limited at the rate it could increase - for it to be approachable infinity would have to be moving slower than advancement is being made.

Where is the mathematical proof that shows that infinity is moving slower than advancement? That is what is missing from these proofs.

They are under the assumption that infinite is a stationary target - and it's growth is slower than the progress being made.


The only "real life" example I could possibly relate this concept (it's a rather hard concept for most to grasp, for some strange reason) is the following:

You are running on a treadmill for eternity - yet you are never making an advancement. While running on this treadmill, your goal was to move ahead by 100 feet. However no matter how long you run during your eternity, you will not advance 100 feet.

 

[Seikend]

I don't see why it is necessary to make a "comparison" between rational and irrational numbers. 0.3.. and 0.9... are both rational (1/3 and 1 respectively, proof for 1/3 can be performed the same as for 1), and I'm also not entirely sure what "against it" I should have used it?

Aah, sorry. I wasn't clear. That part was meant to be for Susa.

She points out there will always be a difference between finite and infinite numbers. I assume this applies to irrational and rational numbers as well.

Buuut, i was just saying that the above point holds zero weight against your proof, as you don't make any comparisons between irrational and rational numbers.

Apologies for the misunderstanding.

 

[SuSa]

I'm a he. :3

My problem with the mathematical proofs, and infinity being approachable, is to approach something. It cannot be moving - and if it is moving, you must be moving faster than it.

Thus for infinity to be approachable - it must be moving slower than advancement, or even stationary. Something that goes on indefinitely is not stationary. Also the idea that one could move faster than something which only begins to exist when you define it means it's impossible to work at a speed faster than the undefinable progress of infinity.

If you count to 1, and your goal is to use whole numbers and count for an infinite amount of time, you will never reach your goal. You will never approach infinity. As much "progress" as you make - you are going nowhere.

Therefore it is impossible to approach infinity.

Another quick example:
Infinity is a great example of "One step forward, two step backs". Except for it's not just two steps back.

It's infinite step backs.
Progress is impossible.


It's 3:32am.
I'm going to bed.

 

[Deleted member]

SuSa, do you have a degree in math to just start yelling "math is wrong and so are the people following it's misconceptions about infinity" (without proper arguments at that), which is pretty much what you are doing now.

If not I strongly urge you to take up this matter with a math teacher/professor.

 

[Seikend]

Math =/= Real Life
Any such analogy is completely irrelevant.


Likewise, stop saying things like you can count to 10, but not to infinity.

Infinity is not a real number. It doesn't obey all the rules of numbers. Of course if you use real numbers you won't reach the concept of infinity. Infinity is a concept outside of the numberline.


The way to "keep up" with an infinite number is to use another infinite number.

If we refer back to this again:

x = 0.9...
10x = 9.9...
10x - x = 9
9x = 9
x = 1

10x - x is possible because the infinite series of 9s is cancelled out by an identical infinite series of 9s.

In maths we don't try to do calculations with an infinite and a finite number interacting (I think anyway). We always compare two finite, or two infinite series. That addresses your issue, whilst keeping maths internally consistent.

 

[SuSa]

SuSa, do you have a degree in math to just start yelling "math is wrong and so are the people following it's misconceptions about infinity" (without proper arguments at that), which is pretty much what you are doing now.

If not I strongly urge you to take up this matter with a math teacher/professor.

A 4th grade grasp of the English Language can show what is at fault here.

Infinity is uncountable. It's incomprehensible. It goes on for an immeasurable distance, time, or whatever unit of measurement you want to even attempt it as. No matter what you use, it is immeasurable.

To approach something - you must be able to come near, or get nearer to. To do so, you are assuming an end or at least a measure to compare your progress to. Infinite is both endless and measureless. How would you judge if you approaching it?

You cannot approach infinity.

Follow the arrow around the circle and let me know when you reach the end:

Oh yah.
You will never reach the end. You may reach a point in which you overlap - but that is not defined as "the end" for the sake of this example. (I need you to truly comprehend what infinite is, and what it actually means)



To assume infinite can be approached jeopardizes our entire mathematical system based upon such assumptions. As, when infinity is not approachable, our mathematical system falls apart. I'll also take this up with my my teacher currently. When I go to college, assuming I have a math professor I will bring it up with them too.

Maybe there's some part about infinity being a set, measurable distance that is approachable that I'm just not getting.

Spoiler

I understand that infinity is not a number, it is a concept. A concept of something ongoing and never ending. Thus in the context of "Continue to add another 9 to 0.999... for infinity" is the ongoing act of adding another 9 at the end. It will go on forever and ever, but no matter how long it goes on for. It still remains 9 at the end. 1 does not end in 9. It ends in 0.

Negligible != Non-existent


Def's used:

Spoiler

Definition of Approach:

to come near or nearer to

Definition of Infinite (as a noun) in regards to mathematics:

Mathematics . an infinite quantity or magnitude.

Definition of Infinite (as an Adj.) in regards to mathematics:

Mathematics .
a.
not finite.
b.
(of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

Definition of finite (as a Noun) in regards to mathematics;

Mathematics .
a.
(of a set of elements) capable of being completely counted.
b.
not infinite or infinitesimal.
c.
not zero.

 

[Deleted member]

this is math, not languague.

 

[SuSa]

Read Seikend's edit. Makes sense now.

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

4am. ****. I need sleep. Past 3 hours were spent on me not grasping the exact concept behind what was happening in the proof. 9.9999....... - 0.999..... = 9......

Took me 3 hours to just read that properly

Willing to bet I made the past 3-4~ hours of your lives filled with frustration. Sorry about that.

*passes out*

 

[Jim Morrison]

Infinity doesn't exist, guys. It's just a concept.

 

[SuSa]

In math perhaps. Although you cannot say for certain it doesn't exist.

Unless the universe was shown that it will at some point end. Oo

 

[Jim Morrison]

No, infinity doesn't exist anywhere, just in your imagination.

 

[A Pimp Named Slickback]

I'm tempted to call you a ****ing ******.

But I'm sure that we both know it.

jk

but really, you're not grasping the concept of Infinity here. May I ask what level of mathematics you've taken? (no offense)

 

[Jim Morrison]

Elementary, I dropped maths after 9th grade.

 

[A Pimp Named Slickback]

nah I was talking to SuSa.

but looking back I sound really condescending. So disregard my being a total ****.

 

[Omni]

That's where you see the problem.

When does 0.999... get the extra push to become 1?

Oh yeah.
Never.

I was merely pointing that out.

This is where I laughed. After figuring out he was wrong he came to a new conclusion and passed it off as if it was his original.