0.9999....(recurring) = 1?

George,Y · 2007-02-25 21:33:49

Completeness property of reals comes from the definition of reals. The definition of reals is Cantor Set -A real is the set of all the rationals before something, or satisfying an inequality or some other conditions.

George,Y · 2007-02-25 21:41:11

I mean, its origin is Cantor's definition of reals as infinite sets. Cantor's reals, or the best reals we have, is equivalent to the points on an axis. (Hilbert proved this) How about the point? A point has no scale, but infinite of them compose a line segment.

How can one point, two points... increase to 1/1000 infinite points, 1/100 infinite points... infinite points? Generally, points, reals are all based on infinity or infinitesimal thus very plausible.

MathsIsFun · 2007-02-25 21:54:11

1/1000 infinite is just infinite, so is 1/100 infinite, so those are not a progression

(Infinity is the concept of endlessness, so "1/1000 endless" is still endless)

George,Y · 2007-02-25 22:09:48

How many points does a linesegment of length 1 have? How many points does a linesegment of length 1/1000 have?

The same? That's indeed the most absurd part of Cantor's theory.

You can cancel infinity/1000, but you cannot have infinity(endless) and finitething at the same time. If "endless" is reachable, you mean infinity is improved by finity. If "endless" is literally unreachable, anything made up of infinite components-an infinite decimals, a line segment- is not even completed, and does not exist at all.

MathsIsFun · 2007-02-25 22:32:36

Both your line examples have an infinite number of points.

But they are not "the same" in the sense of quantity, because infinity is a concept not a quantity.

Think of another concept like "now" - do 1000 people have more "now" than just one person?

George,Y · 2007-02-25 23:04:40

but they are made of points of the same kind while the property of one millinare isn't the same as one of the 1000 ordinary people.

when you say "infinite" points or "infinite" of points, you actually use it as quantity, we can name it as "super quantity", as I named in Post107.

Dross · 2007-02-25 23:08:19

George,Y wrote:

How many points does a linesegment of length 1 have? How many points does a linesegment of length 1/1000 have?
The same? That's indeed the most absurd part of Cantor's theory.

This may be highly counter-intuitive, but then again so is the whole of quantum mechanics, and relativity. Yet these are well established scientific theories with many real-life experiments to back-up those theories. Just because it sounds absurd, doesn't mean it's not true.

Likewise, two segments of the real line having different lengths but the same number of points sounds ridiculous - but that doesn't mean you should dismiss it, and it is in fact demonstratably true.

Question for you - how many natural numbers are there? Infinitely many, of course!

Now, how many even numbers are there? Well, there are infinitely many. Does that mean there are the same amount of even numbers as there are natural numbers?

Well, it seems so - and we can demonstrate this by pairing them up. Write out the set of ordered pairs, the first member of which is the next natural number, the second member is the next even number. It'll look like:

(1, 2), (2, 4), (3, 6), (4, 8), (5, 10), ...

(I've started the natural numbers from 1 purely for convenience - it's irrelevant whether they start from 1 or 0 in this case)

Now you can hopefully see that there are the same amount of even numbers as there are natural numbers - every even number gets paired with a natural number, and vice-versa. No magic in it, either.

Similarly, there are indeed the same number of points on a unit-interval of the real line as there are in an interval of length 1/1000.

George,Y · 2007-02-26 00:28:07

Linesegment of length1= infinite points
Linesegment of length 1/100= infinite points.
They have the same kind of points, so infinite amount=100 infinite amount. (infinite amount in fact equals real infinity-reached endless)

(1, 2), (2, 4), (3, 6), (4, 8), (5, 10), ...

you can do this only on the basis that natural numbers in the end goes to infinity, or are endless. Let's say we have infinite amount of natural numbers, how do 1, 2 ... grow to ∞? Or how does the latter shrink to finite amount?

Again if you mean natural numbers are only unreachably endless, clearly you haven't finished counting (1,2), (2,4), (3,6), ... But how can you state that one by one, same amount? Because you have counted one pair, two pairs, three pairs... so you conclude the same even before you count them down.

However it is naive induction, philosophically flawed. The sun rises today, the sun rose yesterday, the sun rose the day before yesterday, but can I say that the sun will rise tomorrow? David Hume rejected this induction and concluded unless we experience it, we are unsure. Equivalent naive inductions include: A hen saw her master bringing her food this morning, yesterday morning,... She natuarally concludes that he will bring her food next morning.(But probably he will kill the hen, otherwise he will kill some other "next morning) People have lived through 18th, 19th, 20th century although they burn fossel. Thus people can live through the 21st century with fossel still burning. If this is true, people can still live the next century... The universe exist for a long time, one century by one century, so it will exist forever, one century by one century endless...

Okay, you may already be impatient when I again, "confuse the pure math with the reality". Again and again, those who claim 0.999... ban the exploration by the disagreeer of the same concept they have used. First, someone says infinite digits is not finite and does not involve infinity, then many argue that infinity is endless but denies it grows, and then many reject 0.000...1 but believe 0.111... can have infinite digits, and then someone denies infinity as quantity but allows "infinite" to modify somethings... The rule is made by believers, they have many bans to stop rejectors' claim. And all they need to do is just claim one rule, like this pure math rule. (guess they have forgotten numbers come from counting fingers and for what purpose the society force kids to study maths-for practical purpose instead of worshipping maths purpose of course) Interesting, ha? A court banning one side from rufuting... how can the banned win the debate? However here I show how unfair the game is to at least let the neutral readers know.

Back to (1,2), (2,4), (3,6)... I use another naive induction:
{2,(1,2)},{4,(3,4)},{6,(5,6},...
again one even number, one natural number set with two natural numbers. Then without counting over, I conclude one even number, one set of number set, two natural numbers, thus the amount of "all" natural numbers is twice as many(much?) as that of "all" even numbers.

What? you gonna ban me again? you got me, you really got me...

Dross · 2007-02-26 00:49:08

George,Y wrote:

However it is naive induction, philosophically flawed....

What? It's mathematical induction - not flawed at all, and made rigorous through the properties of the real numbers.

And the last section of your post merely demonstrates that:

... well... yeah. I know.

Last edited by Dross (2007-02-26 00:49:43)

Ricky · 2007-02-26 04:04:40

Why? Is it based on the continum of the Reals? or Dedekind's cut?
I guess it's from a theorem stating "Any close number set includes its limit point, either its sup or its inf". But this theorem is a part of Real Analysis, and a natural derivative of Cantor's definition of a real Number as the Set of all the rationals Before it. So basically if one rejects Cantor's definition and accept only rationals, this proof can also be rejected.

The reals, when defined by Dedekind or Cantor (they are provably equivalent), form the smallest complete ordered field containing the rational numbers. By this, I mean any complete ordered field that contains the rationals also contains a subfield isomorphic to the reals. And this is provably so.

So any definition you come up with for a complete ordered field which contains the rationals will have the properties that the real numbers have as they do when going off of Dedekind or Cantor's definition. If you don't accept their definition, it must be that you don't accept the idea of a complete field, which is what calculus is now based off of.

However it is naive induction, philosophically flawed. The sun rises today, the sun rose yesterday, the sun rose the day before yesterday, but can I say that the sun will rise tomorrow? David Hume rejected this induction and concluded unless we experience it, we are unsure.

The 5th Peano Postulate is mathematical induction. In other words, we define the natural numbers to be a successor set which has the inductive property (and we can prove the integers, defined as such, exist in ZFC set theory as well). If you don't accept the Peano Postulates, then we are no longer talking about the same thing when we say "natural number". It's the equivalent of arguing over whether or not an apple is red, when one of us thinks of an apple as a potatoe.

John E. Franklin · 2007-02-26 14:28:23

Now you're spelling pototo like a U.S. vice president did I think, at a spelling bee.
This discussion is riveting. Who was the genius that proved infinity divided by a constant is still the same
infinity you started with?? And I guess 2.2222...22 is different from 2.222222... or 2.22...?
What's so wrong with building a set of discrete things, and filling in the middle, not at the end??
Why are you guys so positive there is only the right way of doing things?

Anthony.R.Brown · 2007-02-27 00:31:57

1 - Infinite 0.9 = 0 is a Contradiction! because the Calculation is saying the .9's End!!

A.R.B

Dross · 2007-02-27 00:46:25

Anthony.R.Brown wrote:

1 - Infinite 0.9 = 0 is a Contradiction! because the Calculation is saying the .9's End!!
A.R.B

So what is 1 - 0.9999... equal to, then?

Maelwys · 2007-02-27 01:28:56

Dross wrote:

So what is 1 - 0.9999... equal to, then?

According to his post above, it's equal to 0.000...1
Of course, that's "a Contradiction! because the Calculation is saying the .9's End!!" (since you can't have a last "1" without the 9s coming to an end) ;-)

George,Y · 2007-02-27 01:44:20

Contradiction Yes, if 0.00..01 is not possible, how can 0.111... possible?
if 0.00..01 with infinite digits means only 0, 0.111... has only finite digits (1 in any infinite digit simply means 0) thus does not equal to 1/9

Ricky · 2007-02-27 01:50:23

Contradiction Yes, if 0.00..01 is not possible, how can 0.111... possible?

It certainly is possible. 0.000...01 = 0. As said before, the digits of a number are countable, and thus, each has a location (finite distance from the decimal point). The 1 does not have a finite distance from the decimal point, and thus, does not appear in that number.

George,Y · 2007-02-27 02:39:17

Ricky, please read my second paragraph from "if 0.00..1" after stating your obstruse rule. I think you have the sense to understand my simple logic now that you know such complex rule, don't you?

Maelwys · 2007-02-27 02:53:55

George,Y wrote:

Contradiction Yes, if 0.00..01 is not possible, how can 0.111... possible?
if 0.00..01 with infinite digits means only 0, 0.111... has only finite digits (1 in any infinite digit simply means 0) thus does not equal to 1/9

I'm not sure I understand your argument here. There are two key differences between 0.000...1 and 0.111...
1/ In order to define the "last digit" is equal to something, there has to be a "last digit" which means it has to be a finite number of digits. It's impossible for a finite number of digits to contain an infinite number of zeros.
2/ 0.111... is a repeating digit, which can repeat for an infinite number of digits without changing. 0.000...01 implies an infinite number of digits that repeat for a while, and then suddenly change. What could possibly produce that change? 1/9 = 0.111... is reached because when you divide 1 / 9 you see that 9 goes into 10 once, leaving a remainder of 1, and then 9 goes into 10 once, leaving a remainder of 1, and then ... etc etc. The same result each time propogating the same result means that the same digit will keep appearing without end.

In order for a "last digit" to be different, that means that at some point, you have to conclude that 9 goes into 10 twice, with no remainder, which would then give you the 0.111...2 result, with a "last digit" that is different from the rest. Of course, math doesn't work like that, the same simple algebra can't spontaneous yield a different result after reaching one result an infinite number of times, nor can ANYTHING happen "after" something else has happened an infinite number of times, because the first thing is still happening.

So, 0.000...1 or 0.111...2 are impossible numbers, but 0.111... is quite possible.

Toast · 2007-02-27 03:06:05

Wait, so is 0.111...2 = 0.111..., or literally, as you put it, 'impossible'?

Maelwys · 2007-02-27 03:19:46

Toast wrote:

Wait, so is 0.111...2 = 0.111..., or literally, as you put it, 'impossible'?

0.111...2 is impossible to write in any form but that one (as opposed to 0.111... which can also be expressed as 1/9). But if you must define it as an understandable number, it would indeed be conceptually equal to 0.111... because as with the 0.000...1 case, if there's an infinite number of 1s, there's nowhere to put a "2" on the end, so it's just 1s forever, which is the same as 0.111...

John E. Franklin · 2007-02-27 04:15:36

Unfortunately, you might be wrong.
Start with an equivalent of 2, which might be 1.999...
Now go to the end, which you may have difficulty getting there.
However, when you find an inhuman way to reach the last 9,
then work your way leftward toward the 9.1 going backwards.
You'll never get there, unless you use an inhuman method we
don't know of. So you might conclude on the way back, that
the 9.1 doesn't exist, until you unlock the inhuman way
of getting to it. Warp drive super power active.

And don't forget that everytime you go for a walk or a jog,
everystep you take, you've jumped over an infinite number
of points, or maybe more, depending how clumsy you are.
A good marcher, tries to hit infinity every time.

Last edited by John E. Franklin (2007-02-27 04:19:31)

Dross · 2007-02-27 04:33:47

John E. Franklin wrote:

Unfortunately, you might be wrong.
Start with an equivalent of 2, which might be 1.999...
Now go to the end, which you may have difficulty getting there.
However, when you find an inhuman way to reach the last 9,
then work your way leftward toward the 9.1 going backwards.
You'll never get there, unless you use an inhuman method we
don't know of. So you might conclude on the way back, that
the 9.1 doesn't exist, until you unlock the inhuman way
of getting to it. Warp drive super power active.

But think about that - go to the end? Even by inhuman means, this is impossible; not because it's very far away, but because there is no end. There are an infinite amount of 9's after the decimal place, which means that for every "9" you point to there will be one after it. Since the last digit would be that digit that has no digits after it, there is therefore no last digit to get to.

All the inhuman devices in the world won't help you tie a knot in a string that doesn't exist.

Last edited by Dross (2007-02-27 04:34:27)

Maelwys · 2007-02-27 04:39:16

John E. Franklin wrote:

Unfortunately, you might be wrong.
Start with an equivalent of 2, which might be 1.999...
Now go to the end, which you may have difficulty getting there.
However, when you find an inhuman way to reach the last 9,
then work your way leftward toward the 9.1 going backwards.
You'll never get there, unless you use an inhuman method we
don't know of. So you might conclude on the way back, that
the 9.1 doesn't exist, until you unlock the inhuman way
of getting to it. Warp drive super power active.

I'm not sure I understand your logic. You're basically saying that if I use some impossible means to get to a place that doesn't exist, and then can't get back without making use of the same impossible means, that means that the place that doesn't exist, must exist?

Maelwys · 2007-02-27 04:45:41

John E. Franklin wrote:

And don't forget that everytime you go for a walk or a jog, everystep you take, you've jumped over an infinite number of points, or maybe more, depending how clumsy you are. A good marcher, tries to hit infinity every time.

That is true. However, that doesn't mean that I'm at the "end" of infinity, because I still have an infinite number of points left infront of me, that I haven't yet jumped over. And no matter how many steps I take, and how many times I walk past an infinite number of points, there is still an infinite number stretched out infront of me, that I haven't yet walked past. That's the nature of infinity.

John E. Franklin · 2007-02-27 04:47:21

Well, maybe you've caught me making a foolish proposition.
Anyhow, the smallest person in the world taking the smallest step he can take, is still stepping over many, many points, I guess an infinite number of points.
So I guess we live in a world where our unit of measure in points, is infinite at the smallest known level, attained in our common consciousness. Our "common consciousness" would be "that which we seem to agree we have experienced".

Okay, so now 0.999... has been thought to be 1, by most everyone, or really close.

But next, what about 0.45454545... Imagine how the each time a five is added, instead of a four, there is a little waver in the space-time continuum. Holy smokes.

Last edited by John E. Franklin (2007-02-27 04:50:32)

Math Is Fun Forum

#426 2007-02-25 21:33:49

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

#427 2007-02-25 21:41:11

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

#428 2007-02-25 21:54:11

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#429 2007-02-25 22:09:48

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#430 2007-02-25 22:32:36

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#431 2007-02-25 23:04:40

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#432 2007-02-25 23:08:19

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#433 2007-02-26 00:28:07

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#436 2007-02-26 14:28:23

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#437 2007-02-27 00:31:57

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#439 2007-02-27 01:28:56

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#440 2007-02-27 01:44:20

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#441 2007-02-27 01:50:23

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#442 2007-02-27 02:39:17

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#443 2007-02-27 02:53:55

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#444 2007-02-27 03:06:05

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#445 2007-02-27 03:19:46

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#446 2007-02-27 04:15:36

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#447 2007-02-27 04:33:47

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