Does .999=1?

[nullPointerException]

well 1/3 = .3 repeating and 2/3 = .6 repeating, but 2/3 +1/3 = 3/3 =1 not .9 repeating. anyway, i do not think .9 repeating =1, i think it = 1 - 1E-∞

[Scarecrow]

My favorite math problem in high school.

This was insightful. (http://www.purplemath.com/modules/howcan1.htm)

We were taught
0,111... = 1/9
0,222... = 2/9
...
0,777... = 7/9
0,888... = 8/9
Hence:
0,999... = 9/9 = 1

Also, I thought of what you would have to add to make a fraction whole. You would have to add 0.000..(infinite digits of zeros)..1 to 0.999... to make it 1. Since you can't do that, 0.999... has to be 1.

...I don't like that, either.

[Dirac]

Holy shit, 26 pages?

[kali]

Yeah, 0.999... = 1. You could liken them to synonyms. If you subtract 0.999... from 1, you get 0, meaning the numbers possess the same value.

[fripping]

basically, it depends on if you have enough time to keep counting more nines or not.

[Scarecrow]

Also, I thought of what you would have to add to make a fraction whole. You would have to add 0.000..(infinite digits of zeros)..1 to 0.999... to make it 1. Since you can't do that, 0.999... has to be 1.


Yeah, 0.999... = 1. You could liken them to synonyms. If you subtract 0.999... from 1, you get 0, meaning the numbers possess the same value.

No, you wouldn't. You'd get 0.000(infinite number of zeros)00001. See two posts earlier.

[Resonance]

No, you wouldn't. You'd get 0.000(infinite number of zeros)00001. See two posts earlier.

taht's the same thing bro

[Dirac]

No, you wouldn't. You'd get 0.000(infinite number of zeros)00001. See two posts earlier.
0.000 (infinite number of zeros) 00001 = 0

Actually, to be honest it just doesn't make sense to have an infinite number of zeros and then have a stop. If it stops somewhere, then it is finite.

[aphemix]

for each .$x...

.$x... + (.$x.../$x...) = 1. Where .$x... is a decimal iterated any number of times.

.999... != 1.

[bass_n_treble]

So, what you're saying is... since they took rocks back from the moon, it's .999 moon now?

[Resonance]

for each .$x...

.$x... + (.$x.../$x...) = 1. Where .$x... is a decimal iterated any number of times.

.999... != 1.

what

".$x... + (.$x.../$x...) = 1"
no

[aphemix]

what

".$x... + (.$x.../$x...) = 1"
no.999... + (.999.../999...) = 1.

[Resonance]

.999... + (.999.../999...) = 1.

no that's 2

[aphemix]

no that's 2I'll crush you!

[porcupine]

The other day may algebra instructor was proving that .9 repeating is equal to 1 because there are no numbers in between the two.

[Resonance]

I'll crush you!

lol too many periods. I see what you're saying now. The thing is, (.999.../999...) is undefined. Lim[x->infinity](.999/x) is 0. So either you're doing it wrong, or you're crazy. :D

[bass_n_treble]

I'm a little rusty, but here it is...























wait for it...




















I'm so nervous...















haven't done math in a while...





but...





I think...
1 = 1

[aphemix]

The thing is, (.999.../999...) is undefined. Lim[x->infinity](.999/x) is 0.I don't know what this means. I don't know any math. I just know .999 divided into 999 pieces is .001, .999 plus .001 is 1, and the formula continues working as you add more digits.

[electric sheep]

I can't believe this thread went on for 27 pages

[DoomBagel]

I can't believe this thread went on for 27 pages

You're just dreaming.

[Resonance]

I don't know what this means. I don't know any math. I just know .999 divided into 999 pieces is .001, .999 plus .001 is 1, and the formula continues working as you add more digits.

yeah but adding an infinite number of digits isn't as simple as saying "and so on."

[kali]

I think...
1 = 1

Please b&t. Everyone knows 1 = 2.

a = b
a^2 = a*b
a^2-b^2 = a*b-b^2
(a+b)(a-b) = b(a-b)
(a+b) = b
a+a = a
2a = a
2 = 1

[bass_n_treble]

Hey guys.

You know I'm pretty sure my answer is correct.

[Dirac]

http://www.stevencarrigan.com/blog/uploaded_images/DivideByZero-780521.jpg

[bass_n_treble]

Please b&t. Everyone knows 1 = 2.

a = b
a^2 = a*b
a^2-b^2 = a*b-b^2
(a+b)(a-b) = b(a-b)
(a+b) = b
a+a = a
2a = a
2 = 1

Actually, this could be your answer as well:

a = 0
b = 0

[bass_n_treble]

I can't believe this thread went on for 27 pages

On my INTPc settings, it's 50 posts per page. Technical getting banned was probably the craziest thread I've ever witnessed. I alone have about 15 rep comments just from one day of being on that thread.

/OT

1 + 1 = Time to drink a beer.

[aphemix]

yeah but adding an infinite number of digits isn't as simple as saying "and so on."yeah, fuck numbers.

[melancholeric]

yeah, fuck numbers.

Numeraphilia? That's gotta be one of the weirdest perversion I've heard of.

[aphemix]

Numeraphilia? That's gotta be one of the weirdest perversion I've heard of.coprophagia? (http://en.wikipedia.org/wiki/Coprophagia)

[melancholeric]

coprophagia? (http://en.wikipedia.org/wiki/Coprophagia)

I said one of the weirdest, not the weirdest. Also, I was thinking strictly sexual perversions.

And I'm still not sure whether eating feces is weirder than sexual intercourse with numbers. Think about it. How exactly would you actually do that?

[rainfall]

This reminds me of the half-distance shit they pull on you in the first few lessons of calculus. Basically, if some miserable bastard approached a wall half the distance at a time, he'd never reach it, which is bullshit simply because of the way universe is - at some point the half distance between the atom in his shoe and the atom in the wall would be sufficient to repel each other, as such, in our material reality the two would *touch* and he would reach the damn wall. And don't give me that shit that we have to calculate the distance between them from the outside of the atom sphere to the outside of the atom sphere - the bastards vibrate and they have electric fields that at some half-distance will start knocking each other off to some degree.

[Resonance]

requesting rule 34 on rule 34

[kali]

requesting rule 34 on rule 34
http://cdn0.knowyourmeme.com/i/2467/original/rule34kp0.jpg

[Limey]

There 10 types of people in this world; those that understand binary and those that do not.

[Resonance]

http://cdn0.knowyourmeme.com/i/2467/original/rule34kp0.jpg

hot.
@melco: any questions?

[Zephyrus055]

If sense experience and math disagree, then too bad for math.

[Scarecrow]

If it stops somewhere, then it is finite.

Yeah, I mentioned that earlier. That's why it's 1.

[vSv]

If "0.999... = 1" would be true, math itself would be flawed.

Either its finite or infinite, it can not be both at the same time unless it's paradoxal, which it of course could be, but you could never prove it.

[bass_n_treble]

There 10 types of people in this world; those that understand binary and those that do not.

And upon learning binary, I then understood why Link could only carry 255 Rupees at a time in the original Legend of Zelda.

Math FTW!

[earwax]

Why does this confuse people? It's just a glitch in base 10 representation of 3/3.

1/3 - .3r
2/3 = .6r
3/3 = .9r = 1

[joft]

If sense experience and math disagree, then too bad for math.

idiot

physicists left behind sense and experience long ago. they've been making more progress than ever before

also, human perception is deeply flawed, and intuition very frequently wrong. a huge number of people were completely wrong about the monty-hall problem, and some innocent people have gone to jail because of a failure to properly apply bayes' theorem

you can take your experience and shove it. I don't need you anyway, I don't need any friends, i've got my precious numbers and they're all i need!

[Ferrus]

http://en.wikipedia.org/wiki/0.999...

It's really not that difficult.

[Zephyrus055]

idiot

physicists left behind sense and experience long ago. they've been making more progress than ever before

also, human perception is deeply flawed, and intuition very frequently wrong. a huge number of people were completely wrong about the monty-hall problem, and some innocent people have gone to jail because of a failure to properly apply bayes' theorem

you can take your experience and shove it. I don't need you anyway, I don't need any friends, i've got my precious numbers and they're all i need!

Yes, and they have committed themselves to idealism, which is why we have string theory and parallel universe theory. None of it is observed, yet according to some true because they are mathematically proven.

I have no problem with math or logic being used to reach derivations from empirically demonstrated premises, but when these derivations make up more than 10% of your world view then that is taking math too far.

Human history offers reasons for being skeptical of mathematicians who depart from observation. Zeno's paradox and Leibniz come to my immediate mind, and I'm sure there are a wealth of others. While the problems of those thinkers can arguably be solved with our refined and modern mathematics, I am not convinced our current level of mathematics is so refined and perfect that it should be granted supreme authority over observation. I think Mathematics should complement observation, not become its substitute.

[Resonance]

when these derivations make up more than 10% of your world view then that is taking math too far.
prove this please

[Zephyrus055]

prove this please
lol...

It's a practical rule for keeping Mathematics on the leash of observation. Clearly, I am going to say it's madness if a mathematician came up to me with an expansive theory about the whole of the cosmos and 90% of it was supported by only math, since I give observation greater authority than math.

[qualia]

Why does this confuse people? It's just a glitch in base 10 representation of 3/3.This. Although shouldn't this just be a lesson in base 10 sucking for some concepts? Kind of like how there's no good word for gezellig in English, so better to switch to Dutch.

[avolkiteshvara]

idiot

I don't need you anyway, I don't need any friends, i've got my precious numbers and they're all i need!

This should be inserted into an INTx profile.

[joft]

Yes, and they have committed themselves to idealism, which is why we have string theory and parallel universe theory. None of it is observed, yet according to some true because they are mathematically proven.

I have no problem with math or logic being used to reach derivations from empirically demonstrated premises, but when these derivations make up more than 10% of your world view then that is taking math too far.

Human history offers reasons for being skeptical of mathematicians who depart from observation. Zeno's paradox and Leibniz come to my immediate mind, and I'm sure there are a wealth of others. While the problems of those thinkers can arguably be solved with our refined and modern mathematics, I am not convinced our current level of mathematics is so refined and perfect that it should be granted supreme authority over observation. I think Mathematics should complement observation, not become its substitute.

let me reiterate: idiot.

you don't know what you're talking about. you can give two examples: "string theory" and "parallel universes" which you conclude are absurd because they have not (possibly cannot) been observed. you seem to have forgotten about quantum mechanics, something which completely defies experience and intuition and yet has been verified and supported possibly more than any other scientific theory in history. you can focus on the fact that it's been experimentally verified and claim that this is somehow consistent with your view that "math cannot depart from observation." but that would be missing the point: in almost all of the groundbreaking accomplishments of modern physics, the math came first and experiments followed. you wouldn't know WTF to experiment with if you weren't following the math.

the scientific method is a combination of rationalism and empiricism. it's not pure statistical machine learning. some data is observed, then extrapolation and derivation allows one to hypothesize a rule, then more data is examined to see if it fits with the rule. in other words, it's not a completely structureless web of statistical associations. such a web would be stupid, like our most elementary attempts at creating AI, it would not know how to ignore uninteresting data and 99%+ of its conclusions would be banal.

Zeno's paradox and Leibniz (not sure what you mean by "Leibniz") are not counter-examples. Neither was Cantor's trouble with transfinite numbers that drove him insane any indication that there was a problem with the mathematics he was doing. Same with the bizarre examples of functions that analysts constructed to poke holes in the non-rigorous formulation of calculus that preceded Cauchy/Weierstrass etc. These were all stepping stones to greater truths and more powerful theories. Mathematical theories which actually turned out to be necessary for the further development of physics.

I can give you any number of examples of mathematicians working on problems that, in their time, had nothing to do with anything remotely related to physical reality or empirical observation, and yet now the theory they developed is used in many applied areas (computer science, statistics, physics, engineering, biology, etc).

I can give you examples of physicists formulating theories through pure rationalism and deriving them mathematically before anyone ever observed anything that would hint at such a theory, and yet now their theories make much of our modern technology possible (e.g. relativity; Einstein can never have experienced relativity, since he never traveled significantly close to the speed of light).

[joft]

let me tell one illustrative story here

Calculus, as conceived by Newton and Leibniz, did not originally have a rigorous mathematical foundation. Too much of it relied upon intuitions which ultimately turned out to be false. The ultimate example of this being the Banach-Tarski theorem (frequently called the "Banach-Tarski paradox" because it seems paradoxical, although it is not: it just proves our intuition is false). Even before that, however, there was a long line of people constructing bizarre counter-examples to various intuitive "truths." I quote Poincare:



Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.

If you lived in such times you would say the mathematicians constructing completely counter-intuitive "monsters" have left the realm of observation. However, if it were not for the insights gained through this pure rationalization (totally unrelated to anything any of these mathematicians every observed), we would not have developed the modern theory of analysis. Without Lebesgue integration, we don't have function spaces, and without function spaces we would lack most of the powerful tools we have now for studying PDEs. Partial differential equations are like the freaking verbs of the language of physics. Today we know much more about how to solve them and so forth because some mathematicians in the past ignored your stupid advice.

On the other hand, it would suck if all mathematicians and all physicists refused to work on problems that are directly applicable or would be applicable in the near term. But nobody is suggesting anything like that. To conclude that we should only focus on things that are immediately applicable would be a false dichotomy, and stupid.

[Limey]

Look, your girlfriend wasn't in a dimension where she was infinitely close to kicking my dog.
She kicked my dog.

[qualia]

I enjoy Joft in this thread.

[euterpenc]

Look, your girlfriend wasn't in a dimension where she was infinitely close to kicking my dog.
She kicked my dog.

Quite so.

[vSv]

Look, the universe isn't infinitely close of being infinite.
It's infinite.

See the difference?
One has an end where the other has none.

[Peguy]

Does .999=1?
Eh...close enough.

[vSv]

Eh...close enough.

How? It's not even close.

[Peguy]

How? It's not even close.

In the grand scheme of things....it's close enough.

[bass_n_treble]

This thread is a prime example of why some INTPs need a reality check and just once and for all ask, "Who cares?"

I understand intellectualism for intellectualism's sake, I really do. But this is ridiculous.

[vSv]

In the grand scheme of things....it's close enough.

Not even close to close enough.

"The law of conservation of energy is an empirical law of physics. It states that the total amount of energy in an isolated system remains constant over time (is said to be conserved over time). A consequence of this law is that energy cannot be created nor destroyed. The only thing that can happen to energy in a closed system is that it can change form, for instance chemical energy can become thermal energy."

If this law would even have the slightest change from inifinite correct to almost correct the outcome would be completely different, do you see why?

[edge walker]

The dumb asses and the retards are having it out with each other in this thread. I was going to make a few points until I read on and realised this is actually a wormhole to the worst part of ESTP Central.

Oh, yeah:

I understand intellectualism for intellectualism's sake, I really do.
Bullshit.

[joft]

The number ".999..." is not "infinitely close" to 1, it is actually EQUAL to 1. They are two symbols with the exact same meaning, two representations of the same quantity.

[zserf]

The sum of an infinite geometric sequence is A_1/(1-r) Where A_1 is the first term and r is the ratio of one term to the next. In this case .9/(1-.1)=1

People say that the difference is infinitesimal. The difference is 1/(10^N) and when N becomes infinity, that number becomes 0.

[edge walker]

infinitesimal
Hey, someone who knows something.

[joft]

The difference between 1 and the FINITE decimal expansion .999...9(end) which terminates after a finite number of places will be 1/10^n.

The difference between 1 and the infinite expansion .999... is zero.

I think people are confused by this because they aren't distinguishing between the limiting process and the quantity that the limiting process is converging to. The number ".999..." isn't changing, it isn't becoming close to 1, it isn't getting infinitely close to 1, it isn't infinitesimally different from 1, it IS 1.

[Hermione]

Hey, cool. I figured you'd explain this to me some day. :shock:


[holy fuck. this really is a thing. was sure it was just made up. winks]

[larry lurex]

0.999 is not 1, but for all practical purposes, it might as well just be.

[Resonance]

0.999 is not 1, but for all practical purposes, it might as well just be.
Hey I'll trade you .999 megatons of gold for 1 megaton of gold

[larry lurex]

Hey I'll trade you .999 megatons of gold for 1 megaton of gold

deal... pending delivery of the megaton of gold to my doorstep. I'd happily absorb the 0.001 megaton for your troubles. :rolleyes:

You think you'd make about $35m but you didn't take into account the cost of delivery, security and insurance, did you?

[Resonance]

deal... pending delivery of the megaton of gold to my doorstep. I'd happily absorb the 0.001 megaton for your troubles. :rolleyes:

You think you'd make about $35m but you didn't take into account the cost of delivery, security and insurance, did you?

Can't we just simplify and have you give me 1 kiloton of gold?

[larry lurex]

Can't we just simplify and have you give me 1 kiloton of gold?

Don't be silly. Well... my point is that there is time and place for the need for precision.

This is Science 101.

Any smart person who deals in precious commodity such as gold do not trade in decimals of kilotons. To put it in perspective, the more appropriate unit measurement is 999000000 grams (or oz. equvalent), not 0.999 kiloton. It is practical, however, to use 'kilotons or its whereabouts' for commodities such as cement or palm kernel shell. It's all about the cost of the margin of error.

You don't take Coca-Cola to court if the volume of the large bottle of Coke Zero is actually 2.24999999 litres instead of the advertised 2.25 litres, do you?

As I've said, by definition 0.999 isn't 1, but for practical purposes, it is.

[Resonance]

Don't be silly. Well... my point is that there is time and place for the need for precision.

This is Science 101.

Any smart person who deals in precious commodity such as gold do not trade in decimals of kilotons. To put it in perspective, the more appropriate unit measurement is 999000000 grams (or oz. equvalent), not 0.999 kiloton. It is practical, however, to use 'kilotons or its whereabouts' for commodities such as cement or palm kernel shell. It's all about the cost of the margin of error.

You don't take Coca-Cola to court if the volume of the large bottle of Coke Zero is actually 2.24999999 litres instead of the advertised 2.25 litres, do you?

As I've said, by definition 0.999 isn't 1, but for practical purposes, it is.

speaking of kilotons... Your mom isn't one, but for practical purposes, she is.

[Architectonic]

The difference between 1 and the infinite expansion .999... is zero.

I would argue that its not computable with typical mathematical systems.

[joft]

Then you would be wrong; depending on your definition of "computable" and "typical mathematical systems."

If you're referring to digital computation, then you obviously can't store infinitely many bits or perform infinitely many addition operations to "compute" .9 + .09 + ...

But this isn't even really a problem for computers. Most 4th generation computer languages have ways of overriding machine precision to symbolic operations with rational numbers, in basically the same way we do in grade school. Instead of storing .3333333 repeated up to the point of machine precision, it stores the array (1,3).

If it's not a problem for computers then you can bet it's not a problem for typical mathematical systems (since computability has yet to prevent mathematicians from, for example, using the axiom of choice).

In almost any mathematical formulation of the real numbers and their decimal expansions, the number

.999...

is provably equal, exactly equal, to 1. In fact I'm not aware of any logically consistent definition of decimal expansions in which infinitely repeated 9's are not equal to 1.

How are decimal expansions defined? They are defined to be sums of rational numbers, as follows:

The decimal number

0.(d1)(d2)(d3)...(dn)...

Where each (dn) is one of 0, 1, 2, ..., 8, 9

is DEFINED to be the real number given by the sum:

sum from n = 1 to infinity of (dn)/(10^n)

Some examples (http://en.wikipedia.org/wiki/Geometric_series#Repeating_decimals)

An infinite sum like that is DEFINED to be the limit, if it exists, of the "partial" sums:

Denote Sm = sum of the first m terms

Then consider the sequence: S1, S2, ..., S100, ... etc.

If this sequence can be proven to converge to a finite number, then the infinite sum S(infinity) is DEFINED to be that number.

So the definition of the decimal number .999.. is

sum from n = 1 to infinity of 9/10^n
= 9/10 * [sum from n = 0 to infinity of (1/10)^n]
= 9/10 * [1/(1-/10)] = 9/10 * (10/9) = 1

So the difference between 1 and .999... is 1 - 1 = 0. I just computed it in THE typical mathematical system (ZFC set theory + real number system + epsilon-delta limits)

[Dman]

Can't remember if this was already brought up in the thread or not, but what happens if you square 0.99999....? What is the answer? Is it 1? Or is it less than 0.999999....?

If 0.999...=1, then it has to be 1 if squared or cubed or whatever, right? But if you square a fraction, even a tiny fraction, isn't the result less than the root?

[joft]

Let r = .999...

Suppose r^2 < 1. Then there exists a number s such that r^2 < s < 1.

But then r < sqrt(s) < 1.

So sqrt(s) is a number strictly greater than .999... but strictly less than 1.

Which is stupid.

[Dman]

Let r = .999...

Suppose r^2 < 1. Then there exists a number s such that r^2 < s < 1.

But then r < sqrt(s) < 1.

So sqrt(s) is a number strictly greater than .999... but strictly less than 1.

Which is stupid.

You lost me - where the hell does the number "s" come into it?

Patience - I am obviously no mathematician. But the way I read your comment, some magical number "s" pops out of nowhere - what am I missing?

[Dirac]

You lost me - where the hell does the number "s" come into it?

Patience - I am obviously no mathematician. But the way I read your comment, some magical number "s" pops out of nowhere - what am I missing?
By the Archimedean property of the reals, there must exist a number between r^2 and 1 (using the assumptions in the proof). Call this number s. It's a proof by contradiction, he assumes the opposite of what he wants to prove, then shows that this yeilds absurd results.

[Dirac]

Let r = .999...

Suppose r^2 < 1. Then there exists a number s such that r^2 < s < 1.

But then r < sqrt(s) < 1.

So sqrt(s) is a number strictly greater than .999... but strictly less than 1.

Which is stupid.
Is this argument a bit circular? The contradiction only seems to arise if you are assuming that .999... = 1 before you start. Maybe not.

EDIT: I see what you're saying, but it boils down to "if they are not equal then there is a number between them, but there is no number between them. So they are equal." The squaring seems superfluous, but i may be missing something.

[vSv]

I don't understand how something can be something else without first assuming that it should be counted as paradoxal.

As I see it, and I'm not a mathemathican either, is that "0.999... = infinite" just as "0.333..." or any other number with an infinite ammount of decimals would also equal infinite/infinity and therefore an unmeasurable number.

"1" on the other hand is as I see it not infinite as long as it's not the definition of, for example the numbers of infinite universes where "1" defines that there only is one infinite universe and "2" defines that there are two infinite universes and so on.
If we take it as just the number "1", then it's a highly measurable number, therefore it can be divided in half, or pieces of four and so on.


If we take infinity for example, as a time-scale and a measure of ammount of energy in a given place, then it has no beginning nor an end, and thus it's infinite, it can't be divided, and nothing can be added/removed because everything is already in there, it's a closed system.


The error I see in the "0.999 = 1" would be the assuming that the infinite would be measurable.



Now, please correct me, I wan't to understand this.

[Dman]

By the Archimedean property of the reals, there must exist a number between r^2 and 1 (using the assumptions in the proof).

Uhhh... wouldn't that number between r^2 and 1 be r? If .99999...was not 1, that is.

So r^2 < r < 1. That was my whole point. No need for "s", "s" is "r".

Meaning that works: 0.999...is not equal to 1, as I have just proven. Holy shit, I am a fucking wizard.


(Actually I know I am still wrong somewhere, but nothing brings out a good credible response than acting like a pompous arrogant know-it-all ass)

[joft]

The squaring was superfluous, I was replying directly to Dman. And yes, that argument rested entirely upon the Archimedean principle of the real numbers. Every possible way of constructing/defining the real number system includes the Archimedean property. Any number system which includes "infinitesimals" is by definition not the real number system (e.g. the so-called "surreal" numbers).

You can define an infinitesimal number, let's call it s, to be a non-negative number such that

1) s does not equal 0
2) for any number N, no matter how large, 1/N cannot be smaller than s.

The difference between the Archimedean property of the real numbers and the existence of infinitesimals can be phrased as a matter of the order in which things occur:

For the Archimedean property it goes like this, "You pick any number greater than zero, and I can find a smaller one. Given r > 0, there exists s > 0 such that r > s."

For infinitesimals it's the other way around, "I have an infinitesimal number s, such that no matter what other number r > 0 you pick, my number s is less than r. There exists s > 0 such that for any other r > 0 we have r > s."

To try to understand the difference between these two, I suggest this exercise: Argue that the two properties above are mutually exclusive. If one is true, the other is not.

The Newton/Leibniz formulation of Calculus (and some other ancient mathematicians) used infinitesimals. But it was not rigorous. Karl Weierstrass gave the epsilon-delta definition of limits and reformulated calculus in terms of limits, and virtually all mathematicians since then have used this instead. It is possible to define infinitesimals rigorously in a way that does not lead to contradictions, but it's entirely unnecessary since the current system works perfectly well and a system based on infinitesimals would accomplish nothing more.

[30footsmurf]

=p u ppl ignored my beautiful equations



This tomfoolery is one of the reasons why I do not trust mathematics. ^.^



hahaha. I love this.
I recently decided to become a mathematician/math teacher/professor eventually. I haven't taken a math class in 6 years, but I'm gonna do it.
They were very beautiful equations. Thank you.

[joft]

The error I see in the "0.999 = 1" would be the assuming that the infinite would be measurable.

Okay, I think I understand your problem. It seems like you are using the word "measurable" as a synonym for "finite."

There's a lot of math you can do (discrete math) without ever encountering infinity. In principle almost all of physics should be phrased in such ways, since quantum theory says matter and energy are discrete, and there are only finitely-many discrete entities in the observable universe. However, mathematicians have never had a problem with accepting the "existence" of infinity. Some ancient Indian mathematicians, and almost all modern mathematicians after Georg Cantor, even accept the existence of different types of infinities--an infinite hierarchy of them, in fact.

However, none of the systems of numbers that I'm aware of would imply anything like ".999... = infinity". The decimal expansion is infinitely-long, but the number is definitely not greater than 1. Since 1 is certainly less than infinity, it follows that .999... is less than infinity.

I think you may be misunderstanding the nature of decimals. I suggest you try to remember how to do long-division (http://en.wikipedia.org/wiki/Long_division), and use it to divide 1 by 3. The first few steps should give you 0.333, and you should recognize that this pattern will continue forever. So 0.333... does not equal infinity, it equals 1/3.

Another exercise for people who still aren't convinced .999... = 1, first convince yourself that long-division is equivalent to subtracting multiples of the divisor, like this:

To calculate 50/3, first subtract 50 - 3*10 = 20. Then 20 - 3*6 = 2, putting these together: 50 - 3*(10+6) = 2, or 52 = 3*16, so 50/3 = 16+2/3.

Now calculate 1/1 in two ways. Obviously 1/1 = 1. But you can also do it this way,

1 - 1*0.9 = 0.1,
0.1 - 1*0.09 = 0.01,
0.01 - 1*0.009 = 0.001,

So 1 - 0.999 = 0.001. Now keep going. If you trust long-division to tell you that 1/3 = 0.333..., then for the same reason you should trust it when it tells you that 1 = 0.999...

[joft]

Uhhh... wouldn't that number between r^2 and 1 be r? If .99999...was not 1, that is.

So r^2 < r < 1. That was my whole point. No need for "s", "s" is "r".

The Archimedean property of real numbers is this:

You give me any real number r > 0, and I can find a positive integer N such that

0< 1/N < r

That's strict inequality on both sides. The point is that, once you pick r, I start looking at large integers N. If 1/N is not strictly less than r, I just keep making N bigger. Since r is fixed, it isn't changing, so eventually I will reach a (finite) number N such that 1/N is strictly less than r.

So if r^2 < 1, then 0 < 1 - r^2, so I can find a = 1/N for some large N such that

0 < a (strictly) and
a < 1 - r^2 (strictly), therefore (rearrange and fiddle with these)
r^2 < 1 - a < 1

Now let 1 - a = s. Then r^1 < s < 1, and these inequalities are strict.

Now since the function f(x) = sqrt(x) is strictly increasing if x > 0, we can take square roots and the strict inequality remains intact. So
r < sqrt(s) < 1.

If I haven't emphasized it enough yet, remember the inequalities are strict, so r does not equal sqrt(s), and r^2 does not equal s. To summarize:

0 < r^2 < s < r < sqrt(s) < 1

But it's just silly for sqrt(2) to be strictly less than 1 but strictly greater than 0.999...

[Dman]

In other words, the number 0.999... is in a different category, a completely different concept than the number 1, due to infinity being involved in one and not the other.

Which means they are not "equal", because each lies in a different conceptual reality. I win!

[Utopmk]

If you're pumping gas, then it does.

[earwax]

This thread confirms my suspicion that being INTP has nothing to do with intelligence.

1/9 = .1r
.1r * 9 = .9r

1/3 = .3r
.3r * 3 = .9r

This isn't that difficult people. It's a minor glitch in decimal representations of fractions.

[Utopmk]

This thread confirms my suspicion that being INTP has nothing to do with intelligence.

1/9 = .1r
.1r * 9 = .9r

1/3 = .3r
.3r * 3 = .9r


I am terrible at math, honestly. I think mathematics has nothing to do with intelligence. It's just a symptom of OCD...silly people...trying to number EVERYTHING.

:p

[Dman]

This thread confirms my suspicion that being INTP has nothing to do with intelligence.

This coming from someone who, in the same post, wrote "represenations". The only thing better would have been if you wrote "intellegence" :)

By the way, your formula proves nothing, which confirms my suspicion that you haven't read the entire thread.

[30footsmurf]

By the way, your formula proves nothing, which confirms my suspicion that you haven't read the entire thread.

you always have the best burns. ;)
I want that on a T-Shirt.

[earwax]

This coming from someone who, in the same post, wrote "represenations". The only thing better would have been if you wrote "intellegence" :)

By the way, your formula proves nothing, which confirms my suspicion that you haven't read the entire thread.
I did have a typo - my bad. But that does not change the fact that .9r (note "r" means repeating) is equal to 1.

[30footsmurf]

I didn't read this whole thread, but it seems really silly to argue that .9r isn't equal to one. The examples on this or the previous page using the 1/3=.3r are pretty straight forward.

You still know that .3r is equal to one third, and that 3*1/3 is 3. That being the case, and the derived .3r being a representation of of the fraction 1/3 means that they are being interchanged solely for the purposes of working the problem out in decimal notation. They are still the same number, written differently. Getting lost in that change doesn't mean the .3r isn't still 1/3, its just that method of viewing or the perspective of that type of notation keeps it hidden for anyone that forgets where the .3r came from. The .3r is the same as 1/3 treating them any differently because you can't see the end of the line even though it is clearly marked 1/3 is just a lack of ability to see where the story is going.

To put it another way, uno=one. Just cause it looks different doesn't change the fact that they both mean 1.

[Dman]

I did have a typo - my bad.

I was just bustin' your balls, we all have teh typos and grammatical fuck-ups, you just asked for it that's all.


But that does not change the fact that .9r (note "r" means repeating) is equal to 1.

Prove it!

[Chunes]

for each .$x...

.$x... + (.$x.../$x...) = 1. Where .$x... is a decimal iterated any number of times.

.999... != 1.

'Any number of times' has quite literally nothing to do with the mechanism behind .999... (instant infinite iteration).

[Dman]

Dang.

I blame Wikipedia for this. Admit it!

Bah, killjoys with yer fancy internet research tools

[joft]

The fact that real numbers may have more than one decimal representation is not even a remotely controversial issue in mathematics. The issue is entirely settled. All that's left to do is for people who don't believe it to find a way of convincing themselves it's true, because it IS true. To argue against it is literally arguing that "1 does not equal 1."

[!diom]

The fact that real numbers may have more than one decimal representation is not even a remotely controversial issue in mathematics. The issue is entirely settled. All that's left to do is for people who don't believe it to find a way of convincing themselves it's true, because it IS true. To argue against it is literally arguing that "1 does not equal 1."


http://upload.wikimedia.org/math/9/9/9/9999f0ace517cc0702d78a67675f14b0.png

Looks convincing to me...

[earwax]

Ooh! Here's a good one!

10/11 = .909090(90)r
1/11 = .090909(09)r
------ -------------
11/11 = .999999(99)r

[!diom]

Sweet.

[Architectonic]

Then you would be wrong; depending on your definition of "computable" and "typical mathematical systems."

I should have been clearer, my point was that a simple arithmetical system cannot compute such an answer. Nevertheless, you proved my point. To solve the problem you had to introduce new mathematics eg the limit. It is an obvious tautology in that case.

[joft]

Such an arithmetic system would indeed be simple, for it would not even be able to expand 1/3 as a decimal.

[earwax]

Here's another fun one.

2/7 = .285714(285714)r
5/7 = .714285(714285)r
---- ----------------
7/7 = .999999(999999)r

[Chunes]

I'm not siding with those who think 0.999... != 1, but I never understood why anyone would ever think that someone who thinks 0.999... != 1 would think 0.111... = 1/9. Makes zero sense to me. Their basic problem is their refusal to believe any repeating decimal could be rational. Using other repeating decimals in a proof may be the only sort simple enough for them to understand, but unfortunately does nothing because it uses the concept they don't believe.

[Utopmk]

2+2=5.

[cripple]

2+2=5.
Of course.

++ + ++ = +++++

[Resonance]

I'm not siding with those who think 0.999... != 1, but I never understood why anyone would ever think that someone who thinks 0.999... != 1 would think 0.111... = 1/9. Makes zero sense to me. Their basic problem is their refusal to believe any repeating decimal could be rational. Using other repeating decimals in a proof may be the only sort simple enough for them to understand, but unfortunately does nothing because it uses the concept they don't believe.

hmm. I'm in a bit of a pickle, here, because your argument is based on the selective application of logic; the use of an isomorphic concept to prove the original is only inappropriate if one is already able to process logic. But then, this same counterargument applies to this very argument, forming a sort of recursive paradox.

Goddamn feelers :wub:

[Eric B]

Any time you mess with zeroes and infinities in the same problem, you get paradoxes. And part of it is the distinction between "countable" and "uncountable" infinities. The number line itself is a countable infinity, because you can count each number on the way up; as are the number of digits in the .9999... notation. However, "infinitessimals" are uncountable, because you basically have an infinite number of "zero" entities squeexed into a finite area.

I tackle this same problem in defining a circle as a polygon with an infinite number of sides. http://www.erictb.info/math&science.html#math (I should make a separate anchor to link directly to that essay).

You can fix the length of sides, and then as n approahes infinity, you get an infinite straight line tiled with a countable infinite number of line segments at 180&#176;. However, if you fix the radius, then all the sides shrink towards zero, and you end up with a circle, with the infinite "points" of its perimeter as its "sides". Since the points are zero length (infinitessimal), they are not countable; and yet, they make up a finite perimeter.

In the same section of the link, to throw another wrench into the equation, what about
an alternate negative number system that simply continues in positive order backward from zero, rather than creating a mirror image of the positive number line?

So if you subtracted 0-1 the same way you would subtract 1000000-1, then you get ...999999! (like the mirror of the .9999... being discussed, and it would be equal to -1). The number also would appear to be divisible by 9, but since it's -1, then it really isn't. It would be 9 &#215; ...11111111. and the next lower number divisible by 9 would be ...999991. In fact, that number would be the product of ...999999 and 9!

Someone told me that Riemann actually came up with a system like this, but I can't find it mentioned anywhere.

[Professor Chaos]

Wouldn't this depend on the context in where either can be used.

1 may not be as precise as .999 is n certain situations where rounding up would effect the minute difference. This would happen in situations where precision is everything.

On the contrary, in some cases rounding up would not make a difference, especially if one rounds up the final answer.

[larry lurex]

speaking of kilotons... Your mom isn't one, but for practical purposes, she is.

I am impressed with your comeback. The very best I've seen yet. :thumbsup:

[Spookygray]

to repeat what has already been said

http://img.photobucket.com/albums/v259/tocca/mathy2.png




http://img.photobucket.com/albums/v259/tocca/mathy1.png

:rofl:

If you EVER bring Calculus into my life again, I WILL come and find you!

(I'm crouching in a corner right now in fear)

[atom]

summation != integration

.999... equals 1. Stop arguing.

Numbers are different than the symbols we use to express them. Intuitively, its not hard to understand that fractional and decimal representation represent the same number (eg 2/5 = .4)

The problem here is, intuitively, many of you think there is only one possible decimal representation of numbers. This is not true. Translating from decimal representation to summation representation like Melody did makes this apparent.

Come on, 36 pages arguing on a math question? Seriously?

edit: I regret indulging in this stupid thread. Let it die please.

[zoltan]

more interesting is that 0!=1

[stuck]

hmm. I'm in a bit of a pickle, here, because your argument is based on the selective application of logic; the use of an isomorphic concept to prove the original is only inappropriate if one is already able to process logic. But then, this same counterargument applies to this very argument, forming a sort of recursive paradox.

Goddamn feelers :wub:

:mellow:

[Ferrus]

Isn't 0.999... effectively equivalent to the right hand side of an interval [x,1)?

[puzzled-observer]

Isn't 0.999... effectively equivalent to the right hand side of an interval [x,1)?
Interesting point. The concept of infinitesimal objects kind of destroys the idea of a well defined break between values...such that x<1 is actually no different from x<=1. You need to specify a finite range for the excluded values in order for it to be a meaningful one.

[Ferrus]

Interesting point. The concept of infinitesimal objects kind of destroys the idea of a well defined break between values...such that x<1 is actually no different from x<=1. You need to specify a finite range for the excluded values in order for it to be a meaningful one.
Well, I should've said - the interval defined in R, if I am not incorrect infiniessimals are only meaningful objects in the hyperreal field. Wouldn't the Archimedean property force the two to be equal?

[puzzled-observer]

Well, I should've said - the interval defined in R, if I am not incorrect infiniessimals are only meaningful objects in the hyperreal field.
Well, it strikes me as self-contradictory to have .999... (an infinitely repeating quantity) in a number system that doesn't allow for infinities (or infinitesimals). The maximum quantity in the interval would have a definite value x =1-dx where dx is whatever minimum distance your defining into your number line in R.

On a somewhat related note, I also find it inconsistent to have 0 on a number line and not allow for infinities. Simple algebra requires that you make up some rule where you define dividing by zero as a disallowed maneuver...but as far as I'm aware, this only serves a practical purpose. It fails to capture the logical consequences of the existence of zero...which is the existence of infinity. The two are inverses of each other and where one is included, so should the other be. I'm no mathematician so I may not be aware of all the reasons that it's not included, but it seems illogical to me.

Edit:

Wouldn't the Archimedean property force the two to be equal?
You meant not equal. Right? Where "the two" is x<1 and x<=1... if I'm reading you correctly.

[Ferrus]

Well, it strikes me as self-contradictory to have .999... (an infinitely repeating quantity) in a number system that doesn't allow for infinities (or infinitesimals). The quantity itself would have a definite value x =1-dx where dx is whatever minimum distance your defining into your number line in R.
I don't think it is self-contradictory if you consider the Least Upper Bound (http://en.wikipedia.org/wiki/Supremum) property of the real numbers.

The real numbers ARE infinitesimal, in the sense that they can be made arbitarily small - only, unlike the hyperreals, which correspond to the 'dxs' of calculus, they have the archimedean property.

[puzzled-observer]

I don't think it is self-contradictory if you consider the Least Upper Bound (http://en.wikipedia.org/wiki/Supremum) property of the real numbers.

The real numbers ARE infinitesimal, in the sense that they can be made arbitarily small - only, unlike the hyperreals, which correspond to the 'dxs' of calculus, they have the archimedean property.

Ahh, ok, I understand you now. I had to wiki the archimedian principle.

"Roughly speaking, it is the property of having no infinitely large or infinitely small elements (i.e. no nontrivial infinitesimals (http://en.wikipedia.org/wiki/Infinitesimal))."
Nontrivial infinitesimals being the key to my misunderstanding of what you're saying. So, we're in agreement that x<1 = x<=1... but then I return to my original point:
I would say that the fact that they're infinitesimal forces them to contribute no finite value...and thus be meaningless on the number line. Archimeadian principle or not, I don't see how an infinitesimal difference could ever be anything but insignificant on a finite scale. But, i haven't read anything about hyperreal mathematics... so maybe I'm wrong.

[Ferrus]

I've been thinking about this. The most convincing proof requires the nested interval theorem. Which in turn is implied by Cantor's intersection theorem. Understand the contradiction that leads to the proof of Cantor's theorem and then you'll see that 0.999... not being equal to 1 is a logical contradiction.

Anyway, with my question about intervals. I've thought about it and realised that actually, the number IS the limit point of a closed interval.

[puzzled-observer]

I've been thinking about this. The most convincing proof requires the nested interval theorem. Which in turn is implied by Cantor's intersection theorem. Understand the contradiction that leads to the proof of Cantor's theorem and then you'll see that 0.999... not being equal to 1 is a logical contradiction.

Anyway, with my question about intervals. I've thought about it and realised that actually, the number IS the limit point of a closed interval.
I'm not sure if this is addressed to me.
I agree with all of your conclusions on here. The main disagreement I had was using the archimedean principle to justify the conclusion.

[Ferrus]

I'm not sure if this is addressed to me.
I agree with all of your conclusions on here. The main disagreement I had was using the archimedean principle to justify the conclusion.
It's required in the algebraic proof, but granted, it is not needed in an analytic proof.


Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."

:happpy:

[joft]

anyone who thinks the prohibition against dividing by zero is artificial or simply a matter of convention needs to explain this picture to me

http://img52.imageshack.us/img52/5718/dividebyzero.jpg

if the picture doesn't work, here's my objection in words: why should 1/0 = infinity instead of 1/0 = -infinity? 1/x goes to +infinity if x goes to zero from the right, but -infinity if x goes to zero from the left. it seems we have a choice, so why do we pick one (usually people say +infinity) instead of the other? why is this choice any less arbitrary than insisting that division by zero is undefined?

it's not that mathematicians are avoiding the question or stubbornly insisting on a convention. it's just that leaving it undefined really is the most natural and non-arbitrary way of resolving the issue. it makes sense algebraically (in other number systems--"rings," the additive identity--zero--is never a multiplicative unit--never invertible). it makes sense analytically--if we tried to define 1/0 as the limit of 1/x as x goes to 0, this definition would be ambiguous because it approaches two separate values. it makes sense topologically: when a space (such as the complex numbers) is not compact then we should not expect it to have all the same properties as its compactification (such as the riemann sphere, in which division by zero IS well-defined).

if you want to define division by zero, that's fine. but whatever you define will not be part of the real number system. it will be another system, and you will be forced to adopt many additional rules in order to make that system self-consistent (such as defining 0*infinity or specifying that it's undefined, etc). similarly, the real number system does not include infinitesimal numbers. if you want to include infinitesimals that's okay, but you'd then be working with a different number system. this whole problem of .999... = 1 is also not just a matter of convention. it really is the one answer that makes the most sense, for the purposes of the real numbers.

and mathematicians have explored those alternative systems. it's not like we are constrained by some orthodoxy or mainstream of mathematics. on the contrary I would say mathematicians are the least constrained out of all scientists or disciplines by anthropocentric things like convention.

[NoahFence]

RE: .9 repeating, two questions:

If .3 repeating is 1/3, and .6 repeating is 2/3, how is .9 repeating not 3/3, which would clearly be 1?
If it is not 1, what would you subtract from 1 to achieve .9 repeating?


RE: Division by 0

If one divided by zero is X, then zero times X is one. Solution set: null

If zero divided by zero is X, then zero times X is zero. Solution set: everything

[Ferrus]

(such as the riemann sphere, in which division by zero IS well-defined).

What is the Reimann sphere and how does that work, out of interest?

[puzzled-observer]

if the picture doesn't work, here's my objection in words: why should 1/0 = infinity instead of 1/0 = -infinity? 1/x goes to +infinity if x goes to zero from the right, but -infinity if x goes to zero from the left. it seems we have a choice, so why do we pick one (usually people say +infinity) instead of the other? why is this choice any less arbitrary than insisting that division by zero is undefined?
Why can't it take either value... just as the square root of x^2 is either x or -x.
taking the equation:
1/infinity = 0 just as
1/-infinity = 0.
It's true that most people would just give the answer "infinity" without thinking about it, but most people say the sqrt(144) = 12



it's not that mathematicians are avoiding the question or stubbornly insisting on a convention. it's just that leaving it undefined really is the most natural and non-arbitrary way of resolving the issue. it makes sense algebraically (in other number systems--"rings," the additive identity--zero--is never a multiplicative unit--never invertible). it makes sense analytically--if we tried to define 1/0 as the limit of 1/x as x goes to 0, this definition would be ambiguous because it approaches two separate values. it makes sense topologically: when a space (such as the complex numbers) is not compact then we should not expect it to have all the same properties as its compactification (such as the riemann sphere, in which division by zero IS well-defined).

if you want to define division by zero, that's fine. but whatever you define will not be part of the real number system.
My position is that this follows from the idea that the number line is perfectly continuous. So, I would argue that it's not me defining division by zero, it has already been defined. I can't offer any rigorous mathematical proof, but conceptually, I can't find where I'm wrong. Maybe you can show me my error.
So, for the function Y=X, for example, i can plug in any arbitrary point between 0 and 1 for x and get a defined value for y. There are no undefined values for Y between 0 and 1.
This means that, because individual points are objects which have 0 length, there are an infinite number of defined points for Y between 0 and 1.
This means that the finite distance of 1 is entirely composed of points.
That is: 1 = 0*infinity an infinite number of zero length objects comprises a finite length of 1. Now, that argument will work for any finite distance... And once you accept that premise, it can be shown that 0*infinity can also take the values of 0 and infinity.



the real number system does not include infinitesimal numbers. if you want to include infinitesimals that's okay, but you'd then be working with a different number system. this whole problem of .999... = 1 is also not just a matter of convention. it really is the one answer that makes the most sense, for the purposes of the real numbers.
I'm not sure if this was aimed at me or other people, but I'm not contending that it includes infinitesimal numbers. I think if it did, .999... would NOT be equal to 1.
I'm only contending that a point is infinitesimal(=0) in the length that it comprises on the number line.



and mathematicians have explored those alternative systems. it's not like we are constrained by some orthodoxy or mainstream of mathematics. on the contrary I would say mathematicians are the least constrained out of all scientists or disciplines by anthropocentric things like convention.
I'm assuming that I set off this tirade. I didn't mean to imply that mathematicians are SJ's. I just don't understand why it's better to say this is undefined (i.e. we don't know anything) rather than it's either X or Y (something we can say something about). It seems like you're just throwing out information.

[Ferrus]

http://en.wikipedia.org/wiki/Projectively_extended_real_numbers

and

http://en.wikipedia.org/wiki/James_Anderson_(computer_scientist)

Have some good stuff.

It strikes me on the hardest conceptual jumps is going from lower level maths, where the formalisations are presented as if God himself had came down upon high and ordained the axioms as engraved on tablets (which is how societies and governments probably prefer their members to think of the features); and the realisation that all mathematics consists of logically consistent structures.

[puzzled-observer]

It strikes me on the hardest conceptual jumps is going from lower level maths, where the formalisations are presented as if God himself had came down upon high and ordained the axioms as engraved on tablets (which is how societies and governments probably prefer their members to think of the features); and the realisation that all mathematics consists of logically consistent structures.
I don't think I can identify a specific point when I made this realization. I agree, though. When you look at the power of mathematics and you think of the handfull of (seemingly) basic premises you started with...It's kind of mind blowing how much nested information and structure you can pull out of it.

[atom]

. . . infinity*0 = 1 ...

0 + 0 = 0

no matter how many zeros you add, you never get more than zero.

Intuitively, you think an "infinite amount of points" make up the real number line. Your intuition about the nature of infinity is wrong. The number line is "continuous" and no matter how close together you put two points they are still discrete and do not touch.

If you take each real number between to be a point and the number of counting numbers to be infinitity then the number of points comprising the line is more than infinity. It can be considered the cardinality of the power set of whole numbers or 2^infinity which should ring true to your intuition if you think of irrationals infinite as strings of whole numbers after a decimal. In fact, you can continue this process and you will find that there are a countably infinite varieties of infinity.

There's a neat proof I can share too if you're interested.

[zoltan]

seems like people dont understand the concept of limits. i dont see how this stretched on to 38 pages.

[atom]

Everyone has their own mathematical intuition and no one's intuition is complete. Limits aren't part of the intuition of a number of people and so seems an arbitrary, unconvincing construction. In terms of courses, you see them in pre-cal but it isn't until analysis that they become fundamental.

Yeah, its painful. 38 goddamn pages.

[esthim8]

so many things wrong but so little time. I recommend that people who are very interested in these kind of things to pick up a rigorous analysis book and a basic abstract algebra book. Go through them both and then think about these questions.

[Ferrus]

so many things wrong but so little time. I recommend that people who are very interested in these kind of things to pick up a rigorous analysis book and a basic abstract algebra book. Go through them both and then think about these questions.
What element of abstract algebra is relevant here?

And what errors did you pick out in what I said?

[zoltan]

Limits aren't part of the intuition of a number of people.

Yeah but people are thought what rounding and approximations are, especially if you apply it to money.

seeing as how this hasnt died after a few posts, im going to stop now.

[puzzled-observer]

0 + 0 = 0

no matter how many zeros you add, you never get more than zero.
This is unfair. You're only adding a finite amount of zeroes.



Intuitively, you think an "infinite amount of points" make up the real number line. Your intuition about the nature of infinity is wrong. The number line is "continuous" and no matter how close together you put two points they are still discrete and do not touch.
They don't need to actually touch, though, for there to be no finite space between them. You can consider an insignificant, infinitesimal space between them.



If you take each real number between to be a point and the number of counting numbers to be infinitity then the number of points comprising the line is more than infinity. It can be considered the cardinality of the power set of whole numbers or 2^infinity which should ring true to your intuition if you think of irrationals infinite as strings of whole numbers after a decimal. In fact, you can continue this process and you will find that there are a countably infinite varieties of infinity.
I wholly agree with this. I'm not saying there's one kind of infinity. It's just a concept that comes into play when the when the number of objects become too large for the number line to represent. If the real number line contains all finite objects, those which it does not contain are infinite.
In fact, if my intuition is correct, every point can be considered as its own number line (and similarly with points within that number line). Each "level" representing a different (infinitely different) order of magnitude.


There's a neat proof I can share too if you're interested.
Please do.

[esthim8]

What element of abstract algebra is relevant here?

And what errors did you pick out in what I said?

Abstract algebra gives a wider perspective on the things that are being discussed. A lot of the speculations in this thread would not take place if people had read some abstract algebra, set theory and elementary logic. Abstract algebra, after all, concerns itself with structures of sets.

I mostly skimmed the pages, I'm not going to read 39 pages of people discussion mathematics with vague wording, bad definitions and strange conclusions. That is what my comment was referring to. I would be happy to answer some specific questions though, if put forth in a comprehensible way and do not require to much background in math, as that would take forever.

By looking a little back and just reviewing your comments, I didn't spot any errors per se, as they were mostly speculations about whether or not one could apply some well know principles and/or reformulations of something in a new and equivalent way.


The real numbers ARE infinitesimal, in the sense that they can be made arbitarily small - only, unlike the hyperreals, which correspond to the 'dxs' of calculus, they have the archimedean property.

I can take this comment for an example. Saying that numbers can be made arbitarily small implies that it has varying size. Numbers are elements of a set, they of them selfs do not have sizes, except if referred to by another set by way of a function. Example being a set of lines, each line could be given a length by assigning to it a number. All measures are functions and therefore it is strange to talk about a size of a number without talking about the measuring set and the function. What I have said above is in a much more general sense than what is being discussed here. On the reals, a number is a point and has no size. So it is not arbitarily small or with varying size. It's a point.

Speaking of measures and seeing that you are thirsty for juice math stuff, you should check out the Banach-Tarski Paradox, which is pretty amazing. Paradox by name only.

My earlier comment is probably a response to the frustration of seeing this absurd argument about whether or not 0.9999.... = 1 or not. It is self evident from the axioms of the reals.

Also, things like x<1 = x<=1, is rage material. Wrong in so many ways.

[atom]

This is unfair. You're only adding a finite amount of zeroes.


sum from n to infinity of 0 is still zero. Zero is the additive identity. Applying the identity operation ad infinitum will never change the output.




They don't need to actually touch, though, for there to be no finite space between them. You can consider an insignificant, infinitesimal space between them.


The real number line is continuous. Points are discrete. Imagine zooming in on such an "insignificant, infinitesimal space." Points will remain points and the space will become a chasm.

Your conception:

..........(.....).....................

My rebuttal:

(. . . . .)


Actuality:

_________________________





In fact, if my intuition is correct, every point can be considered as its own number line (and similarly with points within that number line). Each "level" representing a different (infinitely different) order of magnitude.

Your intuition is a bit off. Remember what each "point" on a number line represents: a number. For example, 1/4. 1/4 is not a number line, it is a single number, a discrete object. You can't go below zero-dimensional, you can go higher but your intuition is poor guide there.


Please do.

Here we go.

Allow me that if you are able to line objects from two groups side by side and, exhausting both groups, if each object from the first group lies by exactly one group from the second group then there are an equal number of objects in each group. Mathematically, a function that assigns such a pairing is called a bijection.

Lets assume that there are as many real numbers (0,1) as there are whole numbers. Then we could line them up side by side and each real would have a counterpart whole number. This creates a contradictin as I can easily construct a real number, r, not on the list.

The construction is as follows: Look at the nth digit of the nth real number. If it is a 5 make the nth digit of r 3, otherwise make the nth digit or r a 5. r then cannot be in a pair of the bijection because its 1st digit is different than the 1st digit of the 1st number. Its 2nd digit is different than the 2nd digit of the 2nd number and so on.

So I have shown that there are more reals (0,1) than there are whole numbers. BTW this is a standard, classic proof, not something i made up. Let me flesh out the relative size of these sets for you later after I go to the gym.

[puzzled-observer]

sum from n to infinity of 0 is still zero. Zero is the additive identity. Applying the identity operation ad infinitum will never change the output.
what about the sum of (x/infinity) from n to infinity? If you accept that 1/infinity =0, this substitution should work. You get infinity/infinity which is indeterminate. As in, it is not identically 0.





The real number line is continuous. Points are discrete. Imagine zooming in on such an "insignificant, infinitesimal space." Points will remain points and the space will become a chasm.
But you must agree that this "chasm", however large it appears in this zoomed in frame, carries no finite magnitude from the reference frame of our original number line. If it did, then we could specify that region and find that Y is not defined for that position.




Your conception:

..........(.....).....................

My rebuttal:

(. . . . .)

Actuality:

_________________________

I'm saying that your rebuttal and what you label my conception are not mutually exclusive. That the difference between the two is a matter of orders of magnitude. That, from the reference frame of our original number line, yours is an infinitely accurate approximation of mine. If there ARE an infinite number of infinities, then there are an infinite number of 0's.



Your intuition is a bit off. Remember what each "point" on a number line represents: a number. For example, 1/4. 1/4 is not a number line, it is a single number, a discrete object. You can't go below zero-dimensional, you can go higher but your intuition is poor guide there.
Dimension isn't the same thing as magnitude. A 3 dimensional object of 0 magnitude is no different than a 0 dimensional object of 0 magnitude. My contention is that some zero magnitudes are smaller than other zero magnitudes (just as some infinities are larger than others). That "0" and "infinity" represent a limitation of the number line itself to distinguish relative values.





*proof*
very nice.

[joft]

I'm reading as people describe their intuition about this and trying to think of what mathematical concept, if any, corresponds to what they are trying to describe. I've noticed a few things:

When people say "the number line is continuous" or "there are no gaps," the mathematical idea they are referring to is called connectedness. This is a property of topological spaces which, roughly stated, means that the whole space is made of one unbroken piece. In fact, people are probably really thinking of path connectedness, which means that every two points in the space can be connected by an unbroken path (a path that does not leave the space). Connectedness and path connectedness are actually not the same thing, which may seem bizarre but is true.

When people talk about the "size of a set," there are two possible meanings. One is the cardinality--how many distinct elements are in the set (if the set is finite, otherwise we must consider cardinality to be defined by bijections as atom wrote about). Another notion of the size of a set is measure--this is like the length of a line segment, the area of a region, or the volume of a solid.

These three concepts are not equivalent to each other, and in general are not even related to each other.

To see that cardinality and measure are (usually) not related, consider the following facts:

1) The entire real number line, the unit square, the unit interval, and the Cantor set all have the same cardinality.

2) The unit square has 2-dimensional measure equal to 1, but the unit interval has 2-dimensional measure equal to 0.

3) The unit interval has 1-dimensional measure equal to 1. The real number line has 1-dimensional measure equal to infinity. The Cantor set has 1-dimensional measure equal to 0.

Caveat: Any countable subset of real numbers has measure equal to zero. But the converse is not true, as the Cantor set shows (it has measure zero but cardinality of the real number line).

To see that cardinality and connectedness are not related, consider:

1) The Cantor set and the unit interval both have the same cardinality.

2) The unit interval is connected, but the Cantor set is totally disconnected; meaning every two distinct points of the Cantor set belong to two distinct "pieces" of the set (or, if you like, no two points can be connected by an unbroken path).

Caveat: Any countable subset of real numbers is disconnected. But the converse is not true, as the Cantor set shows.

To see that measure and connectedness are (usually) not related:

1) The unit interval with one point, 1/2, removed, has the same cardinality as the whole unit interval.

2) The former is disconnected, the latter is connected.

Caveat: If a set of real numbers has measure 0 and at least 2 elements, it is disconnected.

[atom]

oye.

i spent 30+ minutes on a nice explanation of the mechanics of the different infinities but hit shift backspace by mistake and lost it all.

I'm a math teacher and love trying to help, but at some point you need to realize your intuition is incorrect. Intuition and creativity are important in math but, among the many disciplines, math has the highest standard of knowledge: absolute proof.

Please drop the idea of an unzoomable infinitesimal gap. If there's a gap, it can be zoomed into. Scale is relative. In fact, there are as many real numbers between 1*10^-1000 and 2*10^-1000 as there are (-infinity, infinity).

There are no degrees of zeros. There are different objects in different sets that can be called zero though. A 3 dimensional object of 0 magnitude is different than a 0 dimensional object of 0 magnitude but its a difference in type not of magnitude.

I'll try again tomorrow.

[zoltan]

i spent 30+ minutes on a nice response but hit shift backspace by mistake and lost it all.


ctrl-z

[atom]

you don't get to undo going back a page in the browser

[Resonance]

firefox and chrome both cache your form entries so that going 'forward' again will solve that

heck even my little shitty phone browser does that

[Ferrus]

I can take this comment for an example. Saying that numbers can be made arbitarily small implies that it has varying size. Numbers are elements of a set, they of them selfs do not have sizes, except if referred to by another set by way of a function. Example being a set of lines, each line could be given a length by assigning to it a number. All measures are functions and therefore it is strange to talk about a size of a number without talking about the measuring set and the function. What I have said above is in a much more general sense than what is being discussed here. On the reals, a number is a point and has no size. So it is not arbitarily small or with varying size. It's a point.
Okay, point taken... hmm that sounds unintentionally punning, oh well. The real line is the set of all points defined by Dedekind cuts, right?

Actually, what you have said has just made it obvious to me now why exactly the concept of a measure developed in the first place.

I'll grant I do need to brush up my my knowledge of logic. I find my lack of facility with it is an impediment in both philosophy and maths.

[esthim8]

Okay, point taken... hmm that sounds unintentionally punning, oh well. The real line is the set of all points defined by Dedekind cuts, right?

Actually, what you have said has just made it obvious to me now why exactly the concept of a measure developed in the first place.

I'll grant I do need to brush up my my knowledge of logic. I find my lack of facility with it is an impediment in both philosophy and maths.

It is in my view much more informative to think of the real number line as the completion of the rationals via Cauchy sequences. But yes, equivalently, the real line is the rational numbers with the addition of dedekind cuts, if memory serves me right. I actually think that the way natural numbers, integers and rationals are defined and constructed is much more interesting. Using functions and sets as the "natural" things in mathematics, Natural numbers can be constructed with the Peano's axiomatization.

[zoltan]

you don't get to undo going back a page in the browser

understood

[atom]

Here we go again, I assume others are still interested:

Infinities

Before we get started, lets clear up some ambiguity. There are different infinities, so lets call the number of whole numbers "aleph nought."

So I've shown there is at least one more real number than the naturals. That may not be very satisfying but we can generate many more.

Consider the list of reals we constructed for the proof. To construct a number not on that list, consider the nth digit of the nth number on the list and choose a different digit. Lets not choose 9 in order to avoid the problem that started this thread. For each of the aleph nought digits of the number we're constructing we can choose from at least 8 other digits (9 when the digit replaced is a 9). This means that, hypothetically, we can construct at least 8^aleph nought additional reals.

Let's call the number of reals C (for continuum). Can we get C from aleph nought?

Lets try adding aleph nought to itself and see what happens. If there are aleph nought whole numbers (the positives) then there are twice as many integers (positives and negatives). However it is straightforward to construct a bijection from the integers to the whole: the negatives are mapped onto the odds and the positives are mapped onto the evens. Because a bijection exists, there are the same number of integers and whole numbers.

So 2(aleph nought) = aleph nought. It follows that taking away a aleph-nought cardinality subset from the whole numbers leaves an aleph nought-sized set.

What if we squared it?

[atom]

Imagine an infinite list of lists of all the whole numbers arranged checkerboard style:

1 2 3 4 ...
1 2 3 4 ...
1 2 3 4 ...
. . . .
. . . .
. . . .

We could label each integer as an ordered pair (row #, column #). Even in such an arrangement, an aleph null by aleph null grid, there are the same number of items as whole numbers. The bijection is as follows: trace out each item one at a time along downward and leftward diagonals:

1 -> (1,1)
2 -> (1,2)
3 -> (2,1)
4 -> (1,3)
5 -> (2,2)
6 -> (3,1), etc.

So aleph nought^2 = aleph nought.

[atom]

So how do we make sense of C and greater infinities?

I gave a hint earlier with the method for creating additional reals.

Let me introduce the idea of a powerset. A powerset of a given set is the set of all possible subsets of the set, denoted P(set name).

Let A = {1,2,3}
Then P(A) = {empty, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

The cardinality of the powerset is always greater than the original set.

Proof: Assume there is a bijection from a set to its powerset. One of the subsets (call it B) of the original set will contain all the items that are not members of the sets they are mapped onto.

A bijection means that one item of the original set (call it y) maps onto B.

If y is a member of B then y can't be a member of B.

If y is not a member of B then y should be in B.

This is a contradiction and so no such bijection can exist.

(Your head should be spinning a bit. This is another classic proof.)

Considering the combinatorics of selection subsets, it should be apparent that |P(A)| = 2^|A|. Thus C, the next bigger infinity after aleph nought is sometimes writtin as 2^aleph nought. It is also written aleph 1.

So |P(aleph nought)| > |aleph nought|. You can again take the power set of P(aleph nought) to construct an even larger set ad infinitum. These are usually denoted aleph 1, aleph 2, etc.

[atom]

The number line

Puzzled observer, I've been thinking about your arguments and I want to say a few things.

Your intuition may better reflect the rational numbers in that the rationals are grainy and the numbers don't touch (disconnected as Joft says). The set of rationals in some sense may approximate the number line as they are "dense" but they are not the same.

I think you are missing the fact that for any given real number, there is no "next" real. For rationals this is easy to demonstrate: given a/b let the next rational be x/y. No such x/y can exist because (ay+bx)\2by (the average) will be between the two and must be rational as well.

Its also true that between any two rationals there is at least one irrational (the converse is true as well). I can provide proof of this as well if youre curious.

It should be ovious then that there is no infinitesimal gap on the real number line. Between any two reals there are, in fact, C other reals.

There you go, the first few weeks of analysis I.

Ferrus, Esthim8 I'd love to contribute to y'alls convo, but I've only taken analysis I.

[Ferrus]

Moved some off topic discussion to Joft's thread.

[puzzled-observer]

First of all, thank you atom for these explanations, I'm grateful.



Please drop the idea of an unzoomable infinitesimal gap. If there's a gap, it can be zoomed into. In fact, there are as many real numbers between 1*10^-1000 and 2*10^-1000 as there are (-infinity, infinity).
I'm not sure what i said to give the impression that i think there is an "unzoomable" scale. I didn't mean to indicate that.
As to the second part: Don't you mean, to be precise, that the cardinality of the two is the same? Would it be wrong to say that cardinality (when applied to infinities) represents the order of magnitude of the infinity?



So how do we make sense of C and greater infinities?

Proof: Assume there is a bijection from a set to its powerset. One of the subsets (call it B) of the original set will contain all the items that are not members of the sets they are mapped onto.

A bijection means that one item of the original set (call it y) maps onto B.

Do you mean that B is a subset of P(A)? And that y is a member of A? (Assuming A is the original set)
I'm not clear on what you mean here.



Considering the combinatorics of selection subsets, it should be apparent that |P(A)| = 2^|A|. Thus C, the next bigger infinity after aleph nought is sometimes writtin as 2^aleph nought. It is also written aleph 1.

Question: Would aleph nought^(aleph nought) be of the same cardinality as to 2^(aleph nought)?


The number line

Puzzled observer, I've been thinking about your arguments and I want to say a few things.

Your intuition may better reflect the rational numbers in that the rationals are grainy and the numbers don't touch (disconnected as Joft says). The set of rationals in some sense may approximate the number line as they are "dense" but they are not the same.

I think you are missing the fact that for any given real number, there is no "next" real. For rationals this is easy to demonstrate: given a/b let the next rational be x/y. No such x/y can exist because (ay+bx)\2by (the average) will be between the two and must be rational as well.

Its also true that between any two rationals there is at least one irrational (the converse is true as well). I can provide proof of this as well if youre curious.

I think I've done a bad job of communicating. Can I ask what leads you to believe that i consider there to be an actual "next real" point. One of the main points I tried to get across was that infinitesimal distances are those that can't be evaluated meaningfully on the number line. That you would need an infinitely smaller reference frame to do so.



Between any two reals there are, in fact, C other reals.


So for two points 0 and 1, doesn't this imply that from 0, there exists a real number that is a distance of 1/C away from 0 (at least in order of magnitude)? Then, applying this principle again, between 0 and the point 1/C away, there are again C reals. Because 1/C is actually equal to 0, between 0 and 0, there are C reals. There are only two ways i see to resolve this paradox: Either the points at 0 on the real number line can be considered an infinitesimally smaller number line (and similarly for the points on that smaller number line). Or that your premise only applies for reals that are a finite distance away from each other.
I must apologize for being as stubborn as I am. But it can't be helped.

[atom]

First of all, thank you atom for these explanations, I'm grateful.

No prob. I enjoy having the opportunity to explain it. Your questions are making me think.



Don't you mean, to be precise, that the cardinality of the two is the same? Would it be wrong to say that cardinality (when applied to infinities) represents the order of magnitude of the infinity?


It is correct to say that the cardinality of any interval [a,b] is equal to that of (-infinity,infinity). It is incorrect to ascribe a cardinality to an infinity. Cardinality is a property only of sets. We are getting to the limits of my knowledge but, as far as I know, the best we can do is to put the alephs in order.

It would be improper to describe an order of magnitude of an aleph. Order of magnitude usually connotes the greatest power of ten that divides a number. Also it may be worth pointing out that the alephs are not continuous like numbers are; there is nothing between aleph nought and C.



Do you mean that B is a subset of P(A)? And that y is a member of A? (Assuming A is the original set)
I'm not clear on what you mean here.


B is a member of P(A). P(A) is a set of sets, every member of P(A) is a set itself. See my example.

Let me define B explicitly. Assume f is the bijection mapping A onto P(A).

Let B = {x: x is not a member of f(x)}

So x is a member of A and f(x) is some subset of A which is a member of P(A) by the definition of P(A).



Question: Would aleph nought^(aleph nought) be of the same cardinality as to 2^(aleph nought)?


I'm not sure how to think about that. I'll do a bit of research but I'm pretty lazy and am not promising anything. Perhaps joft or esthim8 knows.



I think I've done a bad job of communicating. Can I ask what leads you to believe that i consider there to be an actual "next real" point. One of the main points I tried to get across was that infinitesimal distances are those that can't be evaluated meaningfully on the number line. That you would need an infinitely smaller reference frame to do so.


Such an infinitesimal distance only makes sense between one number and the next so I assume your thinking something like that.

Distance on the metric space of real numbers is a property only of pairs numbers. The common (but not only) measure of distance can be intuited as a distance along an interval of the number line like distance along a ruler. Any such interval has the same cardinality as the entire number line. If the interval doesn't have two endpoints, its singleton (single number) or empty.

You are intuiting there are two types of distances: zoomable that will contain C elements and infinitesimal which differ in that you must zoom "infinitely" to "see". I am trying to convince you infinitesimal distances is not a coherent idea and there is one kind of interval.



So for two points 0 and 1, doesn't this imply that from 0, there exists a real number that is a distance of 1/C away from 0 (at least in order of magnitude)? Then, applying this principle again, between 0 and the point 1/C away, there are again C reals. Because 1/C is actually equal to 0, between 0 and 0, there are C reals. There are only two ways i see to resolve this paradox: Either the points at 0 on the real number line can be considered an infinitesimally smaller number line (and similarly for the points on that smaller number line). Or that your premise only applies for reals that are a finite distance away from each other.
I must apologize for being as stubborn as I am. But it can't be helped.

C is a cardinal number. The cardinal numbers are 1,2,3,..., aleph nought, aleph 1.... The cardinal numbers have no multiplicative inverse, ie 1/C is meaningless. It then follows there are no numbers such that |a-b| = 1/c. Distance in any metric space is measured as a real number and C is not a real.

I believe that you believe your idea of infinitesimal distances allows you to define 1/C as zero which leads to the contradicition you have stated, that there are numbers between 0 and 0. This is not sensible and so your premise is flawed.

My premise only applies for reals that are a finite distance away from each other.

Let me know if I left any loose ends.

[puzzled-observer]

No prob. I enjoy having the opportunity to explain it. Your questions are making me think.
Very good.



Such an infinitesimal distance only makes sense between one number and the next so I assume your thinking something like that.

Distance on the metric space of real numbers is a property only of pairs numbers. The common (but not only) measure of distance can be intuited as a distance along an interval of the number line like distance along a ruler. Any such interval has the same cardinality as the entire number line. If the interval doesn't have two endpoints, its singleton (single number) or empty.
Where "empty" corresponds to "0 distance", or to frame it in a way that better supports my argument: "no finite distance".




You are intuiting there are two types of distances: zoomable that will contain C elements and infinitesimal which differ in that you must zoom "infinitely" to "see". I am trying to convince you infinitesimal distances is not a coherent idea and there is one kind of interval.

I'm saying that in the case that you have C objects confined to your field of view, in order to be able to differentiate 1 element from another element, you need to have a magnification that is proportional to C. If I could zoom in any finite degree (say 100 times zoom) and be able to accurately differentiate one element from another element, I could say with confidence that there were not C objects in that field.





C is a cardinal number. The cardinal numbers are 1,2,3,..., aleph nought, aleph 1.... The cardinal numbers have no multiplicative inverse, ie 1/C is meaningless. It then follows there are no numbers such that |a-b| = 1/c. Distance in any metric space is measured as a real number and C is not a real.

I believe that you believe your idea of infinitesimal distances allows you to define 1/C as zero which leads to the contradicition you have stated, that there are numbers between 0 and 0. This is not sensible and so your premise is flawed.

My premise only applies for reals that are a finite distance away from each other.

Let me know if I left any loose ends.
So, I've been thinking about this a lot today. I'm going to summarize my perception of the argument here. I hope I'm not misrepresenting you.
My argument
1. There is either a finite or non-finite distance between points.
a. If the distance between elements is finite, then obviously having C elements causes the sum to explode to infinity.
b. If the distance is not finite then we have the problem where multiple elements will exist for the same real number value. As, by a limiting argument, I think most would agree that 1/infinity = 0 (even if it's not technically meaningful to use "C" in this case, it's conceptually the same idea).

Your response to this seems to be to deny the premise behind my dilemma all together. That, in fact, the spacing doesn't need to be either finite or non-finite because specifying a distance in that way is meaningless. That points kind of exist as independent objects from the number line all together. That there is no "next" point because I cannot ever specify the next point. This, it seems to me, is essentially a consequence of using sets and bijections to define the intervals and their respective cardinality.

But if we are to say that the line is actually made up of points strung together (as in, assuming path connectedness), then there IS a "next" point. Isn't there? That next point must be infinitely close to this one. Meaning, that the result of 1b. must be true, however senseless it may seem.

It simply doesn't make sense to my mind how one can have C elements in a finite distance and not have elements existing infinitesimally close to one another.

[atom]

Where "empty" corresponds to "0 distance", or to frame it in a way that better supports my argument: "no finite distance".


No. An interval is the set of real numbers with magnitudes between the endpoints of the interval.

A singleton: [1,1] = {1}

Empty: (1,1) = {} empty

There are no numbers between 1 and 1, exclusive, and so the interval is empty. There is nothing there.




I'm saying that in the case that you have C objects confined to your field of view, in order to be able to differentiate 1 element from another element, you need to have a magnification that is proportional to C. If I could zoom in any finite degree (say 100 times zoom) and be able to accurately differentiate one element from another element, I could say with confidence that there were not C objects in that field.



The second part is true and good.

The first part is wrong.

Again: _____________

You cannot differentiate between the points visually. You want to find a scale at which you can count them and confirm there a C numbers there. But C is not countable, there is no bijection from C to the counting numbers; its not an issue of scale.



So, I've been thinking about this a lot today. I'm going to summarize my perception of the argument here. I hope I'm not misrepresenting you.
My argument
1. There is either a finite or non-finite distance between points.
a. If the distance between elements is finite, then obviously having C elements causes the sum to explode to infinity.
b. If the distance is not finite then we have the problem where multiple elements will exist for the same real number value. As, by a limiting argument, I think most would agree that 1/infinity = 0 (even if it's not technically meaningful to use "C" in this case, it's conceptually the same idea).

Your response to this seems to be to deny the premise behind my dilemma all together. That, in fact, the spacing doesn't need to be either finite or non-finite because specifying a distance in that way is meaningless. That points kind of exist as independent objects from the number line all together. That there is no "next" point because I cannot ever specify the next point. This, it seems to me, is essentially a consequence of using sets and bijections to define the intervals and their respective cardinality.

But if we are to say that the line is actually made up of points strung together (as in, assuming path connectedness), then there IS a "next" point. Isn't there? That next point must be infinitely close to this one. Meaning, that the result of 1b. must be true, however senseless it may seem.

It simply doesn't make sense to my mind how one can have C elements in a finite distance and not have elements existing infinitesimally close to one another.

Your intuition of the number line as a "string of numbers" is incorrect. There is no next point.

Also, the set theoretic account of reals is just as fundamental as the number line. Using a set to understand an interval doesn't invalidate any conlusion just as using arguments based on decimal representations wouldn't.

Consider: What is the first number after zero?

Consider the list of all numbers of the form 1/n. As n gets bigger, 1/n gets closer to zero. Think of any candidate as the next number after zero and there will be some n such that 1/n is less than that number.

The limit as n goes to infinity of 1/n is zero but 1/infinity is meaningless because infinity is not a number.

Do me a favor and try to define infinitesimal for me.

[Polemarch]

Is this all some kind of joke? This thread was completely resolved by post # 6:

http://forums.intpcentral.com/showpost.php?p=22364&postcount=6

The Titanic sunk during the opening credits...

[atom]

You're preaching to the choir.

We're just shooting the shit after the party's over.

[puzzled-observer]

I'm putting this first as i see it as the crux of the debate.


Do me a favor and try to define infinitesimal for me.
I suppose you could define it a number of ways. Simply stated something which is infinitesimal has 0 length.

To use a more conceptually sensible definition either I would borrow from your proof:


Consider the list of reals we constructed for the proof. To construct a number not on that list, consider the nth digit of the nth number on the list and choose a different digit. Lets not choose 9 in order to avoid the problem that started this thread. For each of the aleph nought digits of the number we're constructing we can choose from at least 8 other digits (9 when the digit replaced is a 9). This means that, hypothetically, we can construct at least 8^aleph nought additional reals.
And say that if n = aleph nought then n-1 is infinitesimally different from n regardless of the digit you assign either. And I'm stating that the point corresponding to n-1 exists on the number line.

Or from limits in general (essentially the same idea): infinitesimal = lim x->infinity (1/x)


No. An interval is the set of real numbers with magnitudes between the endpoints of the interval.

Ahh, I see. Well that makes sense.




You cannot differentiate between the points visually. You want to find a scale at which you can count them and confirm there a C numbers there. But C is not countable, there is no bijection from C to the counting numbers; its not an issue of scale.
I'm only talking in a visual sense because we're talking about zooming in. Conceptually, if i were to zoom in to where points are differentiable, then (assuming i understood your earlier proofs correctly), you're right, I could only observe aleph nought numbers. That there would be infinitely many numbers (C numbers) which are infinitely far from any point in this scale.



Consider: What is the first number after zero?

Consider the list of all numbers of the form 1/n. As n gets bigger, 1/n gets closer to zero. Think of any candidate as the next number after zero and there will be some n such that 1/n is less than that number.
This proof only shows that the next number isn't distinguishable (A statement which i agree with). I find this interesting because it brings to mind the idea of a theory vs a fact. Maybe what I'm claiming would be better categorized as a theory of behavior rather than a mathematical fact. The nature of the idea is one that's, as far as i can tell, unprovable. From what I can tell of discussion we don't disagree on any claims which are actually demonstrable. The only thing we disagree on is the meaning of 1/0 (which WILL always be either infinity or -infinity) and 0*infinity (which, in case's where I've seen it resolved, can yield any value from 0 to infinity). Now, don't consider this as me giving up, I haven't yet. I still think i may be right




The limit as n goes to infinity of 1/n is zero but 1/infinity is meaningless because infinity is not a number.

So what does it mean for something to "go to infinity" then? Are you saying it's improper to graph the function on the number line when you're showing it go to infinity? Whether or not it's on the number line (and there DOES exist the extended real line where infinity is tacked on the end via the ... mechanism), it can still have a meaningful value in an equation with numbers.



Also, the set theoretic account of reals is just as fundamental as the number line. Using a set to understand an interval doesn't invalidate any conlusion just as using arguments based on decimal representations wouldn't.I have to go, i'll respond to this later.

[Eric B]

Your intuition of the number line as a "string of numbers" is incorrect. There is no next point.

Also, the set theoretic account of reals is just as fundamental as the number line. Using a set to understand an interval doesn't invalidate any conlusion just as using arguments based on decimal representations wouldn't.

Consider: What is the first number after zero?

Consider the list of all numbers of the form 1/n. As n gets bigger, 1/n gets closer to zero. Think of any candidate as the next number after zero and there will be some n such that 1/n is less than that number.

The limit as n goes to infinity of 1/n is zero but 1/infinity is meaningless because infinity is not a number.

Do me a favor and try to define infinitesimal for me.

The "next point" would actually be in the same exact place as the first point, or basically, would be the same point. A line is defined by points in that at any interval (1/1, 1/2, 1/100, 1/1000000, etc) there is a point. They really are not "next to" each other, unless you set a particular scale as your smallest distance. In an infinitessimal scale, the concept of "next to" breaks down. It's infinity squeezed down into finite lengths.

[bass_n_treble]

I would be pissed if I was the .001 right now. Nobody thought of them, did they? :(

[atom]

I suppose you could define it a number of ways. Simply stated something which is infinitesimal has 0 length.


0 != anything that's not zero.

If infinitesimal has any meaning, it can't mean the same thing as zero.




To use a more conceptually sensible definition either I would borrow from your proof:

And say that if n = aleph nought then n-1 is infinitesimally different from n regardless of the digit you assign either. And I'm stating that the point corresponding to n-1 exists on the number line.



False.

aleph nought - 1 = aleph nought





Or from limits in general (essentially the same idea): infinitesimal = lim x->infinity (1/x)


False. this limit is equal to zero.




I'm only talking in a visual sense because we're talking about zooming in. Conceptually, if i were to zoom in to where points are differentiable, then (assuming i understood your earlier proofs correctly), you're right, I could only observe aleph nought numbers. That there would be infinitely many numbers (C numbers) which are infinitely far from any point in this scale.


__________ -> no "differentiable" numbers at any scale. There is no "graininess."




This proof only shows that the next number isn't distinguishable (A statement which i agree with).


False. The proof shows no such number exists because it would be simple to construct a number that is even closer to zero.



So what does it mean for something to "go to infinity" then? Are you saying it's improper to graph the function on the number line when you're showing it go to infinity? Whether or not it's on the number line (and there DOES exist the extended real line where infinity is tacked on the end via the ... mechanism), it can still have a meaningful value in an equation with numbers.


What you are looking for I think is a formal definition of a limit.

Lim f(x) as x -> infinity = r means that for for all e > 0, there exists a N such that |f(x) - r| < e when x > N.

Notice that there is no mention of infinity in the definition.

You want to use infinity likes its a number when its not.

[^_\\]

Here are some of my thoughst. Sorry for terrible notation, I haven't done much limits. I'm gonna try and extrapolate from n up to infinity. Or rather find for n approaching infinity as n cannot reach infinity (infinity not being a number.)

For a capital letter followed by a number or lower case letter, read the second as subscript.


series 1: An=3/10^N
Sum of series converges towards 1/3. (but doesn't reach it.)
Assume sum of series from A to infinity =0.3`

A1=3/10^N or 0.3
A2=3/10^(N+1) or 3/10^2 or 3/100 or 0.03
A3=3/10^(N+2) or 0.003

A1+A2+A3+...=0.3`

3+ A1+A2+A3...Ainfinity=3.3` 3 is not A0 here (okay so it is but try to think of 3 and the sequence as seperate) I should have picked better numbers but it's a bit late now

let X= A1+A2+A3+...=0.3`
let Y=3+ A1+A2+A3...=3.3`



if 3.3`=10/3
and 0.3`=1/3
0.3`*10=3.3`
or expressed differently 10x=y

for sum An up to n=1
X= 0.3=0.3
Y=3 +0.3=3.3
10x=3=3
Y=3.3=3.3
y-10x=0.3

for sum up to n=2
X= 0.3+0.03=0.33
Y=3+0.3+0.03=3.33
10x=3.3
Y=3.33
y-10x=0.03

That's me convinced anyway. Looks to me like 10x never catches Y.
y-10x= 0.(n-1 zeros)3
Where does this change? If we treat infinity as a number, it never changes. If we don't treat infinity as a number n just continues to get bigger, so the difference continues to get smaller forever. So, lets treat infinity like a number. As N approaches infinity, the difference becomes infinitely small, and at no point becomes zero (and does not skip past the point of becoming zero, to suddenly be zero.)

To put it another way:

For any value of n
10Xn=10A1+10A2+10A3+etc if you prefer.

Every term is ten times bigger than the preceding one so...
10Xn=A0+A1+A2+etc
yn=3+ A1+A2+A3 etc

for any given value of n y is An bigger than 10x. in this case n=3. y is A3 bigger than 10x. Y is always An bigger than X.

I'm not sure how I came to this conclusion as I'm tired but the way I see it 1/3 is always 0.0`3 bigger than 0.3` and 0.0`3 is an infinitessamally small number.

From what I can tell either point of view is recursive (they come down to the definition of X/infinity.) Mine has the advantage of not making 1/a concept or a number or something 0, yours has the advantage of being easy to work with, and being the established way of doing things.

As far as I know (which isn't very far) everything that can be done with standard analysis can be done with non-standard analysis and then some. This seems to me like evidence that non-standard analysis is fundementally more accurate.

I haven't read all of the thread. sorry if I missed some important things. I am basing all of this on an hour or so of scribblings so sorry if I'm full of shit.

edit: tl:dr an infinite sequence that converges on a number never reaches that number. Limits are for approximation: the number is never reached. In the same way that a graph that approaches 0 never touches the X axis assuming infinitely small or (0 sized) axis and line.

[artifact]

In my algebra 2 class a kid was saying it does. If I remeber correctly he said something about the fraction 1/3. I wasn't paying much attention because I was doing the assignment but I find the question quite intresting. I was hoping some of you math wizards could clear this up for me.

.999... = 1

Melody basically hit the nail on the head mathematically speaking. You need the infinite sum.

[Reepicheep]

No, not even infinitely recurring.

As you have no doubt already been informed, your answer is incorrect.

In consideration of repeating numbers in pre-calculus class lo these many years ago, we proved that 0.999... (repeating an infinite series of 9s to the right of the decimal) was precisely equal to 1. I no longer remember the details of the proof, but I remember that it was so.

[GoetheWasGreat]

9820

Here is my proof, since I cannot link an image.

[sandwich]

LOL.

I would berate you for necromancy, but this is one of my most favorite threads.

[Professor Chaos]

In most cases, I think so.

However, wouldn't this in context dependent, in that the level of precision could potentially matter?

[GoetheWasGreat]

LOL.

I would berate you for necromancy, but this is one of my most favorite threads.

Sorry about that, I didn't even think to check dates. :stupid:



In most cases, I think so.

However, wouldn't this in context dependent, in that the level of precision could potentially matter?

It only works in cases where the trailing series of 9's are infinite. Anything less and it is absolutely not equal to 1.

[sandwich]

Ha, no worries. Though I do suggest you attempt to read this thread. The whole thing.

[Skinart]

To hell with suggesting. I double dog dare you.

Side note...have we never had a thread about a^0 = 1?

[A Schnitzel]

To hell with suggesting. I double dog dare you.

Side note...have we never had a thread about a^0 = 1?

It's as good a time as any to start it.

We can only hope it's as successful as this one.

[GoetheWasGreat]

I skimmed the thread and it looks like there are simply a lot of imprecise definitions and faulty intuition. My proof was rather simple, but straightforward. I assumed that if .999... doesn't equal 1, then there must be a real number k such that .999... + k = 1. I then went on to show that such a number k must be equal to 0, and therefore that .999... = 1. Of course the proof rests upon some other axioms and proven statements about infinite summations, but such is the nature of mathematics. On the shoulders of giants and all that stuff.

[Hexchild]

I skimmed the thread and it looks like there are simply a lot of imprecise definitions and faulty intuition. My proof was rather simple, but straightforward. I assumed that if .999... doesn't equal 1, then there must be a real number k such that .999... + k = 1. I then went on to show that such a number k must be equal to 0, and therefore that .999... = 1. Of course the proof rests upon some other axioms and proven statements about infinite summations, but such is the nature of mathematics. On the shoulders of giants and all that stuff.

My guess is that EZ, were this still 2004 and were he not banned a few times over, would have argued something along the lines of either that 10 krunk indeed equals krunk, even though krunk != 0, or that you can't multiply by 10 in the first place because of krunk being unpredictable in such cases, much in the same way that you can't expect equations to match up if you divide by 0 somewhere.

In case you missed it, krunk is supposedly (unless I misinterpreted it) the number you get if you divide 1.0 by infinity, or subtract 0.999... from 1.0, so I think it pretty much fills the role of k in your proof.

[Polemarch]

In case you missed it, krunk is supposedly (unless I misinterpreted it) the number you get if you divide 1.0 by infinity, or subtract 0.999... from 1.0, so I think it pretty much fills the role of k in your proof.

Is that distinct from

http://facepwn.com/posters/crunk.jpg

[Arcades]

I think this thread gets resurrected more than any other. Hell I was brand new here when it first popped up.

yes... still lurking.

[square]

This may have already been posted, but I have seen a proof along these lines before.

Let 0.9` + k = 1
3(0.3`) + k = 1
3(1/3) + k = 1
1 + k = 1
k = 1 - 1
k = 0

Therefore 0.9` = 1

Alternatively, 0.9` = 3(0.3`) = 3(1/3) = 1

If the proof is sound then there are two possibilities. 0.9`= 1, or 0.9`!= 1. The only way for there to be a sound proof of equality yet 0.9`!= 1 is for 0.9` to not exist. It is not possible for a non-existent thing to be equal to something that exists despite a proof that they are equal. If 0.9` does not exist then that is because infinity does not exist.

It would seem to come down to a choice. People can either chose to accept that 0.9`=1 and accept the existence of infinity along with it, or reject that 0.9`=1 and reject the existence of infinity along with it.

Another way of looking at it is to ask if 0.9`!= 1 then what is the difference between them. The difference would have to be 1/infinity. The question then becomes what is the value of 1/infinity. Is it zero, or is it not zero. If you define infinity such that 1/infinity = zero, then 0.9` = 1. If infinity exists, then I am happy to define infinity such that 1/infinity = zero. If you want to claim that 1/infinity is not equal to zero then it can't be some other real number because that would make infinity finite. You'd have to create a new type of number to contain the number 1/infinity, but I don't see much value in doing that.

I think the answer is that 0.9`= 1 because of the strange properties of infinity, or 0.9`!= 1 because 0.9` and infinity do not exist.

[Hexchild]

People can either chose to accept that 0.9`=1 and accept the existence of infinity along with it, or reject that 0.9`=1 and reject the existence of infinity along with it.

As you nearly stated, there's also the option to declare the proof unsound. This would require some further definitions of course, such as what exactly it is that makes the proof unsound, and by extension what it is one must avoid to be able to produce sound proofs.


As for the value of creating new types of numbers, IMO there is always an inherent value in exploring the possibilities of alternative math systems. Without the desire to do that, I doubt we'd have complex numbers and quaternions, or even negative numbers for that matter.

[square]

As for the value of creating new types of numbers, IMO there is always an inherent value in exploring the possibilities of alternative math systems. Without the desire to do that, I doubt we'd have complex numbers and quaternions, or even negative numbers for that matter.

Would infinity + 1 be included in the set of this new type of number? ;)

[ApeTheDog]

This may have already been posted, but I have seen a proof along these lines before.

Let 0.9` + k = 1
3(0.3`) + k = 1
3(1/3) + k = 1
1 + k = 1
k = 1 - 1
k = 0

Therefore 0.9` = 1

Alternatively, 0.9` = 3(0.3`) = 3(1/3) = 1

If the proof is sound then there are two possibilities. 0.9`= 1, or 0.9`!= 1. The only way for there to be a sound proof of equality yet 0.9`!= 1 is for 0.9` to not exist. It is not possible for a non-existent thing to be equal to something that exists despite a proof that they are equal. If 0.9` does not exist then that is because infinity does not exist.

It would seem to come down to a choice. People can either chose to accept that 0.9`=1 and accept the existence of infinity along with it, or reject that 0.9`=1 and reject the existence of infinity along with it.

Another way of looking at it is to ask if 0.9`!= 1 then what is the difference between them. The difference would have to be 1/infinity. The question then becomes what is the value of 1/infinity. Is it zero, or is it not zero. If you define infinity such that 1/infinity = zero, then 0.9` = 1. If infinity exists, then I am happy to define infinity such that 1/infinity = zero. If you want to claim that 1/infinity is not equal to zero then it can't be some other real number because that would make infinity finite. You'd have to create a new type of number to contain the number 1/infinity, but I don't see much value in doing that.

I think the answer is that 0.9`= 1 because of the strange properties of infinity, or 0.9`!= 1 because 0.9` and infinity do not exist.

The problem with this proof is that the piece of paper / textbox in which you're writing is not infinite in size, so it cannot accomodate every 3 that needs to be written down for the proof to hold ground.

So it is impossible to prove infinity does not exist without an infinitely big piece of paper.

[MacGuffin]

I am sick of proofs that postulate there is a real number between 0.99999999... and 1.

The whole damn point is there isn't one!

[Skinart]

A basic method of mathematical proofs is to prove by contradiction. You begin by postulating that what you are about to prove is false is true, and then demonstrate it can't be true.

The whole damn point of postulating there is a number between 0.999... and 1 is that there isn't one.

[square]

I am sick of proofs that postulate there is a real number between 0.99999999... and 1.

The whole damn point is there isn't one!

Clearly the difference between 0.9` and 1 is either zero or 1/infinity. Being equal means that the difference between two numbers is zero.

My point was that it comes down to choices rather than a proof. I am saying that people can chose to say that 0.9` and infinity do not exist so 0.9`!= 1. Alternatively they can chose to say that 1/infinity != 0 so 0.9`!= 1 or chose to say that 1/infinity = 0 so 0.9`= 1.

Maybe we could decide the matter by putting it to the vote. I suggest that the most definitive question would be, "Does 1/infinity = zero?" a) No, infinity is a non-existent number b) No, 1/infinity is not the same as zero c) Yes, they are equivalent because of the nature of infinity d) Yes, they are equivalent because 1/infinity is equal to zero by definition

[square]

A basic method of mathematical proofs is to prove by contradiction. You begin by postulating that what you are about to prove is false is true, and then demonstrate it can't be true.

The whole damn point of postulating there is a number between 0.999... and 1 is that there isn't one.

If you are saying that there is no number between 0.9` and 1, does that mean that you are saying that 0.9` = 1?

[nonperson]

Reminds of Koch snowflakes. :yes:

[Frenetic Tranquility]

The limit going to infinity of .9 repeating grows arbitrarily close to 1, but never reaches it. For the sake of mathematics we call them equal, because the arbitrarily small value between becomes too small to be of consequence.
.3 repeating does not equal 1/3rd in much the same way, except we use the values interchangeably because the arbitrarily small value between them becomes inconsequential.

The argument of multiplying by 10, then subtracting, is actually incorrect. You can't subtract .9 repeating from 10 times .9 repeating, because the 10 times .9 repeating has 1 more digit involved, even though both stretch to infinity.

[Hexchild]

The limit going to infinity of .9 repeating grows arbitrarily close to 1, but never reaches it. For the sake of mathematics we call them equal, because the arbitrarily small value between becomes too small to be of consequence.
.3 repeating does not equal 1/3rd in much the same way, except we use the values interchangeably because the arbitrarily small value between them becomes inconsequential.

What you call "limit" is not the same as what is defined as a limit by standard mathematics. Limits (http://en.wikipedia.org/wiki/Limit_%28mathematics%29) don't grow, they are fixed (for a specific range and function or sequence). In this particular case, the limit is equal to 1.0..., whether that value is ever reached or not.

[joft]

Would infinity + 1 be included in the set of this new type of number? ;)

You have no idea. Allow me to blow your mind: http://en.wikipedia.org/wiki/Ordinal_number


Maybe we could decide the matter by putting it to the vote. I suggest that the most definitive question would be, "Does 1/infinity = zero?" a) No, infinity is a non-existent number b) No, 1/infinity is not the same as zero c) Yes, they are equivalent because of the nature of infinity d) Yes, they are equivalent because 1/infinity is equal to zero by definition
"Infinity" is not a member of the set of "real numbers." Some conventions include +infinity and -infinity and might be called "extended real numbers." And they usually assume 0*infinity = 0, and maybe even 1/infinity = 0. But one might remain skeptical of such things, saying that they are only convenient notations that make certain formulas work even in the "bad" cases, e.g. some inequality like "f <= g" might be true even if f and g are "infinity"



The limit going to infinity of .9 repeating grows arbitrarily close to 1, but never reaches it.
As Hexchild has pointed out, a limit is a fixed quantity. But the intuition behind what you said is correct- the sequence is getting closer to 1, so the limit of the sequence is 1.



The argument of multiplying by 10, then subtracting, is actually incorrect. You can't subtract .9 repeating from 10 times .9 repeating, because the 10 times .9 repeating has 1 more digit involved, even though both stretch to infinity.
The decimal notation is just that- a notation. The number "exists" (in whatever sense we don't know, that's an issue for the philosophy of mathematics) independent of what we call it. So 10 times .9 repeating is just 10, and I certainly can subtract 1 from 10 (I hope!). It seems like your mistake here may be that you are confusing (1) the algorithm that we use for doing subtraction on paper with (2) the actual mathematical operation of subtraction defined on the set of real numbers.

[Psycholitics]

The fraction represents a ratio. 1/3 is defined as one of three equal parts, that when added make up a whole. 1/3 of 9 is 3 exactly, because 9 is divisible by 3. The number 1 is not divisible equally by 3, hence the repeating decimals. 1/3 is a ratio, representing the theoretical amount that is one of three equal parts that when added equal 1. But that number is not truly quantifiable. .333 is just an approximation of that ratio.

[Skinart]

Erh... .333... is a true quantification and an exact number. You are correct that .333 is an approximation, your error is that 1 is not divisible equally by 3. It is not divisible evenly by three--different. If it were not divisible equally, we would be unable to divide anything into thirds.

[Hexchild]

Looking back at my last post in this thread, I see that I probably should have stated more clearly that, according to standard mathematics, the limit of the sequence [0.9, 0.99, 0.999, 0.9999 etc.] is exactly equal to the limit of the sequence [1.0, 1.00, 1.000, 1.0000 etc.].

However, if you define 0.9... as "one minus the smallest possible number greater than 0" as has been suggested previously (thus diverging from standard mathematics), then it makes sense to also define those two limits as being different. They could (and should, IMO) still both be fixed quantities. But if you do this, the definition of what a limit is would have to change enough that methods currently used for calculating them would not be consistent with that new definition.

[euterpenc]

Maybe this is off topic, but does 0 = ∞?

[nfinityi]

Maybe this is off topic, but does 0 = ∞?
No, 0=0. Infinity is not a number, it is a condition unbound by time or space... Or spelling.

[Dman]

I think this thread gets resurrected more than any other. Hell I was brand new here when it first popped up.

yes... still lurking.

I'll say. I thought wikipedia came along and that killed it. It lives! Pretty impressive.

Even more impressive is that I still remember my password to log in here.

Speaking of which, I won't say there is a lot of drivel here now, but I will say where did everyone go? Seems real quiet.

I bet there's a thread discussing that, isn't there.