/sci/ - Science & Math

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Your premise is wrong. You, like most everyone who thinks that 0.999... does not equal 1, assume that these numbers *are* their decimals representations. That's not how real numbers work. Rather, real numbers can be *represented* by decimal expansions. A priori there is no reason why a given real number cannot be represented by two different decimal expansions, and it turns out that uniqueness does not hold.

The point is, real numbers exist independently of knowing anything about decimals.

>>12501874
I do not disagree that 1, a real number, exists.
I hold that to refer to 0.999... as 1 is to fundamentally ignore the issue at hand, the existence of an infinite sequence. You may say there is no literal difference, but if I want to discuss an infinite sequence of numbers I would never refer to 1 because it is just a whole number, not an infinite sequence of numbers. Since I cannot use both words (1 and 0.999...) interchangeably in a meaningful way, 1 =/= 0.999...
This is not true of other things. Say 5 = 2 + 3. I can say "I have five apples altogether" or "I have two apples here and I have three apples there". But if I say 1 = 0.999... it would have to follow that I can say "1 is a decimal involving an infinite series of 9's".
To be clear, if both were just terms for the same object, I would not see an issue with saying 1 = 0.999... But since there are clear differences (e.g. not being able to refer to one as an infinite sequence) they clearly are not interchangeable terms. And if they're not interchangeable, x =/= y.

>>12501874
>there is no reason why a given real number cannot be represented by two different decimal expansions
What decimal expansions can 1 be written as?

>>12501884
The number "1" has an infinite sequence of zeros after the decimal point. Zero is not a "more natural" numeral than 9

>>12501901
An infinite sequence of 0's is not an infinite sequence of 9's.

>>12501855
1/inf=0

>>12501915
No, it's 1/inf with no equivalent expression.
0 is nothing.
1/inf is an infinitely small object.

>>12501917
>infinitely small object
zero, yes

>>12501884

If you clarify for yourself what decimals actually mean, you will see that the distinctions you are making disappear, and that the "fundamental issue at hand" is not fundamental.

A.BCDE... means the real number

A + B/10 + C/100 + D/1000 + E/10000 + etc.

Here the infinite sum is defined as the limit of the sequence of numbers A ; A + B/10 ; A + B/10 + C/100 etc. which you can prove always exists.

In its decimal expansion, 1 is properly written as 1.00000... which really means

1 + 0/10 + 0/100 + 0/1000 + etc.

and you can show that this is the same as

0 + 9/10 + 9/100 + 9/1000 + 9/10000 + etc.

which is represented by the decimal 0.999...

>assume a≠b, then a≠b

>>12501920
Let's say you have a normal human range of vision. You're in a rocket ship leaving earth. You look back and see light reflecting off the planet. Eventually, the light fades. Eventually, you do not see the earth. It becomes infinitely small in your field of vision but it never ceases existing. You can say with confidence, almost always, "earth still exists".
The earth doesn't become nothing. 0 is nothing.

>>12501928
You say that, but I just don't see it as applicable to reality. What I see when I see "0.999..." is "this is a placeholder term for an object which is not whole, but which is infinitely close to being whole." When I see "1" I see "this is a placeholder term for a singular whole object on its own". While "1" can also be "1.00..." it still is referring to a whole object.
So when you say "0.999... = 1" what I see is you saying "an object which is infinitely close to being another object is that object". Now I can see "an object which is infinitely close to being another object is practically the same as that object" as being true, but to say they are the same is to deny the essential nature of the first object.

>>12501884
>But if I say 1 = 0.999... it would have to follow that I can say "1 is a decimal involving an infinite series of 9's".

You can say that. It's a cumbersome way to refer to 1, but there's absolutely nothing incorrect about saying that.

Every time I talk about the ratio of circle's circumference to its diameter, I could say that this ratio is

4(1 - 1/3 + 1/5 - 1/7 + 1/9 - etc.)

and I would be completely correct. Everyone else says this is pi.

>>12501893
1 + 0/10 + 0/100 + ...

>>12501942

0.999... is just as precisely referring to a whole object. Look up the definition of a limit.

>>12501943
But 1 is not a decimal involving an infinite series of 9's. There is not an instance where it is useful to refer to it as such, and if there's anything math has taught me it's that mathematicians can argue ridiculously in favor of nonsense like 2 + 2 = 5. It really seems like math people have a hard time understanding that a term like 0.999... exists for the purpose of referring to an infinite sequence and a term like 1 does not. They're not the same, and the 2 + 2 = 5 kind of sophistry that people post to claim that 0.999... = 1 just doesn't seem convincing. You need to convince me that an object which is infinitely close to being another object is in fact the second object and not its own thing.

>>12501946
Yes, it refers to an object which is infinitely approaching equality with another object. That may be a "whole object", but hear me out.
To say that an object infinitely approaching another object in identity is the same as that object which it is approaching is to fundamentally ignore the defining parts of the two objects. One is definitively infinitely approaching the identity of the other. It may be defined in terms of its functionality, namely "is functioning as an object infinitely approaching the identity of another object". The object which is approached is not infinitely approaching identity with the other object; it's the other way around!

>>12501942

You're trying to do math using your own immature intuition about what it means for a number to be whole and what it means for something to be infinitely close to something else. You're flailing around with imprecise concepts because you haven't thought about them enough to make them 100% rock solid (don't take it personally, it took the efforts of many smart people over many centuries to get us to our current level of logical precision).

Stop appealing to your feels. Stop saying shit like "When I see 0.999... what I see is..." Read about what the real numbers actually are in a mathematically rigorous way (as the metric completion of the field of rational numbers). Read about what limits of sequences are (using the epsilon definition).

>>12501950

Mathematical truth is truth regardless of whether it is useful, dumbass. 1 is a decimal involving an infinite sequence of 9's. 1 is also a decimal involving an infinite sequence of 0s. This was just explained to you and you refuse to accept it.

>You need to convince me that an object which is infinitely close to being another object is in fact the second object and not its own thing.

There is no such thing as two real numbers which are unequal yet infinitely close. If such real numbers existed, their difference would be a real number which is infinitely close to 0, but not actually 0. There does not exist such a real number.

I don't need to convince you of shit because it's a basic fucking fact about real numbers.

>>12501967
>You're trying to do math using your own immature intuition about what it means for a number to be whole and what it means for something to be infinitely close to something else.
Am I supposed to use someone else's immature definition of those things? Is a definition better simply because mathematicians use it? Are mathematicians not capable of being fools?
> it took the efforts of many smart people over many centuries to get us to our current level of logical precision
As someone who has studied formal logic, I'm not quite sure logical precision should be given respect. It's arguable that the classical logic which serves as the foundation for much of mathematics is wrong, i.e. bivalence, as writers who cover problems involving heaps often write about. I hold the position that both epistemic and linguistic vagueness exist and that math terms are not immune to them. If it's arguable that even the syllogism "A > B, B > C, :. A > C" is wrong, how am I to take seriously mathematics which use such reasoning as their foundation?
>Stop saying shit like "When I see 0.999... what I see is..."
Why? If I were wrong you would be able to say with simple accuracy that "0.999... does not refer to an object which is infinitely close to holding the identity of another object" using the definitions you're favorable towards. Everything is arguable, my friend, including the laws of logic.

>>12501942
>Let's say you
...physics isn't math, retard

>>12501976
>Mathematical truth is truth regardless of whether it is useful
Depends entirely on your definition of truth. By a pragmatist (or even pragmaticist) definition, there is no difference at all between what is true and what is useful.
1 is not both of those things. It is clearly not. 1 is a symbol which refers to a whole object. In fact, 1.000.... is a different symbol than 1 as well.
It takes a special kind of stupid to say "1" contains an infinite sequence of 9's. Educated stupid.

>>12501956

>To say that an object infinitely approaching another object in identity is the same as that object which it is approaching is to fundamentally ignore the defining parts of the two objects.

That's not what's going on here. You don't even know what

9/10 + 9/100 + 9/1000 + etc.

means do you? You think it is an infinite process. It is not.

0.999... does not refer to a process. It is a number, not a process. The number that it is, is the same as the number 1.

>>12501982

I don't know how you don't get hit by a bus every day.

>>12502001
Logical realists have nothing going for them at all.
>>12501996
What I am saying is that a number which is infinitely close to being another number is not the other number, it is itself. To refer to it as the number which it is not is just to be ignorant of both terms. An object which holds properties which another object doesn't hold is not that object.
If 0.999... and 1 cannot be referred to as objects, they have no value whatsoever.
You can very clearly visually see that 0.999... and 1 are different things. They also have different grammatical rules when used in language. A ton of properties are different between the two terms.
So what the hell are you saying? "0.999..." is "0.999..." and there's nothing more to its definition? The definitions of the terms we use don't matter? Is that what you're saying?

>>12501855

Your incorrect reasoning has already been pointed out.

Now, if you are correct that 1 =/= 0.999... the difference 1 - 0.999... is not 0. What is the decimal expansion of this difference?

>>12502013

Your arguments have been addressed and answered already.

>"is not a whole number"

stop begging the question.

>>12502013

>You can very clearly visually see that 0.999... and 1 are different things.

Troll harder, retard.

>>12502020
The decimal expansion does not exist because mathematics does not contain appropriate terminology to refer to this phenomenon. I'd posit that it is 0.0....n... or something of the sort.
The only purpose such a term would serve is to say "is infinitely close to being void, but is not".
>>12502027
That's a cop out if I ever heard one. Another mathematician incapable of using language to explain their nonsensical ideas, where have I seen that before?

>>12502042

>The decimal expansion does not exist

Every real number has a decimal expansion. This is a fact which you are ignoring because you are an autistic retard.

>I'd posit that it is 0.0....n... or something of the sort.

You would posit some bullshit like that, you fat gay nigger bitch.

Equivalence itself shouldn't be thought of as a strict term.
In terms of strict mathematics, godel's incomplete theorem is a basis for arguing why equivalence can't be argued for in perfect terms.
In terms of mathematics as a representation of reality, heisenberg's uncertainty principle proposes that quantum units can't be discerned in a strict manner in measuring their speed & position.
The invalid status of equivalence is actually more prevalent if you see mathematics as a modeling exercise for reality. For example, while mass is widely considered a constant, have you ever considered how we measure mass? Every highschool science physics course goes into how weight and mass are different (they are), but the fact of the matter is that we measure mass using gravity and forces - which inevitably presents mass as...weight and forces. What we recognize as reality isn't some kind of involatile set of constant, but rather an integrated system of measurements based on what we can see so far, and the deeper we dive into physics the more we notice that those characteristics are never, EVER, the same for two of any objects in the universe - in part because of the heisenberg principle, and in part because our system itself is one that at some point integrated modeling and measurements to work together to extrapolate possible rules (physics) for how things work. Yet at some point, we have to be able to say that we have two apples, even if their position, colouration, sugar content, weight, speed, mass and so on and so forth are all different. That this phenomena extends to the modeling (math) side shouldn't be surprising.

>>12502038
Okay. So since symbols and their definitions don't matter to you, you agree that 2 = 3. Because the definitions of symbols that we use don't matter. Right?
If they do matter, then explain why "1" and "0.999..." have different grammatical functions and different properties and that there are literal ontic differences between the two symbols.
As I noted earlier, you should be able to use a simple sentence to display that the two are interchangeable terms. Like "I ate 2/6 of the pie" and "I ate 1/3 of the pie" mean exactly the same thing: you had a pie, you ate a certain portion, the numbers exist here only to refer to that portion and both terms are entirely interchangeable.
Now you cannot do the same thing with 1 and 0.999... because they do not have the same referent and do not exist for the same purpose. Hell, I'll make it easier for you and use 1.000... and 0.999... since both are decimals, so they'll at least share ONE property in common. It is proper to say "The gas tank is full; I have 1.000 tank of gas." It is not proper to say "The gas tank is full; I have 0.999... tank of gas", even grammatically. It is proper to say "The gas tank is almost full; it is at 99.99...% capacity" or something of the like.

>>12502052
>This is a fact
Says who? Why should I take them seriously when I just gave a clear counterexample?
>You would posit some bullshit like tha
Same as every mathematician who discovered a novel concept ever.

>>12501942
>What I see when I see "0.999..." is "this is a placeholder term for an object which is not whole, but which is infinitely close to being whole."
You're hallucinating

>>12502121
What does it mean to you when you see "0.999... = 1"?
What does that actually mean to you? Is it a meaningful statement at all? Aren't symbols and expressions supposed to... idk, express something?

>>12502186
>What does it mean to you when you see "0.999... = 1"?
>What does that actually mean to you?
[math]\sup \{ \frac{9}{10},\frac{99}{100},\frac{999}{1000},\dots\} = 1[/math]

>>12502239
That's just a way of expressing the same thing. What is being expressed? What concept is being portrayed in what you just wrote? Explain it. In English.

>>12502248
>math is not good enough answer for a math question
Fact of the matter is. This is a non-trivial result and you won't get it without a fundamental understanding of the construction of the rela numbers, which you clearly do not have since you're doing lookey-see math so how about you just accept you're not smart enough and let it go?

>>12502248
The concept of supremum

>>12502263
I don't think you understand what you're saying. I think you're parroting what you saw in a math class and you actually don't understand the concepts behind what you write. You just know that you're supposed to do it a certain way because that's the way it's done by mathematicians.
Too bad they don't teach you folks about critical thinking.

>>12501904
And 0.5 + 0.5 looks different than 0.9 + 0.1 as well. Still the same number, multiple ways to express it

>>12502042
>I'd posit
lol
your Fields Medal is in the mail

>>12502295
>1 cannot be 0.999... because one has decimal point and the other does not
>"critical thinking"

>>12501893
0.99999999999999999999999999999999...

>>12501855
>brainlet
1=.999....
>normal brain
1 is not the same as .999....
>big brain
1 is not the same as .999.... but 1=.999...

An infinitist category theory professor and transsexual was teaching a class on David Hilbert, known nonconstructivist.

"Before the class begins, you must get on your knees and worship Georg Cantor and accept that he was the most highly-rigorous being the world has ever known, even greater than Archimedes!"

At this moment, a brave, intuitionist, wildbergian euclidean geometer who had produced 1500 constructive proofs and understood the necessity of algorithmic thinking stood up and held up a 0.999... foot ball.

"How wide is this ball?"

The arrogant professor smirked quite Infinitistly and smugly replied "The equivalence class of the sequence (1,1,1...)"

"Wrong. Only three nines were written down. It there were infinity and real numbers, as you say, are real... then i would need infinite paper."

The professor was visibly shaken, and dropped his japanese chalk and copy of Rudin. He stormed out of the room crying those infinist crocodile tears. The same tears infinitists cry for the “non-measurable sets” (who today live in such luxury that they need not be constructed) when they jealously try to take up space in textbooks from the deserving theorems. There is no doubt that at this point our professor, Cardinal Trannystein, wished he had pulled himself up by his bootstraps and become more than a sophist infinity schizo. He wished so much that he had a gun to shoot himself from embarrassment, but the bullet would take infinite steps to reach his head!

The students applauded and all studied Wittgenstein that day and accepted Kroenecker as their lord and savior. An eagle named “Induction” flew into the room and perched atop the number theory book and shed a tear on the chalk. Wildberger's videos were watched several times, and God himself showed up after descending a finite amount from heaven.

The professor lost his tenure and was fired the next day. He died of gay plague AIDS and was expelled from the paradise Wildberger had created for all eternity.

>>12502375
>i have a micropenis

Remember that these people laugh about medieval Christians arguing about pinhead angels.

>>12501855
Correct. You are comparing a circle with a spiral. The spiral does not vecome a circle by simple virtue of running out of room in the middle.

>>12502396
we also laugh at you

>>12502368
>1 is not the same as .999.... but 1=.999...

>>12502368
Anon, "doublethink" is not an indicator of brain capacity

>>12502295
>I'm uneducated in this particular topic and that makes me right

>>12502511
Seems like the so-called "educated" in this topic can't explain a thing about what they're talking about. Doesn't seem like that education is worth much. Just makes you seem like a babbling fool who can't use simple English to express basic concepts.
If "0.99..." is such a lofty concept that it can't be simply explained in basic English, I fear you'd never complete a description of what "=" means!

>>12502530
Go read a book. You can get them in any language!

>>12502542
Yes, but when I read math books the writing is so low quality that it makes me wonder how these people got jobs writing books.

>>12502667
>so how about you just accept you're not smart enough and let it go?

>>12501855
>"is not a whole number"
Wrong.

>If two objects have properties which differ, those objects do not share an identity.
Incorrect. Two words can be different but mean the same thing.

>>12501855
I agree. Another argument is this; Lets assume 1=0.99...999.. then you could do the same with 0.999...=0.99...98.. and then go all the way down to -1. Now we get the expression 1=-1, which is clearly incorrect, since a number cannot be equal to himself with the opposite sign.

10 tenths are a whole. 9 tenths are not.
10 hundredths are a tenth. 9 hundredths are not.
10 thousandths are a hundredth. 9 thousandths are not.
This logical sequence stretches on for just as long as those three dots and as long as it remains true .999... will never equal 1.

>>12502740
>This logical sequence stretches on for just as long as those three dots and as long as it remains true .999... will never equal 1.
schizo shit. why don't you actually learn some math?

>>12502667

Too stupid to read a math textbook, so you shift the blame and say the writing is bad.

>>12501855

The second reply to this thread completely answered the question for you. You refused to accept it.

>>12502740

You are too fucked to read a math textbook on limits so I'm going to spell it right out for you. I'll have none of your idiotic philosophizing. Read what I write and explain where you think I'm wrong.

Let n be a positive integer, and let S(n) be the number 0.999...9, where there are n '9's. For example, S(1) = 0.9, S(2) = 0.99 etc. Are you good so far, retard?

Now, here is what people mean when they say 0.999... because you are too fucking lazy to read a goddamn textbook you stupid bitch.

0.999... is defined to be a number with the following property: for any number ε > 0, there exists a positive integer N such that for any positive integer n bigger than N, it holds that

|0.999... - S(n)| < ε.

Notice, retard, that there is absolutely nothing here involving "infinitely many '9's to the right of the decimal place. 0.999... is a mathematical symbol. The notation is suggestive of an infinite process, but formally it is just notation.

Here is another way of saying it. 0.999... is by definition the limit of the sequence S(1), S(2), S(3), etc. But 1 is also the limit of the sequence S(1), S(2), S(3), etc. It is a basic mathematical fact that limits of a sequence, if they exist, are unique. Therefore, 0.999... = 1, you fucking faggot.

>>12502375
historical truth

Holy shit how is this still going?
Unless the limit is material to the problem at hand, then 0.999... is simply a calculation error.
Either it is important to whatever you are working on, or you've fucked up and look retarded to everyone that sees your shit.

Now, the reason this arises: are you aware of the inherent properties of a base 10 number system? Note the property of decimals never working properly with 3. Shift to base 12 and your misconceived error disappears. Note that the error remains for all primes that are not a factor of the base. Stop being a damn child insisting that infinities arising due to your own ignorance are somehow transcendental.

>>12503691
>are you aware of the inherent properties of a base 10 number system?
yeah, for example 1 = 0.999...

If they are different, then there should be a small number epsilon added to 0.999... to produce 1, or a number between them, but there isnt.

>>12501855
A number is not a string of digits, faggot.