Thread: 0.9999... = 1

It was not a straw man, it was the point I was trying to make from my very first post. I made one simple comment saying that math was imperfect. Xei and Cmind then blatantly attacked me:

Originally Posted by Xei

you fail

Originally Posted by Cmind

makes everyone think you're a moron

So I defended myself. However instead of refuting my argument, Cmind simply tried to insult my intelligence(very childish). And Xei continued to argue that .999 does in fact = 1, which I never disputed. In math .999 does in fact equal 1. However, 1/3=.333*3=.999 has never set well with me. I just feel that it is highly possible there is a much more intuitive way to display integers and do arithmetic that doesn't such require an abstract explanation.
Had Cmind not been a troll, and had Xei not mounted his high horse trying to display his math prowess, my first comment would have been my last.

Nah, what's actually happened here is that I've repeatedly tried to explain that your queries were addressed in the thread already and I also briefly summarised for you, and you've repeatedly ignored me.

When did I get on my high horse and display my leet maths skills? Trust me, if I did that, you'd have died due to excessive awe. Every response of mine to you has been about a sentence long and I didn't write a single equation. Derp.

Originally Posted by LikesToTrip

It was not a straw man, it was the point I was trying to make from my very first post. I made one simple comment saying that math was imperfect. Xei and Cmind then blatantly attacked me:

Your first comment was (paraphrasing) "I've struggled with this since I was little, and it's proof maths isn't perfect". That right there is enough to get alarm bells ringing for me, because this is not a particularly hard concept to grasp, and I certainly do not consider myself an amazing mathematician (though given that I studied Chemistry, I suppose I've got to be reasonably competent at it).

Xei then correctly pointed out that this has been discussed earlier in the thread, and how it actually makes perfect sense. And he's right, it does, even if it seems somewhat counterintuitive at first.

You then claimed "I'm too lazy to read it", yet you decided to post any way. If you're too lazy to bother to read a set of posts that directly deals with what you wrote, then don't bother posting in the first place.

And you're surprised that people have made judgements about you when you've ignorantly used it as "proof" that maths isn't perfect, that you can't grasp a fairly simple concept, and you're too lazy to bother reading an explanation of why you are wrong which has already been provided in the thread?

I can't believe there's an actual argument going on here. Those proofs are faulty, as at some point you either reach an infinite number of decimals, or must have a ...3334 ending to allow it to become 1.

If that were the case, then .999...8 = .999... = 1 = 1.000...1, etc.

In essence, you are rounding just before infinity in the logic of Proof 1. The reason .999... does not equal 1 is because they are not the same number...? THat's the best explanation I have of it. They are not equal, therefore, they are not equal.

.999...8

If you have an infinite amount of nines, then by definition, you cannot have an eight 'past' them; meaning .9..8 is the same as .9...

Why do so many people fail at math when it comes to this?

Originally Posted by ThePreserver

I can't believe there's an actual argument going on here. Those proofs are faulty, as at some point you either reach an infinite number of decimals, or must have a ...3334 ending to allow it to become 1.

If that were the case, then .999...8 = .999... = 1 = 1.000...1, etc.

In essence, you are rounding just before infinity in the logic of Proof 1. The reason .999... does not equal 1 is because they are not the same number...? THat's the best explanation I have of it. They are not equal, therefore, they are not equal.

You fail a large amount.

Please prove that 0.999...9998 = 1, then? I will give you £10,000 worth of sweeties.

Seriously, you're saying that 'they are not equal' is a proof of their inequality? You actually said that without considering yourself to be ridiculous? Honestly, are you being serious?

While it may seem ridiculous... the way I see it is 1 is separate from .999... by the fact that in Euclidean mathematics, they are exactly one infinitesimally small step down or up, similar to dividing a segment infinite times. You can always divide it once more.

Now for the .999...8, that would be (under the presumption that the ellipsis is an infinite count of 9) one infinitesimally small step down from .999... If this makes any sense.

And what is wrong with the algebra and calculus of the previous proofs?

Formally your argument doesn't make any sense because of the Axiom of Archimedes, which is a consequence of the Fundamental Axiom of the Real Numbers.

F.A.R. states that if a collection of real numbers has an upper bound, then that set has a least upper bound.

If you agree with this (and its truth is extremely obvious) then you must also agree with the Archimedean Axiom which can be proven from it, which states that there are no infinitesimally small numbers (and hence there can be no infinitesimally small step, as you put it, between the two numbers).

This is all covered on the first page, check it out.

Your geometrical argument doesn't make any sense because to place a line between the two points anyway, you have to assume that they are not equal (which you can't assume as you're trying to prove it).

Originally Posted by ThePreserver

While it may seem ridiculous... the way I see it is 1 is separate from .999... by the fact that in Euclidean mathematics, they are exactly one infinitesimally small step down or up, similar to dividing a segment infinite times. You can always divide it once more.

Now for the .999...8, that would be (under the presumption that the ellipsis is an infinite count of 9) one infinitesimally small step down from .999... If this makes any sense.

"0.999...8" doesn't exist. There's no such concept, not even in abstract math. The mere fact that you would even bring up a thing like "oh, the ellipsis is an infinite number of 9s" (which implies that the 8 is the "infinity + 1"-th term), is absolutely ridiculous.

Here is another proof. One that anyone who's half-way done high school should be able to grasp:

Geometric series - Wikipedia, the free encyclopedia

This page has a well-accepted proof that the following holds true:



Now, in that formula, let r = "1/10" and let a = "0.9"

that gives you:

0.9 + 0.9/10 + 0.9/100 + 0.9/1000 + ... = 0.999... = 0.9 / (1 - 1/10) = 0.9/0.9 = 1

Originally Posted by ThePreserver

In essence, you are rounding just before infinity in the logic of Proof 1.

I'm surprised that with all the responses, there's still an error to point out here but there it is. Please define "just before infinity". There might still be other errors...


EDIT:

Spoiler for proof from the ground up:

Some people seem to be having trouble with infinity. So I'm going to demonstrate how to deal with it directly. We do this by not dealing with it.

Proposition:
0.999... = 1

Proof:

By definition, we have 0.999... = 9/10 + 9/100 + 9/1000 + ...
This is called a series.

Consider the numbers
a₁ = 9/10 = .9
a₂ = 9/10 + 9/100 = .99
a₃ = 9/10 + 9/100 + 9/1000 = .999
a_n = 9/10 + 9/100 + ... + 9/10ⁿ = .99..9 <-- n 9s

These are called the partial sums of the series. The idea is to show that the partial sums "converge" to 1. Specifically, this means that for any number ε with ε > 0, there exists some N such that

|1 - a_n| < ε

whenever n > N. Put another way, this means that for any ε, however small, every partial sum past some point is less than ε away from 1. Intuitively, we're showing that we can make the value of the series as close to 1 as we like by taking more terms. So, suppose we are given some ε.

Consider the number

1 - a_n = 1 - 9/10 + 9/100 + ... + 9/10ⁿ (by definition of a_n)
= 1 - (9*10^{n - 1} + 9*10^n-2 + ... + 9)/10^b (addition of rational numbers)
= 10ⁿ/10ⁿ - 99...9/10ⁿ
= 1/10ⁿ

We see that |1 - a_n| = 1 - a_n so we can dispense with the absolute value and just prove that there exists an N so that n > N implies that 1 - a_n < ε.

Here's how we can find that N.

1 - a_N = < ε
1/10^N < ε (substitution justified by previous calculation)
1 < ε*10^N (multiply by 10ⁿ)
1/ε < 10*^N (divide across by ε)

There plainly exists an N with 10^N > 1/ε by the archimedean property (otherwise, 1/ε would be infinite with respect to 10 which we know doesn't occur). If anybody has any problems with this, ask and I'll tighten it up.

We can take that N and see that 1 - a_n < ε when n > N. Again, if anybody has any problem with this, ask and I'll tighten it up.

QED

So there you have it. .999... = 1. It's an easy theorem (normally given as an exercise) that if a sequence converges to A and also converges to B then A = B. So .999... doesn't equal anything but 1.

You stand in error. Convention of names. Irrational means the inability to provide a name using the given naming convention. The approximation of a name, is still not a name. When you write 1/3 = .3333, you are actually using the equal sign in error.

When you learn that there are two distinct primitive branches of logic, you may come to realize that no matter what you do, you cannot claim that the one is the other. These two branches are derived from the two basic elements. Two-Element Metaphysics. Material and Form. (Enumeration and Definition). etc.

If you can see the same idea in all I said, you can put it another way. 1/3 is absolute, .3333 is relative. You simply said that the absolute is relative, which is pure non-sense.

To demonstrate the difference between the two primitive branches of logic, I provide a figure on the internet archive which can multiply and divide any two magnitudes using any unit while always giving an exact answer--there are no irrational operations in it. However, using arithmetic, which is not a relatiologic, but a tautologic, you have irrational operations--due to the type of logic you chose to work the problem.

Please point out the error in my proof.

Originally Posted by Philosopher8659

You stand in error. Convention of names. Irrational means the inability to provide a name using the given naming convention. The approximation of a name, is still not a name. When you write 1/3 = .3333, you are actually using the equal sign in error.

When you learn that there are two distinct primitive branches of logic, you may come to realize that no matter what you do, you cannot claim that the one is the other. These two branches are derived from the two basic elements. Two-Element Metaphysics. Material and Form. (Enumeration and Definition). etc.

If you can see the same idea in all I said, you can put it another way. 1/3 is absolute, .3333 is relative. You simply said that the absolute is relative, which is pure non-sense.

To demonstrate the difference between the two primitive branches of logic, I provide a figure on the internet archive which can multiply and divide any two magnitudes using any unit while always giving an exact answer--there are no irrational operations in it. However, using arithmetic, which is not a relatiologic, but a tautologic, you have irrational operations--due to the type of logic you chose to work the problem.

What's 1/3 in base 9?

How do people not get that infinity means never ending?

This thread is funny thought because I just figured this out the other day. My dad told me this years ago, but I didn't pay a lot of attention to why it is true and just said it must be bullshit because it doesn't make sense. But for some reason or another, the other day it just came to me that it must be true. Because of the first example 1/3 = .333 and .333 + .333 + .333 = .999....

I feel that there must be some application for this. Although that maths is probably infinitely beyond me
Do you know (philosopherstoned or anybody else) if this has been put to use for anything?

It has zero applications really. If you tried to manufacture an object that was 0.999... metres across, even if you didn't know it equaled 1 metre, for all intents and purposes you would end up manufacturing an object 1 metre across, because you'd round up or whatever.

The more general knowledge that infinite series can converge to a finite number is extremely important however, and most of physics probably wouldn't exist without it. We wouldn't have a notion of an integral for instance (an object in calculus), the development of which was basically synonymous with the birth of science.

You know, the "1/3 = 0.333..., therefore 0.999... = 3*0.333... = 3/3 = 1" proof says more about people than mathematics.

I don't think that's a "proof" at all, because accepting "1/3 = 0.333..." and accepting "1 = 0.999..." takes exactly the same amount of "faith"... it's just that because "1" is such a nice, round number, there is a different psychology/feeling around seeing "1 = 0.999..." than when you see "1/3 = 0.333..."

I guess what I'm trying to say is, the "3 times 0.333..." proof is NOT a proof.

It is a proof if you accept that 1/3 = 0.333

Also, Xei just says that we wouldn't have integrals without power series cause he's a Brit and Newton originally conceived of integrals as power series We could still integrate a large class of functions. Leibniz did it in closed form like we do now.

But yeah, convergence of infinite series is probably one of the most important things in mathematics and this is just a special case of that. My last post with a proof shows how it's actually handled. If you had questions than I'm sure they'd get answered.

So.... does this mean that infinite doesn't exist?
Or is this the reason why there is two types of infinity?

One that is infinite but converges to a finite amount somehow and
one that just keeps going forever?

There's nothing here about infinite converging to a finite value. What's converging to a finite value is an infinite amount of finite numbers. For this to work, the numbers have to get smaller "faster" than the sum gets bigger.

So if there's an infinite set of numbers {a1, a2, ..., an, ...}, we can "add" them by looking at the sequence of partial sums

s1 = a1
s2 = a1 + a2
...
sn = a1 + a2 + ... + an

If this sequence converges then we say that that the a's have a sum.

But we're only ever looking at finite numbers.

What I mean when I say "converge" is that they get closer and closer to some definite, finite value. I've never had to do it but I'm pretty sure that this could be made to work for any "type of infinity". There's actually more than two types. There's at least as many "sizes" of infinity as there are integers.

Originally Posted by tommo

So.... does this mean that infinite doesn't exist?
Or is this the reason why there is two types of infinity?

One that is infinite but converges to a finite amount somehow and
one that just keeps going forever?

"Infinity" is an abstract concept.

Come to think of it, any regular number is ALSO an abstract concept. I'm pretty sure you've never seen "the number 5" anywhere. Sure, you've seen "5 of something" but that's not the same as the abstract concept that is the number 5. Like, you've never been walking around, and you trip on it, and you're like "damn, fucking number 5!"

Yes but infinity is still real. I know what you're saying completely, but the universe/nature/whatever this is will go on forever, even though forever is just a concept. The concept is just explaining a real occurrence.

I'm not sure I get your reply philosopherstoned, I'm really very average at maths, it's about the only thing I can't grasp very well. So I'll just take your word for it lol.

Originally Posted by PhilosopherStoned

It is a proof if you accept that 1/3 = 0.333

Also, Xei just says that we wouldn't have integrals without power series cause he's a Brit and Newton originally conceived of integrals as power series We could still integrate a large class of functions. Leibniz did it in closed form like we do now.

But yeah, convergence of infinite series is probably one of the most important things in mathematics and this is just a special case of that. My last post with a proof shows how it's actually handled. If you had questions than I'm sure they'd get answered.

Definite integrals are defined in terms of areas under curves, which are in turn defined as limits of Riemann sums as the number of rectangles go to infinity... the fact that this is equivalent to antidifferentiation, which I suppose is what you mean by Liebniz, is something that requires proof. Definite integrals are fundamentally an infinite sum, hence the elongated S that represents them.

Right but none of that happened until the 1800s or so. It is intuitively very clear. I'm not saying it's right. It was the confusion that resulting from doing things that way that led us to rigour outside of geometry. But yeah, you're pretty much right.

You're right about the definite integral being an infinite sum. It had actually occurred to me right after I posted my last message in this thread that the're just a sum over {a_i} with i in R. So the next problem becomes summing a set that's indexed by the power set of the reals. Not that I can think of a practical reason or anything.

Originally Posted by Replicon

You know, the "1/3 = 0.333..., therefore 0.999... = 3*0.333... = 3/3 = 1" proof says more about people than mathematics.

I don't think that's a "proof" at all, because accepting "1/3 = 0.333..." and accepting "1 = 0.999..." takes exactly the same amount of "faith"... it's just that because "1" is such a nice, round number, there is a different psychology/feeling around seeing "1 = 0.999..." than when you see "1/3 = 0.333..."

I guess what I'm trying to say is, the "3 times 0.333..." proof is NOT a proof.

What's 1/3 in base 9?