1=.999999...?

[Leifmk]

However, if one actually adds or multiplies 0.999... to or by 0.999... you get something like 1.999...8 or 0.999...8.

No you don't, there's no last digit.

 

[TheMeInTeam]

Why not?

The definition of infinity requires that it goes on forever. The notation used is to denote infinity.

If you don't refute either of the above points, having *anything* follow a string of infinitely many digits is absurd. Unless you change the definition of infinity, you're basically claiming infinity and then instantly turning around and claiming not infinity.

Something like .999...8 is an overt self-contradiction. If there's an 8 on the end, it's not an infinite string of numbers, in that case it really is just a number very close to 1 (but not one) and using notation that implies infinite digits is wrong.

In other words, I'm having a hard time grasping why everyone seems to have a hankering to first define an infinite string of digits then instantaneously self-contradict themselves. The only way this habit makes sense to me is if people aren't conceptualizing infinity properly.

It's kind of like defining a color as blue, accepting it as blue, then asking why it's green while still accepting it as blue...nonsense .

 

[Atticus]

I had a nightmare that this thread was on the 50th page and it would took me only one post to reach 1000 and get this closed. After writing the typical "lol ur wrong" I hit the submit reply button, but managed to post only 999.9th post in the thread.

Ps. Not really

Terax: I came to think of this example that could be illuminating: If you don't accept that 0.999... = 0.9 + 0.09 + 0.009 + ... equals 1, you probably won't accept similar for any other series either. For example, while 1/2 + 1/4 + 1/8 +... equals 1 in maths, you'd claim that it in reality doesn't ever quite reach the 1.

Now, you wanted a real life example, and this is the Zeno's arrow paradox: To reach one you have to reach first 1/2, then 3/4 and after that 7/8 and so on. Zeno's paradox is paradox because the real life events show us that arrows do hit their targets. Since that happens, the reasoning must be wrong.

If you think it a little further, you'll see that the question about 0.999... is the same thing with just different distances: To reach 1 you have to reach first 9/10, and then 99/100, after that 999/1000.

What I'm actually more interested in: what is the point of 0.999...? Since it has been proved to be 1. And, since equality goes both ways: what is the point of 1=0.999...? In what calculations would this be useful? (I tried adding/multiplying 0.999... with 0.999... by calculator, but the calculator can't handle it.)

It's not useful in anyway at all. It's completely useless by-product that comes if you want to express numbers like 1/3 as decimals. While it's useless, it doesn't have really any downsides either. Except that arguing about it wastes people's time.

(What you said in your post, if you think of adding 0.999... + 0.999... decimal by decimal, it obviously becomes 1.999... = 2).

 

[TheMeInTeam]

I'm sure. I noticed another problem with the 'proof' that 0.999...=1.

1+1=2

It follows that 0.999...+0.999...=2 (since 0.999...=1)

1x1=1.

It follows that 0.999...x0.999...=1 (since 0.999...=1).

However, if one actually adds or multiplies 0.999... to or by 0.999... you get something like 1.999...8 or 0.999...8.

Now you will say, an infinite number doesn't have a last number. Which would obviously be correct.

However, no matter how far you add 9s, the addition (or multiplication) simply won't add up to 2 (or 1).

I haven't tried, but I would assume that it is equally possible to prove 0.999... does not equal 1.

What I'm actually more interested in: what is the point of 0.999...? Since it has been proved to be 1. And, since equality goes both ways: what is the point of 1=0.999...? In what calculations would this be useful? (I tried adding/multiplying 0.999... with 0.999... by calculator, but the calculator can't handle it.)

Make up your mind. Are you dealing with an infinite string of 9's or not? Saying "no matter how far you add 9's" is showing a misconception of infinity. There is no "how far". It's forever. Infinity has no distance measure, don't try to fabricate a finite distance and still claim you're working with infinity! It's a self-contradiction.

The evidence we have observed suggests that .999... = 1, just as the evidence we have observed is that 2 + 2 = 4. You claim it's possible to prove .999... =/= 1, which is similar to attempting to prove that 2 + 2 =/= 4. In theory, it could happen, if you started putting two sets of two things together and didn't wind up with 4 things after doing so.

The annoying issue in this thread is the repeated usage of infinity followed by the immediate application of something other than infinity to refute infinity. This doesn't seem to be an argument over definitions, but rather a difficulty in conceptualization.

The 9's don't stop, ever. That means that even if you double or quadruple them up, they'll still never stop.

 

[Akka]

The definition of infinity requires that it goes on forever. The notation used is to denote infinity.

If you don't refute either of the above points, having *anything* follow a string of infinitely many digits is absurd. Unless you change the definition of infinity, you're basically claiming infinity and then instantly turning around and claiming not infinity.
[...]

Make up your mind. Are you dealing with an infinite string of 9's or not? Saying "no matter how far you add 9's" is showing a misconception of infinity. There is no "how far". It's forever. Infinity has no distance measure, don't try to fabricate a finite distance and still claim you're working with infinity! It's a self-contradiction.
[...]

I've been trying to explain this self-obvious part and arguing that the whole problem for people refuting the thread subject is that they don't get what "infinity" means for several pages I think. Good luck

 

[El_Machinae]

You can't actually add 0.999... to itself.

Addition requires starting on the right side of the numbers. So, to add 0.999..., you need to convert it to its more useful format, 9/9.

 

[azzaman333]

I've been trying to explain this self-obvious part and arguing that the whole problem for people refuting the thread subject is that they don't get what "infinity" means for several pages I think. Good luck

To be fair, it's not easy for most to conceptualise what infinity actually means, but you'd think they'd start to get it after being beaten over the head with it so many times.

 

[Bootstoots]

I love how the "1 = 0.999999..." subject is a controversial one on the internet. Not only does it not involve deep-seated political, moral, or religious convictions like most things that spark controversy, it actually has an objective truth value. It's just a true statement. But infinity is hard and I'm not much better than average at wrapping my head around it either, so it still makes for an interesting discussion.

 

[hoplitejoe]

I wonder if a Monty Hall problem thread would cause as much issues...

 

[TheMeInTeam]

I've been trying to explain this self-obvious part and arguing that the whole problem for people refuting the thread subject is that they don't get what "infinity" means for several pages I think. Good luck

I'm just trying a different angle.

But infinity is hard and I'm not much better than average at wrapping my head around it either, so it still makes for an interesting discussion.

My guess is that it's a difficulty in accepting an abstract concept. You can't see infinity or know what it looks like. In our observational experiences even things that give the impression of being infinite are "only" ridiculously huge finite numbers. Even the length of the entire universe is finite...but we're still talking about putting 9's on the end of something that keeps on going relative to that size just like it would keep on going past 10 digits.

I can't picture the length of the universe or more. I can't even reasonably picture what 10000000 oranges stacked in different shapes would be like. I doubt anybody else can either.

So for a lot of people it appears like they refuse to actually accept infinity. They hear and agree to the definition, but when they go to apply it in practice they instead conceptualize a very large finite number rather than something that is impossible for them to experience.

I wonder if a Monty Hall problem thread would cause as much issues...

I don't doubt you'd get arguments over it, but it's conceptually easier to grasp and rather unlike infinity, you can experience the reason the solution is correct through trials if so inclined.

 

[trader/warrior]

I think it's understandable that to people with only a basic understanding of math it can seem like a fundamental breakdown of the logic of numbers. That -0.999...+1 would equal one infinitesimal "infinityth" digit would to most people seem to must hold true just as much as -99+100 would equal 1. It doesn't quite make sense to me logically that you can have infinite numbers, but you cant increment the numbers by infinitesimal quantities (and that's where I lose grip of the concept of infinity, I suppose).

 

[Akka]

I think it's understandable that to people with only a basic understanding of math it can seem like a fundamental breakdown of the logic of numbers.

Of course it's understandable. The fact this thread popped everywhere a few years ago, the fact that the thread even exists at all, is based on this.

The problem comes when their ignorance becomes arrogance and they feel they know better than the entire history of mathematics. It's of course pure Dunning–Kruger effect at work, but it's still pretty stunning to see people refusing to accept a proved true answer, despite the wealth of information and evidences available at a click of a mouse, and the fact it's a known and extremely documented mathematical oddity.
It's just, basically, conspiracy theory, but fueled by misplaced hubris ("I know better than the world !!!") instead of paranoia.

 

[brennan]

Keep up the circle jerk if it makes you feel good.

 

[hoplitejoe]

Sorry, clearly we need to start talking about how with no formal training forum goer brennan manages to to prove a fundamental concept wrong with his repeated refusal to accept the use of standard mathematics!

 

[brennan]

Actually, since you've obviously not been following, we've seen from people who know enough to give useful responses that there is at least one academically rigorous formulation of mathematics in which 0.999... =/= 1.

I like the way you know my academic history simply by the fact i'm on this forum btw. Highly amusing.

 

[TheMeInTeam]

Actually, since you've obviously not been following, we've seen from people who know enough to give useful responses that there is at least one academically rigorous formulation of mathematics in which 0.999... =/= 1.

Citation needed . I'm not sure which post you're referring to. I'm not interested in an appeal to authority but maybe someone really did that in this thread. It'd at least be worth a look if identified.

 

[brennan]

We've brought up finitism and non standard analysis. It seems that 0.999...=1 arises from the axioms chosen in the definition of real numbers. If you change the axioms you can do perfectly valid maths without the apparent paradox this thread is about.

 

[onejayhawk]

I've been trying to explain this self-obvious part and arguing that the whole problem for people refuting the thread subject is that they don't get what "infinity" means for several pages I think. Good luck

The problem is that the concept of infinite you present is countable. There are deeper depths. If you think 'infinite' is difficult, try 'continuous'.

J

 

[onejayhawk]

Sorry, clearly we need to start talking about how with no formal training forum goer brennan manages to to prove a fundamental concept wrong with his repeated refusal to accept the use of standard mathematics!

You need to try harder. If it happened, Brenner would not be the first.

J

 

[Akka]

Keep up the circle jerk if it makes you feel good.

You mean the people who think they know better than the whole world because they are too limited to understand infinity and pat each other on the back about how they see the flaw that Nobel prizes couldn't ?
Because THAT looks much more of a circle-jerk.