1 + 1 + 1 = ?

[DrRocket]

I actually thought, at first, that you are all yanking my chain. This entire thing sounds ridiculous to me, honestly, but I must admit, mathematics has this tendency to have paradoxes that I simply don't get.

 

Useful link to anyone who's as confused as I was/am: http://en.wikipedia.org/wiki/0.999... (thanks swanstont)

 

 

So now that I know this is actually REAL, I have a few questions of my own:

 

First this seems to talk about infinite 9s after the decimal. Does this mean that the original post, with

 

as in, there's a 1 at the end and it's not infinite 9s, is *not* the same as infinite 9s? The wiki has examples with limit goes to infinity, and this one doesn't...

 

Second, why is this different than the mathematical paradoxes out there, where I can get a nonsensical result out of mathematical manipulation? Is this not nonsensical? The definition of .999 is that it's not yet 1, isn't it?? so isn't this manipulation resulting in a nonsensical result?

 

I saw a math paradox where 1=0, simply by stating something like

1 = 1 + 0 + 0 + 0 + 0 + ...

and using rules of math, so I can change order of addition:

1 = 0 + 1 + 0 + 0 + 0 + ...

1 = 0 + 0 + 1 + 0 + 0 + ...

1 = 0 + 0 + 0 + 1 + 0 + ...

1 = 0 + 0 + 0 + 0 + 0 + ...

1=0

 

This isn't REAL. It's nonsensical, it's just abusing the laws of math to reach a nonsensical result, hence being called a PARADOX. the 0.999... thing is also a paradox, isn't it? (btw, it's under "math paradoxes" category in wikipedia, if it helps my point).

 

So.. what's the difference between producing nonsensical results that we laugh about and NOT treat seriously like the 1=0 one and this 0.999... one?

 

 

This makes no sense to me.

 

There is no paradox. .9999999............ = 1

 

 

There is a big difference between a finite decimal expansion and an infinite one.

 

 

1 = 0 + 1 + 0 + 0 + 0 + ... true

1 = 0 + 0 + 1 + 0 + 0 + ... true

1 = 0 + 0 + 0 + 1 + 0 + ... true

1 = 0 + 0 + 0 + 0 + 0 + ... false , you can't just make the "1" disappear

1=0 ridiculous

Edited April 19, 2011 by DrRocket

[Xittenn]

There is no paradox. .9999999............ = 1

 

Definition number two suggests that this is in fact a paradox because most people feel that it is a contradiction but is in fact true by rigorous mathematical proof.

[mississippichem]

[math] \frac {1}{3} = 0. \overline{333} [/math]

 

[math] \left ( \frac {1}{3} \right ) 3 = 1 [/math]

 

[math] 0. \overline{333} \times 3 = 0. \overline{999} = 1 [/math]

 

Not the most rigorous of proofs. Seems sound to me though at first glance.

[Hal.]

 

2.99999999999999999999999999999999999999999999999999999999999999999999991 is not equal to 2.9999999999.......................recurring

 

 

I put 2.99999999999999999999999999999999999999999999999999999999999999999999991 in the original post because I was asking if it would be an alternative correct answer to the question , is 1 + 1 + 1 = 3 ?

 

Thus giving , is 1 + 1 + 1 = 2.99999999999999999999999999999999999999999999999999999999999999999999991 ?

 

 

[mooeypoo]

2.99999999999999999999999999999999999999999999999999999999999999999999991 is not equal to 2.9999999999.......................recurring

 

 

I put 2.99999999999999999999999999999999999999999999999999999999999999999999991 in the original post because I was asking if it would be an alternative correct answer to the question , is 1 + 1 + 1 = 3 ?

 

Thus giving , is 1 + 1 + 1 = 2.99999999999999999999999999999999999999999999999999999999999999999999991 ?

 

 

 

seems to me that if this "paradox" is the way to go, then it only works with infinite decimals. That would mean that 1+1+1=2.999... and not 2.999...1

 

Am I right?

[Klaynos]

seems to me that if this "paradox" is the way to go, then it only works with infinite decimals. That would mean that 1+1+1=2.999... and not 2.999...1

 

Am I right?

 

Indeed, the infinities are important here.

[mooeypoo]

[math] \frac {1}{3} = 0. \overline{333} [/math]

 

[math] \left ( \frac {1}{3} \right ) 3 = 1 [/math]

 

[math] 0. \overline{333} \times 3 = 0. \overline{999} = 1 [/math]

 

Not the most rigorous of proofs. Seems sound to me though at first glance.

 

Actually, oddly enough, that makes more sense than any of the other proofs I've seen.

 

It still seems odd but I guess it works.

[Hal.]

But if a person calculated an answer to a problem as being 2.99999999999999999999999999999999999999999999999999999999999999999999991 the likelihood is that they would round it to 3 and then assume that 1 + 1 + 1 = what they calculated .

 

 

[swansont]

But if a person calculated an answer to a problem as being 2.99999999999999999999999999999999999999999999999999999999999999999999991 the likelihood is that they would round it to 3 and then assume that 1 + 1 + 1 = what they calculated .

 

There are times when it is appropriate and reasonable to round off an answer. In many instances in application of math, using 2.99999999999999999999999999999999999999999999999999999999999999999999991 and not rounding would be a failure to understand the importance of significant digits.

 

Do you have a point?

[Hal.]

An accumulation of errors is the point , swansont .

[Cap'n Refsmmat]

The fact that floating-point errors can occur does not make something like 1 + 1 + 1 = 2.99999 mathematically true. It just means they made errors.

[steevey]

Why does there' have to be infinite decimals? Does cutting something into thirds or 5ths or 7ths not actually exist in nature? There is nothing that is actually exactly 1 third of something else, but it seems like making halves or 4ths is pretty easy in the universe.

Edited April 19, 2011 by steevey

[mooeypoo]

Why does there' have to be infinite decimals? Does cutting something into thirds or 5ths or 7ths not actually exist in nature? There is nothing that is actually exactly 1 third of something else, but it seems like making halves or 4ths is pretty easy in the universe.

 

We're talking about math, not apples.

[Cap'n Refsmmat]

Why does there' have to be infinite decimals? Does cutting something into thirds or 5ths or 7ths not actually exist in nature? There is nothing that is actually exactly 1 third of something else, but it seems like making halves or 4ths is pretty easy in the universe.

It's strange, but required. If I have nine apples and I take away three, I have taken away exactly [imath]\frac{1}{3}[/imath] of the apples, but in decimal that's 33.3333333333...%. Thirds certainly exist in nature, but in decimal form they're infinitely wrong. There's nothing wrong with an infinitely long number having a role in nature (see also [math]\pi[/math] and e).

[steevey]

It's strange, but required. If I have nine apples and I take away three, I have taken away exactly [imath]\frac{1}{3}[/imath] of the apples, but in decimal that's 33.3333333333...%. Thirds certainly exist in nature, but in decimal form they're infinitely wrong. There's nothing wrong with an infinitely long number having a role in nature (see also [math]\pi[/math] and e).

 

 

 

You say I have taken away 1/3 of the apples, but does anything in the universe actually account for that? The only thing thats happening is the distance between some particular atoms has suddenly increased, and thats it, we just happen to label it as "1/3" to recognize it as a distinct pattern, even if nothing else really cares.

 

 

[DrRocket]

Why does there' have to be infinite decimals? Does cutting something into thirds or 5ths or 7ths not actually exist in nature? There is nothing that is actually exactly 1 third of something else, but it seems like making halves or 4ths is pretty easy in the universe.

 

Most real numbers have infinite decimal representations. Without the real numbers you could not do the bulk of calculus, or physics.

 

If you don't want to worry about infinite decinals, become an accoutant.

[Cap'n Refsmmat]

[imath]\frac{1}{3}[/imath] is not intended to describe the physical state of the system and its locations. If I define an arbitrary boundary, exactly [imath]\frac{1}{3}[/imath] of the apples are on one side of it. The universe doesn't need to "account for it"; my number is merely a way of describing the universe.

[keelanz]

1 + 1 + 1 = 11

[Shadow]

Precisely; the problem is not with the Universe, but with our methods of describing it. People realized long ago that .3333...just wasn't going to cut it, and thus fractions were born (probably not true historically, but you get my point). Today, we use fractions to avoid (some) infinite decimal expressions. Think of decimal numbers as being a less perfect "language" than fractions with which one expresses amount.

[keelanz]

Precisely; the problem is not with the Universe, but with our methods of describing it. People realized long ago that .3333...just wasn't going to cut it, and thus fractions were born (probably not true historically, but you get my point). Today, we use fractions to avoid (some) infinite decimal expressions. Think of decimal numbers as being a less perfect "language" than fractions with which one expresses amount.

 

where does the simplest numbering system fit into all this jargon?

[ewmon]

People realized long ago that .3333...just wasn't going to cut it, and thus fractions were born (probably not true historically, but you get my point). Today, we use fractions to avoid (some) infinite decimal expressions.

Yeah, fractions predate decimal places.

[Xerxes]

Think of decimal numbers as being a less perfect "language" than fractions with which one expresses amount.

 

If it's all the same to you I shan't.

 

Or rather, if by "fraction" you mean a rational number, then these barely scratch the surface of all possible real numbers. And to claim that the whole set of real numbers (including the irrationals) are rather useful in describing what another poster somewhat pretentiously called "the universe" would be something of an understatement.

 

Though to claim that they tell the full story, when the complex numbers are available, would just be wrong

[Shadow]

As far as I can see, it's a perfectly accurate metaphor; I'm not sure why you disagree. You can never exactly express a third using decimal numbers; you can with fractions. Also, what's pretentious about calling the universe "the universe"?

I'm well aware of the different number systems. However, since introducing all that extra information wouldn't do anything to help the attempted simplicity of the explanation, I refrained from giving it. Also, since it might not have been obvious from my previous post, I was responding to the following:

 

You say I have taken away 1/3 of the apples, but does anything in the universe actually account for that? The only thing thats happening is the distance between some particular atoms has suddenly increased, and thats it, we just happen to label it as "1/3" to recognize it as a distinct pattern, even if nothing else really cares.

 

However, I might have misinterpreted the meaning; the way I understand it, and for me the post is not clear enough to be completely sure, the author was saying that numbers with infinite decimal places are "unreal". I was attempting to point out that there is a way to represent them finitely and thus helping the author accept their "realness", evidently not doing a great job of conveying that either.

 

A piece of trivia, which you might already know, not even the complex numbers occupy the highest level; see quaternions, octonions and sedenions.

 

 

keelanz: What numbering system would that be? The naturals?

[Xerxes]

You can never exactly express a third using decimal numbers; you can with fractions.

Which gets to the heart of this totally pointless thread, One third is exactly 0.333...., just the same way that 0.999... is exactly 1.

 

There are no approximations, no equivocations, these facts have been rehearsed over and over, here and elsewhere.

 

the author was saying that numbers with infinite decimal places are "unreal".

Agreed, if that is what was said it's nonsense - I have no idea what an "unreal number" is

I was attempting to point out that there is a way to represent them finitely

But what does this mean? Is one third aka 0.333... not finite? I am pretty sure that both these numbers, whether or not you (or anyone else, for that matter) agrees they are equal, they are still both greater than zero and less than 1 i.e. finite numbers.

[Shadow]

You misread; I'm not referring to the finiteness of the numbers themselves, but of their representations; and while [math]\frac{1}{3}[/math] is a finite way of expressing one third, [math]0.333\bar{3}[/math] most certainly is not.