Poll: Does 0.999.. equal 1 ?

[Boris Goodenough]

It's more like Lim(f(x))=1 for x->k, and we're trying to decide if that means f(k)=1.

Doesn't make a difference though, because there's no real infinitesimal number in our system, and so it must be true.

Although mathematically sound, I think the point I am trying to make is that if it keeps going towards 1 the limit will be 1. Which is a bit easier on the untrained mind than your setup.

This discussion always reminds me of
http://www.thinkgeek.com/tshirts-apparel/unisex/generic/60f5/?i=front

 

[JackEmpty]

This should not be a poll. This is not an opinion question. The answer is fact. It IS equal, there is no doubt. There is no disproof. Anywhere.

I'll refer you here for a whole barrage of wonderful proofs, and quote a bit of it: http://qntm.org/pointnine

Common counter-arguments and my responses
"0.9999... and 1 are obviously different numbers."
In mathematics, "obvious" means "a proof immediately springs to mind". If you don't have a proof in mind, then unfortunately no mathematical statement you make carries any weight.

"1 and 0.9999... are written differently, therefore they are different numbers."
There are many ways of writing any number. You could write 1/1, or 3-2, or 1.0, or 1.00, or 1.0000... or any number of other expressions, and all of them ultimately have the same meaning, "one".

"0.9999... is a concept, not a number."
All numbers are concepts.

"0.9999... can't exist in reality, but 1 can, therefore they are different."
Firstly, just because a number can't exist in reality doesn't mean it can't exist in mathematics.

Secondly, because 1 can exist in reality and 1 = 0.9999..., that means that 0.9999... can also exist in reality.

"There is a rounding error. 0.9999... and 1 are approximately equal."
Rounding errors only occur when we truncate a decimal expansion after a finite number of digits. All of the proofs above use the "..." notation at every step, which means that we always take into account all of the infinitely many decimal digits. There is no rounding, which means there is no error.

"0.9999... gets closer and closer to 1, but never reaches it."
0.9999... is a single number. It doesn't move, so it can't get closer and closer to anything. It is where it is.

"0.9999... is a decimal representation of infinity, not a number."
0.9999... is definitely less than 2, so it can't be infinitely large.

"Humans can't comprehend infinity, and not being able to comprehend infinity means you can't do mathematics with it."
Firstly, humans can comprehend infinity. Mathematicians do it all the time. It may impossible to literally "conceive of" infinite values, whatever that means, but that doesn't stop mathematicians from dealing with them without going nuts.

Secondly, infinity obeys rules. If something obeys rules in a consistent fashion, then you can do mathematics with it. Ordinal arithmetic is a good example.

In case the connection isn't clear, what is true of infinite values is equally true of infinite decimal expansions. There are rules and procedures and they work and give meaningful results. See "The Real Proof" above for a relatively tame glimpse of this, which is actually a vast region of mathematics known as "analysis", naturally based on rock-solid fundamental axioms.

"My mate/my dad/my mathematics teacher/Professor Stephen Hawking told me that 0.9999... and 1 were different numbers."
They were wrong.

"But they proved it, too!"
The proof was fallacious. Send it to me and I'll show you why.

"I still don't believe it and I'm entitled to my own opinion."
In regular science, we have theories. A theory is proposed in order to explain observations, and can be overturned in light of new, inexplicable observations. Multiple theories and opinions may compete with one another. There are fashions. There is room for debate.

In mathematics, we have theorems instead of theories. A theorem is the result of a mathematical proof. A theorem is a fact. A theorem cannot be overturned and is not a matter of opinion. Once proven, a theorem stands for eternity. Mathematics is not ideological.

Thanks to the many proofs above, "point nine recurring equals one" is just such a theorem. So, your opinion is wrong. And sorry, but no: you're not entitled to be wrong in mathematics. That's not how it works.

 

[Torrasque]

Worst. Mathematical. Problem. Ever.

The Agnosticism of Maths.

It's equivalent but not equal.

Also, this.
Tried to post earlier, but it seems Escapist accidentally forgot to pay the bill? /shrug

Edit:
Also, this is the reason significant digits was invented.
To preserve sanity.

 

[TiefBlau]

No, they are not the same.
Yes, math is flawed.
You can see this especially in fractions.

disclaimer: fractions are a poor representation of non-fractions
1/2 = 0.5
1/4 = 0.25
1/3 = 0.333...
2/3 = 0/666...

The simple fact that it repeats forever and ever, means one thing: that it repeats forever.

Here's a fun fact: If you add up 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..... it would equal 1.

Infinity has a funny way of making numbers approach stuff sometimes. And sometimes the difference between the numbers and the stuff it approaches gets really, really small. So small, so infinitely small, in fact, that mathematicians give this gap a name: zero.

Just like Pi does not exactly equal 3.14, it equals 3.14159265358979323846264338327950288419716939937510582097494459...

As it turns out, Pi is irrational, so it can't really be represented by either fractions or non-fractions.

and then mathematicians throw their hands up in the air,

Saying aaaaayoooh, gotta leeeet go.

and say it equals 3.14 for the sake of sanity.

Aw.

Still wrong, though.

To put this into perspective, lets say you live on Earth, and I live 3 light years away.
That is an extremely long ways away, and only an idiot would say that I am touching you.
One day, I decide to teleport back to Earth to visit you, and teleport within 1 centimeter of you.
Compared to the ridiculous distance that separated us before, I am practically touching you. But I am not.
I move to within 1 nanometre of you, just because I am creepy that way, but do not touch you.

Think of 0.999... as that. The difference between 0.999... and 1 is so insignificantly small that depending on the case, you'd just ignore it. But there is still a difference between touching you, and not touching you, whether that is 3 light years, or 1 nanometre.

And what if the distance between us were infinitely small? That's not small; that's not even existent. It's zero.

 

[JackEmpty]

I paraphrase my post above:

This is not an open question.

.(9) = 1 is a theorem.

If you say it is incorrect you are wrong.

 

[Thaliur]

Apparently, at this point the poll was answered by 118 people with a mathematical mind and 101 with an engineering-compatible (AKA pragmatic or realistic) mind.

 

[maninahat]

I originally picked no, despite seeing all the various proofs to the contrary (which are infact correct). I, like a lot of people, was incapable of understanding how 0.999... = 1

Human perception makes it hard to accept the idea. We think to ourselves, "but the 9s go on forever! They will always get closer to 1, but never reach it." Unfortunately, the nature of infinity and infinity small numbers works in a highly counter-intuitive fashion: it is entirely because the 9s go on forever that you can't say it is any less than 1.

The straight dope article gives a practicle example of this problem, known as Zeno's paradox: If a man is racing against a tortoise, and the tortoise has a ten meter head start, you would expect the man to reach the tortoise very quickly. But by the time the man has run the ten metres, the tortoise has moved foward 1 meter. By the time the man has run another meter, the tortoise will have gone on another .10 meter, and so on and so forth. If this goes on for infinity, how can the man ever overtake the tortoise?

In the real world, we know that the man can obviously overtake the tortoise in barely any time, so what does this say about the nature of infinity? Just because it goes on forever, doesn't make it an unachievable task.

 

[Zantos]

Most analytical proofs rely on the fact that you can make small perturbations to an object without it altering it (these are usually denoted epsilon or delta). So by the same logic any infinitessimal difference between 0.999... and 1 is mathematically insignificant. Although if you study analysis you should already know decimals are a really bad way of trying to express irrational numbers anyway.

 

[Lazy Kitty]

No, it's very close to 1, but not 1 itself.

 

[graverobber2]

x=.9999...
10x=9.9999...
10x-x=9.9999...-.9999...
9x=9
x=1
.9999...=x=1
.9999...=1

this
It?s the complete mathematical proof that 0,999999 = 1

 

[Dave Davey]

0.999... = 1.000...

This is universally agreed upon by mathematicians and scientists - basically, people who are experts in the nitty gritty of mathematics.

As everyone else has said, Wikipedia is your go-to place for proofs of this fact.

Common arguments against it are things like 'Oh, but there's an infinitesimally small amount between them' - NO THERE'S NOT, that's the whole point.

Whether you like it or not, mathematics allows us to prove things that our little fleshy brains might not like. Did you know you can draw a shape with finite area and infinite perimeter?

Infinity is not for the faint of heart. 0.999... is as equal to 1 as 4/8 is equal to a half. They are numerically identical, two difference ways of writing the same scalar value.

 

[Volkov]

Apparently, at this point the poll was answered by 118 people with a mathematical mind and 101 with an engineering-compatible (AKA pragmatic or realistic) mind.

Incorrect. Vast majority of engineers know math past 5th grade, and so will correctly answer that the two numbers are equal. Again, this is not an opinion, an open discussion, or a debate.

"Pragmatic/realistic" mind will also realize, that a *number* and a *usable in practice* number are two different things, and in this question *usable in practice* was never raised. Therefore, *numbers* are being considered.

 

[Volkov]

Also, there is no such thing as "equivalent" in mathematical terms. The two numbers are EQUAL, not equivalent.

 

[The_root_of_all_evil]

The straight dope article gives a practicle example of this problem, known as Zeno's paradox: If a man is racing against a tortoise, and the tortoise has a ten meter head start, you would expect the man to reach the tortoise very quickly. But by the time the man has run the ten metres, the tortoise has moved foward 1 meter. By the time the man has run another meter, the tortoise will have gone on another .10 meter, and so on and so forth. If this goes on for infinity, how can the man ever overtake the tortoise?

Zeno's paradox confuses by talking about distance, but measuring with time. Given the Paradox as it stands, the Man can never pass the tortoise because the time at which he does is never reached.

This is why the equivalency was brought in, but also why 0.9 recurring cannot equal 1. Because .9 recurring cannot finitely exist; it is, in itself, an irrational number - therefore it has a rational equivalency of 1. It can't equal 1, because 1 is rational.

If people want to argue and throw wiki's at me, then that's fair enough. But I'd also challenge you to find a definition of a solid material that's over 99% space - which is what it's like in real physics (Atomic Theory).

Also, there is no such thing as "equivalent" in mathematical terms. The two numbers are EQUAL, not equivalent.

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

*rolls eyes*

 

[Volkov]

Equal, and so close we can treat them equal, are not the same.

Equal, and different but base 10 has trouble displaying the difference is not the same.

.9999 is not = 1.

So what's the difference between the two? Express it in any way you like, doesn't have to be base 10.

 

[rokkolpo]

No it doesn't. The gap is so infinitesimal that it hardly counts, but 0.9999 recurring does not equal one.

Untrue. It goes 0.999... on to infinity. You'd think logically, you would just add a number that went 0.000...1. However, you can't have a 1 at the end of an infinite number of zeros, for that assumes there is an end to infinity, which there isn't. In mathematics, everything needs to be proven, and all the proofs say that 0.999... = 1. Whether you use rounding, limits, or just absolute practical value, they are identical.

Also, you have to remember that such a number would be identical to a level so infinitely small that no physical measurement could possibly amount to it. The decimal limits go down to a point that is smaller than any subatomic particle, and therefore, non-existent for any argument.

I like this explanation.^^

It seems to make sense, though I wouldn't know.

 

[Volkov]

The straight dope article gives a practicle example of this problem, known as Zeno's paradox: If a man is racing against a tortoise, and the tortoise has a ten meter head start, you would expect the man to reach the tortoise very quickly. But by the time the man has run the ten metres, the tortoise has moved foward 1 meter. By the time the man has run another meter, the tortoise will have gone on another .10 meter, and so on and so forth. If this goes on for infinity, how can the man ever overtake the tortoise?

Zeno's paradox confuses by talking about distance, but measuring with time. Given the Paradox as it stands, the Man can never pass the tortoise because the time at which he does is never reached.

This is why the equivalency was brought in, but also why 0.9 recurring cannot equal 1. Because .9 recurring cannot finitely exist; it is, in itself, an irrational number - therefore it has a rational equivalency of 1. It can't equal 1, because 1 is rational.

If people want to argue and throw wiki's at me, then that's fair enough. But I'd also challenge you to find a definition of a solid material that's over 99% space - which is what it's like in real physics (Atomic Theory).

These are several incorrect statements.

1. 0.(9) can finitely exist. It exists as 1.0.
2. It is NOT an irrational number.
3. There is no such thing as "rational equivalency" of an irrational number.

This is not an argument. This is a precise mathematical statement based on direct, unambiguous conclusions from founding axioms of real number arithmetic, versus numerous incorrect statements using incorrect terms. Only one side is correct.

Solid matter is matter which can sustain shear, tensile stress, and therefore can hold shape without boundary forces acting on it. It has nothing to do with atom density, mass density, or anything similar. Therefore asking for "a definition of solid material that's over 99% space" is like "asking for definition of the term 'video game' that necessarily mentions erythrocytes." The two are entirely unrelated, and therefore the latter will not be in the definition of the former.

 

[tjarne]

I would say it's a good enough approximation for practical purposes. What else is needed? Just as small angles are treated as 1