Thread: .999~=1!

It's one of the first things you learn when dealing with calculus. I find it rather logical.

I mean, let's divide 1 by 3.

1/3 = 0.333...

Multiply it by 3 and you should be back to 1:

3 * 1/3 = 3 * 0.333... = 0.999... = 1

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Oh yeah, this is SB. My post needs some senselessness or it'll be kicked out:

Whenever green fairies blink, the rush of adrenaline makes half-full cups of tea fly aimfully to the right ear of the person sitting by the door.

We've been through this.

[Edit] All of what I said was false. Joke's on me.

Invader, just go ask any maths teacher. 0.999~ equals one. Proof is on the first post. You forget that the nines go on endlessly. The limit of a number *is* the number.

Today, you buy a cheese. Tomorrow, you will buy half a cheese. After tomorrow, you will buy half of half of a cheese, and so on. If you continue to do this endlessly, how much cheese will you have at the end? Answer is 2 whole cheeses. Not nearly 2 whole cheeses, but 2 whole cheeses exactly.

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0.333 is not equal to one third, but the repeating decimal 0.3333333~ does equal 1/3.

Originally Posted by hellohihello

Proof:

lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1

fixed. All that stuff that comes after is unnecessary. the fact is that every real number is defined to be an infinite sum as above. So the left hand side is just the definition of .99999...... If you don't want to parse that, it's just .9 + .09 + .009 + .0009 + .......

The value of this infinite sum is defined to be the limit of its sequence of partial sums. You get the i'th member of this sequence by adding the first i terms of the infinite sum. so it is {.9, .99, .999, .9999, ..... } the limit of this sequence is 1 so by definition, the value of the infinite sum is one and again, by definition, the value of .99999...... is 1. Opposing arguments RIP.

@invader
I think you meant hyberbolas and that can't be represented by an elementary function in the way that I think you are thinking, it can be represented as a vector valued function or as the set of zeros of a polynomial. If we adjoin the "line at infinity" and pass to projective coordinates, then it does have a value at infinity, namely 0, and is really just an elipse that intersects the line at infinity in two places. The fact that it doesn't have a value at infinity is really just a defect of the non-projective coordinates. A parabala, in that case also has a value at infinity and is just an ellipse that intersects the line at infinity twice at the same place in the same way that (x - 0)^2 = x^2 intersects the x axis twice at 0.

Originally Posted by PhilosopherStoned

The fact that it doesn't have a value at infinity is really just a defect of the non-projective coordinates.

But it is exactly why 0.999~ does not equal 1.
It is used for all intents and purposes to equal 1.
But by its very definition it "infinitely approaches 1, never reaching it".

So I don't get why this whole debate ever comes to be.

0.999~ is just that. And 1 is just that.
They are not equal, but the former is rounded to the latter.
Because during application the difference is negligible.

However, when you say "something is equal to something else", that isn't about application.
It's about static conditions.
So when saying 0.999~ is equal to 1, you're saying two different values (no matter how small the difference is) are the same.

And that's just obviously bogus.

In other words, you don't even have to know any math or anything about parabolas or any such to realise that if two different values are defined to be different, they are different. Doesn't matter how they are applied in calculations and all. Conceptually (not mathematically), they are different.

Originally Posted by Black_Eagle

We've been through this.

.

Originally Posted by Merlock

But it is exactly why 0.999~ does not equal 1.
It is used for all intents and purposes to equal 1.

two different contexts.

Originally Posted by Merlock

But by its very definition it "infinitely approaches 1, never reaching it".

You're confusing the definition of an infinite sum with an infinite sequence.

Originally Posted by Merlock

So I don't get why this whole debate ever comes to be.

because people that know nothing about math like to feel like they have any input to offer.

Originally Posted by Merlock

0.999~ is just that. And 1 is just that.

precisely. They are both equal to 1.

Originally Posted by Merlock

They are not equal, but the former is rounded to the latter.
Because during application the difference is negligible.

See above. Is there something you don't understand? I would be more than willing to help you.

Originally Posted by Merlock

However, when you say "something is equal to something else", that isn't about application.

It's about static conditions.

from a formal perspective, all mathematics is about static conditions..

Originally Posted by Merlock

So when saying 0.999~ is equal to 1, you're saying two different values (no matter how small the difference is) are the same.

see the proof in my previous post.

Originally Posted by Merlock

In other words, you don't even have to know any math or anything about parabolas or any such to realise that if two different values are defined to be different, they are different. Doesn't matter how they are applied in calculations and all. Conceptually (not mathematically), they are different.

reread my post. .9999...... is defined to be the limit of the sum which is 1. You don't need to know anything about math but it helps if you want to understand stuff. This is math, not literature. There is one truth value that can be assigned to any statement.

Hope that clears stuff up.

Originally Posted by PhilosopherStoned

Hope that clears stuff up.

It doesn't clear stuff up, since you yourself agreed with my stance.

Originally Posted by PhilosopherStoned

The fact that it doesn't have a value at infinity is really just a defect of the non-projective coordinates.

You agree that it's a fact that there is no value for it at infinity.
No value. Not 1.
And if it isn't 1, it isn't equal to 1.

Doesn't matter how it's applied. Conceptually, it never reaches 1.

Originally Posted by Merlock

It doesn't clear stuff up, since you yourself agreed with my stance.

Quote me agreeing with your stance. Do you mean about it being static? you're right. That has no bearing on the question at hand.

Originally Posted by Merlock

You agree that it's a fact that there is no value for it at infinity.
No value. Not 1.
And if it isn't 1, it isn't equal to 1.

Doesn't matter how it's applied. Conceptually, it never reaches 1.

I was talking about an algebraic curve which is a function. It was an aside for invader. It's a cool fact. This is totally different.


Here's another way to think about it from a topological perspective. The real numbers form a continuum. This means that given two real numbers, a and b, I can always find a third real number c, such that a < c < b. I can always stick a number in between them. All you have to do to prove that .9999...... is not equal to 1 is find some number that you can stick between the two.

here's yet another way to think about it. What's 1 - .99999999.........?


essentially what you are doing is saying that 4 + 1 =/= 3 + 2 because I am writing them differently. 1 = 1.000000....... = 1 + .0 + .00 + .000 + ....
is an infinite sum as well. It just has trivial terms after the first but is really just another way of writing .99999..... = 0 + .9 + .99 + .999 + .....

Originally Posted by PhilosopherStoned

The real numbers form a continuum. This means that given two real numbers, a and b, I can always find a third real number c, such that a < c < b.

This is what I meant when I said I don't understand why this whole debate comes up.

What you said in the above quote has no relevance whatsoever.
There is simply no need to prove or disprove that 0.999~ = 1.
By definition, 0.999~ infinitely reaches towards 1 but never gets there.

And calculations like those limits up there or formulating things like a < c < b...
None of it matters.
If something is defined specifically to never be equal to something else...it isn't.

In other words, what I mean to say is that I don't see any reason for any of the above proof to even exist.

And as such, I shall withdraw from this topic so as to avoid taking part in trying to prove or disprove something, heh. Just wanted to get my position out there, as it's always misinterpreted as me trying to disprove something. When in fact, I fully realise that 0.999~ and 1 are two different values by definition, and I simply wonder where anyone got the idea to start a debate about their equality.

It's like someone seeing 5 apples lying near a pile of 476 metal bolts, then suddenly getting up and stating, "I'm going to prove that 5 = 476!"

To all of you who can't understand how .999~ = 1, keep in mind you can't argue math in words.

So, provide a formal proof showing .999~ =/= 1 or GTFO.

Edit: Also, I find it hilarious that this was re-posted in Senseless Banter.

Ahrgh!

These threads are like an infinitely recurring nightmare.

The .9999~ is not equivalent to the value that comes after the decimal in 9.9999~.

NO.

But it is exactly why 0.999~ does not equal 1.
It is used for all intents and purposes to equal 1.
But by its very definition it "infinitely approaches 1, never reaching it".

NO.

Numbers don't APPROACH anything, they are FIXED values. Note that 0.999~ represents an INFINITE string of 9s, not ROUGHLY 0.999.

The fact is that it has been proved in the first post and you can't argue, as has been said above, with a mathematical proof. Especially not with words and analogies.

it's widely accepted in mathematics that the function that represents a parabola can approach but never be equal to it's asymptote at any point on the graph, even though it can close the 'space' to infinitesimally small distances.

Lord, you sound extremely intelligent. I'm in awe at the largeness of your words.

Except parabolas don't even have any asymptotes.

Xei speaks truth.

We had two threads going for a while where we discussed this. My conclusion was that the two figures are equal and that it has been proven but that it has never been fully explained how the 9's go on infinitely yet comprise a value of 1, even though calculus says they do and there are terms for the phenomena involved. I know the 9's are not moving, but they do "go" on forever. There is no end to infinity. The situation seems paradoxical.

To prove it in a different way, you can actually use the formula for calculating the sum of an infinite geometric progression. The initial term would be 0.9 and the ratio would be 0.1:

S = (0.9 + 0.09 + 0.009 + ....)

S = a1 / (1 - q)

S = 0.9 / (1 - 0.1)

S = 0.9 / 0.9

S = 1

You deny Blizzard? They will ban you from the internet.

Originally Posted by Universal Mind

The situation seems paradoxical.

It really comes down to math vs. logic. Math tells us that it does equal 1, but logic tells us just the oposite.

I think I've figured out how to use what Kromoh posted to argue that .999~ is not equal to 1. I'm sure my argument will be immediately proven wrong by someone far better at math than me, but I think I'll post it anyway, just for the hell of it. I mean, this is SB after all.

It is generally accepted that 1/3=.333~. So, if 1/3x3=1, then .333~ should also equal one. But it comes out to .999~. This would seem to say that .999~=1. However, consider this: no matter how many 3's you add on to that .333~, it is never exactly equal to 1/3. Therefore 1/3≠.333~, meaning that 1/3x3≠.333~x3, and .999≠1.

I think that the key to grasping it intuitively is to consider the quantity 1 - .999~

If we can show that 1 - .999~ = 0 then this is equivelent to saying that 1 = .999~

Lets go ahead and say that d = 1 - .999~ just so we don't have to write it out every time.

There is a theorem that says that if x is a number and x is not negative and x is less than any positive real number, then x = 0. Does that make sense?

It should be obvious that d is not negative because for 1 - b to be negative, b has to be bigger than 1. But .999~ is at the very least not bigger than 1 so d is not negative.

next, we need to prove that d is smaller than any positive real number. We can then apply the above theorem to conclude that d = 0.

We will do this by approximating d. Let b(1) = 1 - .9 = .1, b(2) = 1 - .99 = .01 and so forth.
For example, b(14) = 1 - .99999999999999 = .00000000000001. The bs are all exact numbers and each one is closer to d than the last one so they are good approximations.

Also, each b is larger than d. So given a positive real number, simply choose a number j so that b(j) has more zeros after the decimal place than the number in question. It is then smaller then the number in question but still larger than d. For example, given the number .00005, just notice that it is non-zero in the fifth decimal place and take b(5) = .00001 < .00005. This shows that d is smaller than that number. Because all we assumed was that the number is positive, we can do this for any positive number, however small. We can then apply the theorem to conclude that d = 0.

This means that d = 1 - .999~ = 0 so 1 = .999~

Originally Posted by Supernova

It really comes down to math vs. logic. Math tells us that it does equal 1, but logic tells us just the oposite.

The word you're looking for is not logic, but intuition. Mathematics is the very essence of logic.

I think I've figured out how to use what Kromoh posted to argue that .999~ is not equal to 1. I'm sure my argument will be immediately proven wrong by someone far better at math than me, but I think I'll post it anyway, just for the hell of it. I mean, this is SB after all.

It is generally accepted that 1/3=.333~. So, if 1/3x3=1, then .333~ should also equal one. But it comes out to .999~. This would seem to say that .999~=1. However, consider this: no matter how many 3's you add on to that .333~, it is never exactly equal to 1/3. Therefore 1/3≠.333~, meaning that 1/3x3≠.333~x3, and .999≠1.

Hmm, to use your thought stream:

assuming 1/3 = .333~
3 * 1/3 = 3 * 0.333~
1 = 0.999~


The matter is that it's not an arbitrary amount of 3's you put there. Meaning, you don't just sit down in a table and start writing 3's until you get tired. 0.333~ is nothing more than the decimal representation of the rational number 1/3. The amount of 3's contained in 0.333~ is infinite. That is the reason why 0.333~ = 1/3.

0.333~ is not roughly 0.333, it is the decimal representation of the number 1/3, a so-called repetitive decimal.

The paradox of the cheese illustrates this well. It's in a previous post of mine somewhere.

Sorry for the double post but Supernova posted this when I was working on my last one.

Originally Posted by Supernova

It really comes down to math vs. logic. Math tells us that it does equal 1, but logic tells us just the oposite.

It comes down to math vs. naive intuition. naive is not offensive in this context, it just means what you would think if you weren't a mathematician. Most mathematicians use "naive set theory" for instance, because it's good enough for their needs.

Originally Posted by Supernova

I think I've figured out how to use what Kromoh posted to argue that .999~ is not equal to 1. I'm sure my argument will be immediately proven wrong by someone far better at math than me, but I think I'll post it anyway, just for the hell of it. I mean, this is SB after all.

Great. It's how one learns how to construct a proof

Originally Posted by Supernova

It is generally accepted that 1/3=.333~. So, if 1/3x3=1, then .333~ should also equal one. But it comes out to .999~. This would seem to say that .999~=1. However, consider this: no matter how many 3's you add on to that .333~, it is never exactly equal to 1/3. Therefore 1/3≠.333~, meaning that 1/3x3≠.333~x3, and .999≠1.

I bolded and blued the problem area. The question of if .333 = 1/3 is really the same issue as if .999~ = 1 or 5.999~ = 6 for that matter. The fundamental question is if one number can have multiple decimal representations. It might help to remember the definition of a rational number (a fraction) as a number with infinitely repeating decimals. .333~ is the infinitely repeating decimal expansion of the rational number 1/3, .999~ is the decimal expansion of 1/1 = 1 and so on.

Well I guess you've only proven that I don't really get this kind of advanced math, something I could have told you myself

(yea, maybe logic isn't the best word for what I was trying to describe)

I demand moderator powers so that I may lock this abomination of a thread.

Originally Posted by Black_Eagle

I demand moderator powers so that I may lock this abomination of a thread.

Well then don't read it. You might as well lock every thread about athiesm vs. god. at least this one has a definitive, absolute answer with proof and everything. quit your bellyaching.