i've always respected darkfire

Well you're misconstruing what I was saying pretty badly. The answer to your question is no and the reason is quite easy. Given two numbers a and b in the real numbers, we say that a = b if and only if for all real numbers e, the difference |a - b| < e. So let's take your example: I claim that 1 is a counterexample |4.59238 - 11.283766| = 6.691386 > 1. Therefore there exists a real number e such that the necessary condition fails. By contrast, no real number e satisfies the condition that |1 - SUM_{n,1,infty} 9/(10^n)| >= e. Why? Because the number 0.000...0001 does not exist. Why not? Because no sequence of rational numbers converges to it and that is how a real number is defined. Everything boils down to definitions when it comes to infinite quantities (large or small), because infinity is not an intuitive concept. Things you think absolutely have to be true break down all the time when it comes to the infinite. That is why mathematicians demand mathematical rigor in proofs: so we don't make mistakes.

As to your other complaints, they are perfectly intuitively valid, but in truth incorrect. You don't know enough math to know why you are wrong yet. I imagine my explanation probably doesn't make a ton of sense either (unless you somehow encountered Cauchy sequences in high school?). That's fine. Please stick with the field for a while and I'm sure you'll figure it out

Dude... You can chose any number you want (not any number but lots) as a proof of that. But the fact is, the decimal form of 2/7 plus the decimal form of 5/7 is 1. Not .999999. You merely can't see that it's 1 because you can't ever end the decimal.

Edit: "Darkfire's post"
I know exactly where you're coming from. However, it isn't 1. It's just close enough to 1 we call it one. As with limits/horizontal asymptotes. Ratio of LC because rest stop mattering. (4x+7)/(4x-7) Ratio of LC is 1. The limit as x goes to infinity is 1. Most people are taught, when you divide by x, 4x goes to 4, 7/x gets so close to 0 we call it 0. It doesn't equal 0, but it *pretty much does*.

If in math you are forced to learn that .99999999=1 then math needs to get some shit straight. Might as well start teaching us we can fly, because even though "Not being able to fly" does not equate to "being able to fly" math seems to be able to "prove" otherwise.

This post was edited by Thann on Apr 24 2010 04:47pm

Well it sounds like you are taking a very intuitionistic approach, which is fine. You get a lot from that. You lose a lot too though. It turns out that .999... = 1 is a consequence of every method of proving calculus to be well-defined that I know of. Similarly, it is a necessary consequence of having well-defined irrational numbers. If you are willing to give those up, then that's fine. You have a perfectly consistent if practically useless mathematics.

Tbh I see this more as a question of english, rather than math.
Obviously I take 1/INFINITY to be 0. Because it pretty much is.
The same as I (honestly) do take .999999 to be 1, because it pretty much is.
I only do this because it is impossible for me to work with infinite decimals, as it is for everyone, and "pretty much is" is close enough for me, to say that. It doesn't make it physically the very exact same thing.

Your approach to math is obviously far superior than mine. Mine would in fact get me nowhere in math, and make it impossible to do certain questions. But it's the fact that I acknowledge it is more or less a cheating (albeit it works) shortcut to simplify something that otherwise can't be simplified. Just like square root of -1 is I. It doesn't exist, but we simplify it as I so we can continue to work with it.

EDIT Brought into new post:
2/7+5/7=7/7
7/7 obviously equals 1/1, 2/2, 3/3, 4/4, 5/5 etc etc. So let's take 1/2. 1/2+1/2 = .5 + .5 = 2/2 = 1. 2/2=7/7 therefore 7/7=1. (Which I suppose you could (and should) say you're not trying to disprove that 7/7=1, you're merely trying to prove 7/7 also = .999999 infinite, or even .000000000....1.) So I'll continue.
This logic is saying that any number, is the same thing as it's "next-one-over" infinite decimal. So, 1=.9999999, and .9999999=.9999999998, .999999998=.999999997. Therefore, on an infinite scale. Every number in the world equals eachother. But of course this isn't the case.

Proof:
x = 0.99999...8
9.999999...x = 9.9999....8
9.999999...x - x = 9.99999....8 - 0.9999...8
9.999999...x = 9.999998
9.999999=9.9999998

x=.999999...7
9.99999...8x=9.999999...7
etcetc

This post was edited by Thann on Apr 24 2010 04:52pm

The bold isn't true actually, unless you do something weird called compacitifaction (basically, you say - infinity = infinity). As to the other part, just stick with math and you'll see what I mean. Physically the entire concept of infinite quantities (big or small) is bullshit. Quantum physics predicts discrete minimal units of time, space, and mass, so there comes a point at which there is no meaningful difference between (for the sake of argument,the numbers aren't this small actually) .9999999999524 and .9999999999999 because no physical quantity exists that is actually that small. If you insist that mathematics makes sense physically, then problems like this will keep popping up. Mathematically, however, we can work with infinite decimals so long as we stay within certain boundaries. That's the beauty of math. You can do it and be right regardless of whether it works in reality

edit: the contradiction in your argument is that the logic is not that a number is the same as "its next one over". This phenomenon occurs in every basis with infinite chains of maximal elements in the basis (but only infinite chains of maximal elements).

This post was edited by darkfire on Apr 24 2010 04:54pm

1=5

I suppose. I obviously see through the many proofs that it's *theoretically right*, but in practicality or reality it's not.
I don't understand how you (or others) -Keep in mind through none of this did I intend to offend anyone- can claim it to be the very same thing, when it's (to me) clearly a simplification we make to be able to work with it.
Is it the same thing? Pretty much. But it will always remain that infinite decimal off of the actual value.

I do appreciate the discussion. I know you won't be able to change my mind (unless there was something that clearly pointed to it being the same thing, which is possible, if it weren't impossible to come up with said proof (theoretically the proof exists, but it's impossible to get, which is why I stick by my guns) and I know your thought is based on just as much logic as mine so I don't think I'll be able to change your mind. But... (if that makes no sense, sorry it was hard to explain what I was trying to say)


If you're not oversimplifying math to make it right, you're oversimplifying English (or the word choice) imo.

This thread may as well be about proving a=b, where a=x, b=y and x=/y. I'm sure there is a way to mathematically prove it, but it will never be true.

This post was edited by Thann on Apr 24 2010 05:13pm

response to Bold

Divide 9/9 and forget that it = 1

What answer do u get?

9/9 = .99999999999999...

Regardless of whether I know it's 1 or not, it's still 1.

Let's take a number I've never computed before. 2856764/2856764. I've never computed it before, and I'm not even going off the premise that x/x is always 1. I'll put it into the calculator and see what I get.. I got 1 btw not .999999999. Weird right?

(0+x) + (0+x) + (0+x) =.1 Where x=0. Go.

This post was edited by Thann on Apr 24 2010 05:17pm