Hi explain why 0.999.... != 1 please.

[Old Manus]

Does math make you frustrated? It should.

[Madame Adequate]

Having thought about this with some intensity for the last few hours, it's making a lot more sense right now than it was last night.

It still doesn't make sense that because there's no number between the two, that they are not distinct numbers though. I understand the rest, but I really don't get that.

[bipper]

So is it safe to say that there is an issue with decimals as we know them? Should we revamp math!? Please, I want some new math, the old stuff is so out dated.

[Fatal Impurity]

0.999999999 infinty doesnt = 1

but the little workaround so it can be expressed properly is 0.99' the little apostraphe is the (actually its a dot but hey this is a computer) expression for an infinate number

[Resha]

I think it would equal 1; if you have 0.9999 recurring to infinity, you can't very just add 0.00000000000000001 or 0.0000000000000000000000000000001 to it, because neither number will reach the end of infinity, if it makes sense. If you add 0.0!1 to it somehow, then it would be more than 1. 1 puts the stopper onto the 0.000, and hence it won't reach infinity.

I don't know if this is right or wrong or makes sense or doesn't. It sounded logical to me in my warped mind at time of writing so please don't rip. xD

[Dignified Pauper]

I think it would equal 1; if you have 0.9999 recurring to infinity, you can't very just add 0.00000000000000001 or 0.0000000000000000000000000000001 to it, because neither number will reach the end of infinity, if it makes sense. If you add 0.0!1 to it somehow, then it would be more than 1. 1 puts the stopper onto the 0.000, and hence it won't reach infinity.

I don't know if this is right or wrong or makes sense or doesn't. It sounded logical to me in my warped mind at time of writing so please don't rip. xD

i like this post, except that you'll never have 0.000!1

I miss math. I should start taking some classes again,

[Resha]

Exactly! 0.0000!1 won't be ! anymore.

I always thought ! was for factorial not infinity but whatever.

[Avarice-ness]

Not entirely sure about .999~ = 1.
But if it did it would be because the repeating decimal at the end got hit by a one, then it would knock all the 9's to a one, crossing the decimal into the one's place. But since it would be a never ending decimal place, which would make it very hard to see what happens -after- al the 9's that you still see, It would almost be like trying to find -every single- decimal place to Pi, It's never ending.
So who knows, it might actually end up 1 at the end of the day.

But what if the .99999~ isn't infinitive, and we just think it is. =O

[Raistlin]

Just because there's no number in between doesn't make it equal to it. >:/ Then .9[insert 9s here]8 would be equal to .9[insert 9s here]9 who's apparantly equal to 1. So in the long run 1 would be equal to 9. And that just ain't right, girlfriend.

There's so many things wrong with that, I don't know where to begin.

First off, not only does any two distinct real numbers have to have a number between them, any two distinct real numbers have to have an <i>infinite</i> amount of numbers between them. 1 and 2 have an infinite amount of numbers between them. There exists not even ONE real number between .99999... and 1.

Also, .99999...8 isn't a number. That's saying "an infinite number of nines that never ends, but when they do end, tack an 8 on the end." It doesn't work like that. 0.9999... means "a decimal followed by an <i>infinite number</i> of nines.

[Sefie1999AD]

Hmm, I would probably take a bit harder way and solve this through series.

0.999... = 0.9 + 0.09 + 0.009 + ... = 9*10^(-1) + 9*10^(-2) + ... = 9 * [ 10^(-1) + 10^(-2) + ... ] = 9 * Σ (n=1...∞) 10^(-n) := 9*S

S is clearly a neverending geometric series, with its first term (n=1) being 0.1. q = 1/10 and |q| < 1, so the series converges. S = 0.1 / (1-q), so

0.999... = 9*S = 9*0.1 / (1-1/10) = 0.9 / 0.9 = 1.

[Raistlin]

Having thought about this with some intensity for the last few hours, it's making a lot more sense right now than it was last night.

It still doesn't make sense that because there's no number between the two, that they are not distinct numbers though. I understand the rest, but I really don't get that.

Ok, MILFy, let me try to explain.

Take the numbers 1 and 2. How many numbers are between 1 and 2? The answer is infinite. You have 1.1, 1.01, 1.001, etc.

Now, take 1 and 1.00000001. How many numbers are between those two? The answer is again, infinite. 1.0000000000001, etc.

Between any two distinct, real numbers there exists an infinite number of other distinct, real numbers, no matter how close those two numbers are. It's called the Density Property, and is a fundamental concept of algebra.

There exists no number between 0.999... and 1, and both are real numbers, therefore, they are equal.

[radyk05]

the real proof is kind of idiotic because it involves the limit argument which is, escencially, an aproximation. the algebraical proof (x = 0.99999....) is way more better if you ask me. the only problem that i find is how you pass from 0.9999... to 1.000...01. maybe my brain isn't abstract enough.

[Raistlin]

the real proof is kind of idiotic because it involves the limit argument which is, escencially, an aproximation. the algebraical proof (x = 0.99999....) is way more better if you ask me. the only problem that i find is how you pass from 0.9999... to 1.000...01. maybe my brain isn't abstract enough.

Limits aren't an approximation. It's only somewhat non-intuitive. The limit proof is simply carried out into infinity, which some people have trouble grasping.

EDIT: what I had below this was wrong.

[Zell's Fists of Fury]

You people need to go outside once in a while.

[radyk05]

well, the way i was taught, the limit is when you come close enough to a number/point but never to the number/point (or something along those lines). then again, that was only a calculus class and not a real analysis class.