Does .999~=1?

[DoctorDan]

K heres how we do this algebraically.

(assume the .99 i use is repeating)
Step 1: .999 = x
Step 2: 9.999 = 10x (mult. both sides by 10)
Step 3: 9 = 9x (step 2 minus step 1)
Step 4: 1 = x
And by Substitution: .999 = 1

[GNOME_n00b]

hehe it is true that 0.9999 recurring wil never equal 1. in the same way that 1/x will never equal 0. if it did, then infinity will have been defined......but thats an impossibility. it just 'always approaches' but never equals.

[a12ctic]

what the hell is the relevence of this?

why does it need to be relevant, its a off topic forum...

[TomX]

I believe 0.9999... equals 1.

If there's a correct proof then it is true, we don't get to decide.

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

There's a proof, therefore it's true.

EDIT: I just started reading this thread having forgot that I'd posted on it. I read this post and thought, "wow, what a moron"... oops.

Anyway, no, what I put isn't a proof that 0.999... = 1; rather, a proof I was once stupider than I am now.

[a12ctic]

I believe 0.9999... equals 1.

If there's a correct proof then it is true, we don't get to decide.

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

There's a proof, therefore it's true.

this is wrong, you cant explain 1/3 as a decimal

[jimbaloo]

Following the logic of 0.999~=1, then 3=65.
And if 3=65 then I should have retired over 30 years ago

Can someone please explain how it is possible to define an infinite number with a discrete value?

[hilowe]

I think you all are missing something very important in this one.

The question is does 0.9999....~=1

The ~= means approximately equal if I remember correctly from the math classes I had.

So, to put it in words, does 0.999.... approximately equal 1.

It's already been said that this is as close to 1 as you can get without actually being 1.

[afroguy013]

Well, that depends on it's variable type really:

If it's a double or other floating point type then it is not euqal to 1, it is equal to .999...

However, if it's an int, then it's equal to zero and this whole argument is nuts.

[IsaacKuo]

The answer is yes. .999.... is exactly equal to 1.

The reason for this is actually a bit complex--it's not just basic Calculus but gets into some Real Analysis.

Basically, the important thing is to figure out what the MEANING of ".999..." is. Once you figure out the only thing which it can reasonably MEAN, then you realize that the only reasonable for it to BE is 1.

It's somewhat straightforward to state what it means--it means the limit of the sequence .9, .99, .999, .999, and so on. What's difficult and deep is to show that this is the ONLY thing which it can reasonably mean. For that, you need to take at least one semester of Real Analysis. The entire concept of "limit" and the fact that all decimal sequences converge to a real number is actually very deep and non-intuitive.

[bidi]

As an engineering student, I run across stuff like this almost every day. The truth is that, it depends on many factors.

First, if you're talking about tolerances, it's impossible to have a machine do a .999000000000..... cut, and if cost is a factor, then it's almost impossible. Generally speaking, a machine that can carry out a cut to the .001mm costs around US$12,000, and cost grows exponetially for every decimal place. So if you're talking about a .999 cut with a +/- .001 tolerance, .999 can either be equal to 1.0 or .998.

You then need to factor in machine precision (not the same as tolerance). Let say you add .999 to 5,000,000 (5 x 10^6). Well, if your computer only uses 16-bit numbers (theoretically speaking) it can only account for 5 (I think that's it) significant figures. So when you add the two number, it does 5 x 10^6 + 9.99 x 10^-2, and since it can only go to the 5th sig fig your answer will still be 5 x 10^6. Which concludes, that in this case .999 is actually equal to 0.

You can then make the assumption that .001 is so insignificant to your calculations, that it can be eaten and .999 rounded to 1.0 (which happens all the time in mechanical engineering, eletrical engineers might be a little more consertative about that).