Is 0.999…=1? This is pretty simple math, but still a source of constant arguments. Many people say that it can’t be true, because no matter how many nines there are, a number starting with 0. cannot be as big as 1. I, on the other hand, believe it is true. My proof for it is that there is always an infinte amount of numbers between two different numbers, but there are none between 0.999… and 1, so they have to be the same number. Both “proofs” are pretty convincing, so could you tell me which one is right and why?
BBB
I, for one, believe that you are almost right - so perhaps you are.
We could rephrase your question, ignoring the proofs you present, as - ignoring any rounding error, if two numbers differ by less than the rounding error are they the same? Yes they are.
Russell
Isn’t the standard proof for that:
1/3 = 0.333… = A
1/3 + 1/3 + 1/3 = 1 = B
0.333… + 0.333… + 0.333… = 0.999… = C
Thefore A + A + A = B
And since B = C, 0.999… = 1
The real math gurus can clean this up.
I don’t buy it. I never have.
As for the proof that Monty shared, I’ve seen it before, and I think it’s flawed.
0.333… + 0.333… + 0.333… does not equal 0.999…
It equals 1.
The reason: At some infinitely distant point after the decimal, the sum of the 0.333…'s roll over. If it came to 0.999… then at least one of the 0.333…'s are being shortchanged.
Yes, for all practical purposes they are the same. But technically, no they’re not.
God is my co-pilot. Blame Him.
The best proof that I’ve seen is to use the method for finding the fractions that are equivalent to repeating decimals:
<BLOCKQUOTE><font size=“1” face=“Verdana, Arial”>code:</font><HR><pre>
_
N = 0.9
_
10N = 9.9
_ _
10N - N = 9.9 - 0.9
9N = 9.0
N = 1
------------------
Wrong thinking is punished, right thinking is just as swiftly rewarded. You'll find it an effective combination.You’re still dropping a decimal place. In this instance 9N should be 9.9999…9981. You’re kind of blurring infinity here. If you’re going to say that there is an infinite number of 9’s after the decimal point, I’m going to say that there is an infinitessimal difference between 9.999… and 1.
KarmaComa wrote
No, you see, you can’t blur infinity, because it is already blurred.
I’m serious. When you say things like
you get into trouble because you are treating infinity like a number, which it is not.
Infinity is not a number. It is a concept.
Well that’s kind of my point. When these people are talking about .9999… they aren’t talking about 3 * 1/3, they’re talking about an irrational number, but they don’t realize it.
Sorry, but .99999… is a rational number. That was shown by AWB.
There have been two proofs that .9999… = 1 shown here already. You refuse to accept the proofs, but you haven’t disproven them.
“What we have here is failure to communicate.” – Strother Martin, anticipating the Internet.
I think you may be confused: AWB’s proof relies on .999 (as we will call it) is irrational, otherwise he is ignoring a decimal place, which I pointed out in a previous post. Do you not understand this? .999 is not 3 * 1/3, it is irrational (why do I feel like I’m repeating myself?)
There is an infinitessimal difference between .999 and 1. Peep this: there is an infinite number of rational numbers between any two other rational numbers.
I think this should be cleared up: .333~~ is not 1/3, it is a decimal approximation of 1/3, and will never actually reach 1/3; it will always be smaller, as will .999~~ always be smaller than 1.
Really this is a moot point.
When my calculator comes up with .999999999 I don’t think, “oh, that’s equal to 1”, I think, “oh, that’s a roundoff error. It means 1”.
KarmaComa, you can pick up any math book that deals with this question, and you will see similar proofs that 0.99999… is in fact equal to 1.
All serious mathematicians agree on this.
Of course, you are free to disagree, but you will be hard-pressed to find many people (knowledgeable in mathematics) to agree with you.
isn’t 1/9=.111…
and 2/9=.222…
and so on, so
5/9 + 4/9 = .9999 or 9/9
and 9/9 IS = 1
Is this proof assuming anything wrong about the numbers?
Oh, wait. I see. KarmaComa’s point is that 1/3 does NOT = .333…
Umm… I think that we CAN prove .333…=1/3, If you want that.
Yeah, I think you’ve got the facts right, kbutcher.
One disagreement: as to KarmaComa’s point about 0.999… being irrational, I’m pretty sure irrational numbers only include those that neither terminate, nor repeat. Infinitely repeating numbers are perfectly rational, since we know how much they are equal to exactly. We’re just not willing to right it out.
I think 0.667 would be a decimal approximation of two thirds, and 0.666 would be a bad decimal approximation of it, but 0.666… (I can’t figure out how to put that little “repeats” bar above the sixes) would be two thirds exactly. The “repeats” bar extends it out to infinity. Yeah, I know it’s just a concept … we can’t get there, but we all know what it means.
1 - 0.999… = a number with no magnitude. It certainly has conceptual value (it equals 0.000…1), but that little 1 is infinitely small and thus equals zero. Multiply it by a huge (finite) number, and the product will be the same as the first factor (0.000…1 again), just as if the first factor were zero. That’s my test, although the mathematicians might have a different one.
Am I reading Hippocrates right?
He’s not just a big box for storing large semi-aquatic mammals, is he?
I’m going to try and do the algebra on this puppy just to be clear, and give you guys a bigger target :0
1 - 0.999… = 0.000…1
0.000…1 x 50 trillion = 0.000…1
0.000…1 = 0
Q.E.D. Or so I hope.
I think you’ve missed the salient point. If 0.999… differs from 1.0, then you must be able to say by how much it differs. In fact, there is no number, rational or irrational, that is the difference between 0.999… and 1.0. Given this fact, given that you will be unable to demonstrate by how much 0.999… differs from 1.0, they must be considered identical.
Now, some of you may know about “non-standard” analysis, but that’s a different story.
Well then, exactly how much does 1 differ from root 2?
By exactly sqrt(2)-1