Poll: 0.999... = 1

[Lyx]

0.999... is not 1
It is as simple as that. No matter how you twist and try to prove it.

Can you please provide an argument with your assertion?

The shortest killer-argument is: "0.999..." is a function. "1" is a value.

You just need to get, that infinity is not a value. If any programming language is telling you that infinity is a value, then it is lying to you (because what it will actually do with that "value" is call an infinity-function, that is STOPPED before reaching infinity - namely, the function will stop when the desired precision is reached).

It really is that simple: Infinity is not a value. It is just saying "do this forever" (a function-loop).

For a more detailed explanation, see my previous post (first post on page 3).

 

[emeraldrafael]

Yes, this does work out. My college math professor showed me this. I'm not going to question the guy with a PHD in math. .9 repeating is essential equal to one because it is only off from one by an infinitesimally small amount. The math works, even if some people have some trouble swallowing it.

Essentially (according to Merrium Webster):
1
: of, relating to, or constituting essence : inherent
2
a : of the utmost importance : basic, indispensable, necessary b : being a substance that is not synthesized by the body in a quantity sufficient for normal health and growth and that must be obtained from the diet ? compare nonessential 2
3
: idiopathic
? es·sen·tial·ly \-ˈsench-lē, -ˈsen-chə-\ adverb
? es·sen·tial·ness \-ˈsen-chəl-nəs\ noun
See essential defined for English-language learners »

Essentially is not equal. While he may have a PhD in math, he doesnt not have a basic understanding over the English language.

 

[havass]

Why is it invalid, though?

Because 10x-x isn't 9. It's 8.999999....9991.

But it goes on to infinity, so technically there's no 1.

 

[Rabid Toilet]

2.
x = 0.999...
10x = 9.999...
10x - x = 9
9x = 9
x = 1
0.999... = 1

while the bold is technically correct if you look at it as (9.999...)-(.999...) = 9
the truth is infinite - infinite = 0
you're not looking at an actual number, you're looking at a concept and concepts don't work the same as numbers

Infinity - infinity does not equal 0, because, like you said, it is a concept and not a number, and also because one infinity does not necessarily equal another infinity. You can have infinities that are larger than other infinities, for example (Don't quote this as a reason why you can have infinity - 1 nines, it's still an infinite number).

Also, you actually can subtract infinite concepts like that.

For example:

{ (n + 1) + n + (n - 1) + (n - 2) + (n - 3)... } - { n + (n - 1) + (n - 2) + (n - 3)... } = (n + 1)
Just because there are an infinite amount of numbers in a sequence or a decimal doesn't mean they aren't there in equal number, which means they can be subtracted.

 

[Devil's Due]

1/3 = 0.3333333333

0.333333333333 x 3 = 0.999999999999

= 1

0.99999 is ~ 1, but it is NOT = 1. So it is safe to use in math, yes, and it's pretty damn close, but regardless how much people try and explain, it is not one, or else it would be one, not .999999(infinite).

 

[tthor]

http://en.wikipedia.org/wiki/Invalid_proof
Read

 

[kael013]

.999... and 1 are the same number because .999... is the decimal equivalent of the fraction 3/3 or 9/9. Just like .5 is the decimal equivalent of 1/2 and 1 is the decimal equivalent of 3/3.

That might seem like math magic, but it's really not. They're just different ways of writing the same number.

I'd like to see that first bit written out because my calculator says different. To me, all numbers have a set value so .9(infifnite) may be so close to 1 that people will say it's 1 but it's still missing .0(inifnite)1 (though I could be wrong in taking that stance, in which case I'd love a chance to open my mind some more). In my mind, math is all about logic and logically two different values cannot equal each other no matter how close together they are.

On a side note, this thread has just reinforced my growing belief that math is a religion.

You misunderstand limits.

Also I need to point out that the reals do not permit numbers which are infinitesimally close. I really don't have time to show the proof for that but you guys are welcome to do the research.

Probably. I've only gotten up to Trig, but I've never completely understood anything past my high school Algebra 2 class. But thank you for telling me what I'm misunderstanding, I'll go look up that concept now before my brain completely melts from this thread.

 

[Coldie]

2.
x = 0.999...
10x = 9.999...
10x - x = 9
9x = 9
x = 1
0.999... = 1

while the bold is technically correct if you look at it as (9.999...)-(.999...) = 9
the truth is infinite - infinite = 0
you're not looking at an actual number, you're looking at a concept and concepts don't work the same as numbers.

You're the one mixing numbers and concepts.
0.99999... and 0.3333... are not infinite, they are finite and quite small at that. They have infinite length, not infinite value, and there is absolutely no point trying to manipulate the length of a value. No matter how many digits, infinite or not, are there after the decimal point, it still will be a finite number that will always be bigger than its integer part and smaller than integer part + 1.

Or integer part + 0.(9), for that matter.

 

[Twilight_guy]

Yes, this does work out. My college math professor showed me this. I'm not going to question the guy with a PHD in math. .9 repeating is essential equal to one because it is only off from one by an infinitesimally small amount. The math works, even if some people have some trouble swallowing it.

Essentially (according to Merrium Webster):
1
: of, relating to, or constituting essence : inherent
2
a : of the utmost importance : basic, indispensable, necessary b : being a substance that is not synthesized by the body in a quantity sufficient for normal health and growth and that must be obtained from the diet ? compare nonessential 2
3
: idiopathic
? es·sen·tial·ly \-ˈsench-lē, -ˈsen-chə-\ adverb
? es·sen·tial·ness \-ˈsen-chəl-nəs\ noun
See essential defined for English-language learners »

Essentially is not equal. While he may have a PhD in math, he doesnt not have a basic understanding over the English language.

Please note that those are my words in the post. He had the lecture I added my own commentary for the post. Do not talk about my professors. Please aim all negate feedback at me and do not slam the people I know.

 

[Rabid Toilet]

Actually, the reason the proof is shady is because its first step is x = .99...

Others have already said this, but...what? Why does x = .999... have to be proven? It's an assignment. It's equivalent to the statement "Let x represent .999... in the following calculations." I see several points in the proof that rely on the correct-but-unproven fact that 1 = .999..., but that's not one of them.

Yeah, I think I was tired and thought he had started with something like 1 = .99...

That would be illegal.

 

[Lyx]

Infinity - infinity does not equal 0, because, like you said, it is a concept and not a number, and also because one infinity does not necessarily equal another infinity.

Sorry, but the guy with the zeroes in his avatar has something to say about this.

Zero is a very special "number". Observe for example, WHY dividing by zero makes no sense - what could be so special about that "number", that it is the only one where that happens? Why on earth would it in a computer trigger an.... infinite loop?

There are two kinds of zero. One is if we take it to be a value that by coincidence is identical to a reference value. Example: zero degrees celsius is calibrated to freezing water as the referencepoint. Here, zero indeed is a normal "value", just one that was choosen to represent the referencepoint.

It is a bit different however, if we take zero as a QUANTITY (that IS how it is interpreted in multiplication and division). What could be no quantity? Is Nothingness a value? Hmmmmmm

 

[Rubashov]

1/3 = 0.3333333333

0.333333333333 x 3 = 0.999999999999

= 1

0.99999 is ~ 1, but it is NOT = 1. So it is safe to use in math, yes, and it's pretty damn close, but regardless how much people try and explain, it is not one, or else it would be one, not .999999(infinite).

0.99999 is approximately 1; the difference between it and one is 0.00001. 0.999... is 1; the difference between it and 1 is infinitely small and therefore 0.

As zoulza pointed out:

Here's another way to think of it: if .99999... is not equal to 1, then there must be some other number between the two. What is it?

 

[emeraldrafael]

Yes, this does work out. My college math professor showed me this. I'm not going to question the guy with a PHD in math. .9 repeating is essential equal to one because it is only off from one by an infinitesimally small amount. The math works, even if some people have some trouble swallowing it.

Essentially (according to Merrium Webster):
1
: of, relating to, or constituting essence : inherent
2
a : of the utmost importance : basic, indispensable, necessary b : being a substance that is not synthesized by the body in a quantity sufficient for normal health and growth and that must be obtained from the diet ? compare nonessential 2
3
: idiopathic
? es·sen·tial·ly \-ˈsench-lē, -ˈsen-chə-\ adverb
? es·sen·tial·ness \-ˈsen-chəl-nəs\ noun
See essential defined for English-language learners »

Essentially is not equal. While he may have a PhD in math, he doesnt not have a basic understanding over the English language.

Please note that those are my words in the post. He had the lecture I added my own commentary for the post. Do not talk about my professors. Please aim all negate feedback at me and do not slam the people I know.

Alright, my apologies, you have little concept for what the english language means, as essentially and equal are two vastly different worlds on the simple basis that they are not even Synonyms.

And again, I would have to ask him how he explains this, while the concept of math dictates a theory (practice?) of limits, and that to say .9999999 = 1 breaks that concept of limits.

 

[Georgie_Leech]

Guys, stop trying to use intuition about this. The entire point of Logic, and by extension, the sub-category of mathematics, is to works when intuition fails us. Intuition fails us at infinity, because it is literally impossible to comprehend it in a physical manner.

 

[Rabid Toilet]

Infinity - infinity does not equal 0, because, like you said, it is a concept and not a number, and also because one infinity does not necessarily equal another infinity.

Sorry, but the guy with the zeroes in his avatar has something to say about this.

Zero is a very special "number". Observe for example, WHY dividing by zero makes no sense - what could be so special about that "number", that it is the only one where that happens? Why on earth would it in a computer trigger an.... infinite loop?

There are two kinds of zero. One is if we take it to be a value that by coincidence is identical to a reference value. Example: zero degrees celsius is calibrated to freezing water as the referencepoint. Here, zero indeed is a normal "value", just one that was choosen to represent the referencepoint.

It is a bit different however, if we take zero as a QUANTITY (that IS how it is interpreted in multiplication and division). What could be no quantity? Is Nothingness a value? Hmmmmmm

It doesn't matter what kind of zero I'm talking about. If you subtract a number from another number that is not equal to the first, you do not get zero. Infinity does not necessarily equal infinity, so subtracting the two does not get you zero.

 

[zama174]

Fuck....... Math.........

I looked at the wrong thread...

 

[BENZOOKA]

0.999... is not 1
It is as simple as that. No matter how you twist and try to prove it.

Can you please provide an argument with your assertion?

How I would've explained it: 0.[footnote]It hurts to use a dot instead of a comma, because that's what we use here and it's nothing but an annoying mistake to use a dot there.[/footnote]999... is the closest you can get to 1 (from 0), but it's still not the smallest unit away from it, because the nines go on forever. They never stop, so it has no definite value. As the delightful nines go on infinitely.

Lyx said it far better than I could:

Spoiler: Lyx's post. Spoilered for convenience

You don't really need maths to understand this, but instead just have some background understanding about concepts like "accuracy" and "infinity" - basically, a conceptual understanding what "numbers" are.

First step: Accuracy

Numbers are not absolute - ignoring what they are supposed to represent, they use a "unit" which is the base (compare: binary base and decimal base). Depending on the base, there are some values that can be expressed with *theoretical* ideal accuracy (in practice, that will never work... nothing in practice can be done with perfect accuracy, and doesnt need to). For those values that however do not lie exactly on the base, you need to express it with fractions... but the problem is, that for some values, the fractions will be repeating - so even that way, you cannot reach that "ideal accuracy"....

UNLESS you cheat.

What is infinity? Well, in short, infinity is not a value. Let me repeat this, because this is an annoyingly common myth: Infinity is not a value. What infinity is, is a looped function... for those who understand a bit about programming, infinity looks a bit like this:

1: n = n + 1
2: Goto 1:

The important thing to notice, is that the loop doesn't end, and therefore will never ever return a value, unless we stop the loop at some point - which will NOT return infinity, but just the highest precision reached at the point of the break. Or to phrase it more ironically: Infinity is a function that never returns infinity because it never finishes. The "infinite accuracy" is just a theoretical symbol - it has no corresponding phenomena (this is why mathematical points do not exist).

What does that mean for the 0.99999... = 1 question? Well, notice the "..."? Thats just saying "append another 9 forever" (infinity). In theory, if "forever" could ever be reached (contradiction), the result would be 1. It just will never happen

So, while 0.99999... does conceptually allow to express "infinitely close to 1", it logically makes no sense.

But in practice, this will not matter. Because in practice applications, you won't ever precisely hit "1" anyways... no matter if you use "0.999..." in your design or "1". All that matters in practice, is that the accuracy is high enough - and a "0.999..." function that is stopped when it reached enough precision, will do just that.

 

[Coldie]

Yeah, I think I was tired and thought he had started with something like 1 = .99...

That would be illegal.

Well, actually...

If you start with a statement you assume to be false, you can prove it to be false via Proof by Contradiction. You can't prove anything to be true by starting with a false statement, but you can disprove the original statement by running the proof until you hit a contradiction.

It's called reductio ad absurdum and is definitely legal and quite useful.

 

[Troublesome Lagomorph]

It is one. You would round up to the nearest whole number, which is one.

 

[Atmos Duality]

A non-terminating expression being compared to an integer?
Oh, stop the presses. We've never seen this before.