A Friendly Chat About Whether 0.999... = 1

IT makes me sad to see all of this useless talk
1 does not equal finite number 0.999, there is no tolerance in finite numbere.
if you make the following statement that 1/9=0.111 is again wrong! 1/9=0.(1)
it is a never ending calculation of 1/9… Ex:{10/9=1 r 1}
witch ca be expressed as 1/9. So 0.(1) cannot be multiplied with any number unless expressed as a fraction! The same works for any finite periodical non-complex number. Artifices like error tolerance are needed for complex numbers that cannot be expressed in simpler terms. In other words: Do not try to complicate things. You become smarter by simplifying complicated things

I am putting marks on the number line like this
(d is the distance between 1 and the position of the mark):

time 0: I am putting a mark at position 0 (d = 1)
time 0,9: I am putting a mark at position 0,9 (d = 0,1)
time 0,99: I am putting a mark at position 0,99 (d = 0,01)
time 0,999: I am putting a mark at position 0,999 (d = 0,001)

Where am I putting a mark at time 1?

@Andrei: I’m not sure I understand what you mean by “finite number 0.9999” – the idea is to figure out what the closest number to “0.9999…” (i.e., infinite nines) would be. If we assume infinitely small numbers cannot exist, then the number that statement refers to is 1 since there cannot exist a number between 0.999… and 1. If we allow infinitesimals, then we can represent that “gap”.

We have 1/3 = 0.33333 recurring;
0.333(recurring) * 3 =>1 (exactly)

So what stops the division of 3/3 => 0.99999…

We should be able to get either of these two proper answers from the same division, so what do we have to unlearn about long division to get the answers.

We need that, in the long division, three into three (without any remaining digits to pull down) is not one, but zero with three left over.

We drop down the unlisted trailing zero to get three into thirty, which then gets an answer of nine remainder three, and repeat.

Now these are correct, but unconventional, ways of having the remainders happen, and it is by this method that we can get both
3/3 => 1 (exactly), and
3/3 => 0.99999(recurring)

Without a simple long division example that ‘normal’ folks can do, they will continue to have difficulty with all the logic reasoning and theorem proofs.

Philip Oakley

I missed a step. It (#80) should have read:

We have 1/3 = 0.33333 recurring;
0.333(recurring) * 3 => 0.99999 (recurring);
while 1/3 * 3 => 3/3 =>1 (exactly).

And this leads onto defining a suitable long division for each case.

Philip

Heh, from everything I read I concluded that definition 0.99999…=1 itself is absurd because it all boils down to quantisation of principal for counting for any integer radix. In essence there can only exist integer numbers because dividing 1 allready means that that so called selected 1 is only group of smaller individual even integer units that represent quanta of one unit that holds meaning for practial use.
I may offend every mathematition in the world, but whole math always operates (sum,sub,multi,div)only with integer numbers, its only result that may seem noninteger, all that left is to find smallest (whith what we can practialy operate to put the result for use) radix, where radix can be n->infinity. Maybe someone can prove otherwise.

the answer for 0.9… = 1 stands in ONE question:
WHAT IS BLUE?

now if you prefer 0.9… questions, here we go:
what is water, cat, tree, air, oil, words, egg, chicken, man, what, where, when, how, who, W aka double you,

one there’s just one, but dead can’ t see.

between any two real numbers we can find infinitely many real numbers. But here between .999999999… and 1 we cannot find a number . therefore 0.999999…=1
3.499999999…=3.5
6.899999…=6.9

[…] if you’d like to look further, a good first step is here on Wikipedia or here on Better-Explained  Have […]

I’m not a mathematician so excuse me if this comment appears naive or irrelevant.
How can any infinite number (eg. 0.3recurring) have any mathematical process applied to it when the concept of infinity is beyond human comprehension? Surely a person can only apply a mathematical process to a number of which it knows its totality.

@Joseph: No worries, all questions are welcome! That’s a great question – personally, I’m not sure it’s easy to put things like infinity into a category of things we cannot understand. Concepts like zero, negatives, imaginary numbers, etc. befuddled us for a while (are are extremely unnatural), but we managed to make sense of them. Similarly, there are ways to think about infinity but we may not have found the simplest explanations for them yet :).

Numbers like .333… are only infinite because we count in terms of tens (there is no simple way to represent 1/3 when your counting system is based on 10). But, the babylonians counted in terms of 60 (how we have 60 seconds in a minute, and 60 minutes in an hour!) and when counting with "60"s you can get 1/3 without an issue (and no repeating decimals).

So, I think sometimes our confusion is the result of using an awkward system vs. a limit to our understanding.

Thanks Kalid. That’s a perspective I hadn’t considered, and one to which I’m going to give some thought.

It can possible to find the answer if we device a method to
square,cube,etc. 0.9999… .

If it tends to Zero it should be less than One. If it does not it is equal to One.

Also,we can only logically understand a number if we know its neighbors(Yes even Numbers succumb to relativity!).

Lets call 0.99999… “p” [ “p” for promisingly progressive]

And lets call 0.000…0001 “i” [ “i” for infinitesimally interesting] :stuck_out_tongue:

Now trying to find the “left side neighbor” of p.

lets call it “n”.

p - (1-p) = n

2p - 1 = n

1 - p    = i

[Let me be frank over here… i has to be non zero for n to exist AND n has to exist if we want p to make sense in our realm of rationalization]

(1-p)^2 = 1-2p+p^2 ( If we assume 1=p)

(1-p)^2= 0…that would help if we want to allow to make i^2=0 too. Which will not help as mentioned above.

Thus more helpful(logically) than not is that:

  ________________
 | 0.99999.... < 1|
  ----------------

the multiply/substract with infinite numbers is too difficult for my brain

let’s try it with the first finite decimal (where those operations are defined):
n = 1 (. 10) n = 10 (- 9) n = 1
n = 0,9 (. 10) n = 9 (- 9) n = 0
therefore 1 = 0

multiplying an infinite number is not defined, you either fix it and get a quantity 10 times larger then it’s all finite numbers otherwise the ‘multiplied’ result is another infinite number that you can’t compare and meaning is lost.

if it’s about limits and approximations then it should be written explicitly, not using an equal sign, right ?

The number of replies here, without resolution of determination… imply “absolute truth” is a consequence.

The consequence is not absolute.

If it were… there would be determination of absolute truth prior to absoloute truth being determined…

I post because somewhere else i asked what progress actually meant. and so far as close as i can get… it includes point of reference.

If the points of reference are different in consequence, then progress can be anything… it depends on the points of reference and the consequence.

My idea is the closest positive number to 0 is: 0.0000…01
It would have to include a 1 since our decimal number system we start counting with a 1 and not some other number. This seems to be the only way to represent the infinetly small. I don’t think
0.0000…1 is the same as 0, since is 0 the same as 0.0000…1? It is if to say since integer 1 is so close to zero in our counting system, that it is practically 0 when you consider the infinity of integers.

The two closest numbers to 1 would be:

higher = 1.0000…1
lower = 0.9999…9

If you take the average of the two values above, you get 1.0

Please leave some comments.

(continued)

To say 0.9999…9 = 1 is like saying 1.0000…1 = 1
and therefore you are saying 0.9999…9 = 1.0000…1
which are the closest numbers to 1.0, but their difference
is 0.0000…2 which is greater than the smallest number or
infinitely smallest number of of 0.0000…1

Those doubting Thomases or less gifted ignoramuses or devil-may-care Johns who question the veracity of the elegant equation .99999999…=1 would not have the derring do’s to write 1/3 = .33333…
whereupon their wisdom of writing such equation gone?
Accept it or deny it but you can not ignore the charm of it.

R.Kesavan.
kesavan7777@yahoo.com

This is confusing for me and my type of people. The comments that Chad Groft wrote just might be true you never know whose right and whose wrong. IF ANYONE CAN TECH ME HOW THIS PROBLEM WORKS pleas let me know bye=)?

Hey Kalid I read this somewhere
1/3=.33333333333333…
2/3=.66666666666666…
1=1/3+2/3=.999999999999999999…
how do you explain this