Thread: Is 0.999... equal to one?

Originally Posted by roncj5

i would like to thank homo bibiens for asking this question.at first my answer was no.but that didn't sit well.this caused me to research this and learn something new.for this i thank you.this is why i love this forum.it provokes thought(at least in me).bravo!by the way,i would like to change my answer to yes it does,at least on paper anyway.

Hi roncj5,

Glad to have you on board You shouldn't feel badly about this, some similar issues troubled mathematicians for a long time. The modern convention is that 0.999... is defined as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, etc., and there does not exist a number that is in any sense "infinitely close" to one without being equal to one. People have developed other number systems with numbers like this (for example, numbers smaller than any positive real number, but still greater than zero), but there is a price to be paid - specifically, some of the ordinary rules of arithmetic must be broken. It is impossible to invent such numbers without breaking at least some of the familiar rules of math.

Cheers,

Hb

Originally Posted by Homo bibiens

Hi roncj5,

Glad to have you on board You shouldn't feel badly about this, some similar issues troubled mathematicians for a long time. The modern convention is that 0.999... is defined as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, etc., and there does not exist a number that is in any sense "infinitely close" to one without being equal to one. People have developed other number systems with numbers like this (for example, numbers smaller than any positive real number, but still greater than zero), but there is a price to be paid - specifically, some of the ordinary rules of arithmetic must be broken. It is impossible to invent such numbers without breaking at least some of the familiar rules of math.

Cheers,

Hb

I still don't agree, after reading through this thread I can see the logic behind what is being said but I still feel its more of an expression of the language of the math than what is real. Infinity is not a "real" number for use in any practical terms but rather an expression used by the person formulating. In a sense its a bit of a cheat, a slight of the hand. The term "infinity" can be defined individually by each user to mean different things. Its a bit like the blank letter counter in a game of scrabble, it can be = to nothing or = to everything. But for all practical purposes yes 0.99999 ~ can be regarded as = 1. For example if you were use this system to accurately measure something then the difference between 1 and 0.999~ would be smaller than the Planck scale so would have no real meaning.

Dmw

Infinitely small and nonexisting are mutually exclusive. Something is either infinitely small or none existant.

if I/infinity = 0, then 1/infinity(infinity) = 0(infinity), then 1 = 0. right?

Originally Posted by tom

One other note ...
There are two systems for a lack of better word although related they are different.

There are integers and real numbers. Integers are quantized ( sp? ) while real numbers are not.

In the "system" of integers there are not really limits ... as you are either exactly at 0 or exactly at 1.

In the system of real numbers I would say that 0.0000 ... 1 = 0 just as 1/infinity = 0

There are more systems then you can imagine.
Integers can be seen as a subset, containing the only the whole numbers, of real numbers. Other subsets can be quantitized if they are so defined.

Originally Posted by Max Caley

Let x = 0.999.... then 10x = 9.999... and subtracting gives

10x - x = 9.999... - 0.999...

So 9x = 9

You can argue as much as you want about possible solutions of 9x = 9, but in any reasonable number system (field) x = 1 is a solution (by definition of identity).

Hence, x = 0.999... = 1

You can't do maths by a poll, logic isn't governed by democracy!

9x0.9 = 8.1; 9x0.99 = 8.9; 9x0.999 = 8.999; You can do this forever without ever reaching 9.

10x-x > 9.999... - 0.999...; It is one decimal place off. Try it in scientific notation and remember infinity -1 < infinity, it is safe to say by 1. (...) with nothing following is deceiving. Not all ifinities are created equal. Don't ask Homo bibi. He says he avoids arithmetic and from what he's written it's for the best.

Originally Posted by Homo bibiens

I am very sorry you disagree with yourself:



Self-contradiction seems to be a very common property of many of the arguments I have seen in defense of one side of this argument at other boards, although this particular one, I have not seen before.



Different from what you refer to as 0.000....1? If that is what you mean, then I decline your offer to "fix" my number system, I prefer the one I have right now.

If 0.000...1 is different than zero, does it obey all the usual rules of arithmetic that the real numbers obey? (The reason I ask is related to the above discussion about self-contradiction; if someone invents a number system with contradictory properties, then I cannot agree that they are as right as everyone else.)



I am glad you say elsewhere "Clarify the definitions", because I think definitions are extremely important in this sort of thing. Can you clarify your definition of 0.000...1? Specifically, what is the "sequence" to which you refer? Is it 0, 0.0, 0.00, 0.000, 0.0000, etc., or 1, 0.1, 0.01, 0.001, 0.0001, etc., or something different from either of these?

But either way, my usual definition of the real number zero is a particular equivalence class of Cauchy sequences. There are other ways of doing it, but since all totally ordered complete fields are isomorphic, they get us to the same place.



Same as above. This is not my definition of the real number one. I am not sure what your definition is.



Real numbers are usually defined as limits. (As per above, there is at least one other way to do it, but it gets you the same end result.) Are you arguing that the real number one is not equal to the integer one? I have no problem defining equality between those two, and as per one of your two opinions on the matter, I am as right as everyone else.

Thank you once again for your kind offer, but the one hint about infinitesmal numbers causes me to want to stick with the number system that I have. If I have misunderstood Bruinekool-numbers, a very rigourous definition would be most helpful in setting things straight.

You do not seem to be able to detect contradictions and inconsistancies within your own writing. What makes you think you can detect it in that of others?
1-0.999... = 0.000...1: It can only be equal to zero by mutal agreement. I offered a knowledgible agreement. You declined. Logic is apparanly not your long suite. I do not expect you to be able to understand.
If you think that real numbers are finite, you need all of the help that you can get.

Originally Posted by Homo bibiens

Hi roncj5,

Glad to have you on board You shouldn't feel badly about this, some similar issues troubled mathematicians for a long time. The modern convention is that 0.999... is defined as the limit of the sequence 0.9, 0.99, 0.999, 0.9999, etc., and there does not exist a number that is in any sense "infinitely close" to one without being equal to one. People have developed other number systems with numbers like this (for example, numbers smaller than any positive real number, but still greater than zero), but there is a price to be paid - specifically, some of the ordinary rules of arithmetic must be broken. It is impossible to invent such numbers without breaking at least some of the familiar rules of math.

Cheers,

Hb

I once had a philosophy professor who was so smart that he took a run around the dog house so fast that he ran right smack into himself. After he revived he didn’t know who he had run into, but he did know that he was wrong. We were all forced to agree that he was right, he was wrong, but he never knew it. To this day he thinks he was wrong, he was right.

0.999... Is an open ended phrase, written in a self limiting notation. It has little if any meaning. What number when subtracted from 0,999... would equal 0.999... ending with the numeral 8 instead of 9? How would you write such a number using your beloved notation? Would you contend that such a number does not exist?

0.999...9 - 0.000...1 = 0.999...8 Weeee, wasn't that fun?

Rewrite that in your limited notation. 0.999... is a trick phrase. It is used to confuse and deceive. It eliminates infinite possibilities and generates nothing but circular arguments.

Does 0.111... = 0.112... or 0.110...? What is the limit of 0.111...?

Originally Posted by Wayne Bruinekool

I once had a philosophy professor who was so smart that he took a run around the dog house so fast that he ran right smack into himself. After he revived he didn’t know who he had run into, but he did know that he was wrong. We were all forced to agree that he was right, he was wrong, but he never knew it. To this day he thinks he was wrong, he was right.

0.999... Is an open ended phrase, written in a self limiting notation. It has little if any meaning. What number when subtracted from 0,999... would equal 0.999... ending with the numeral 8 instead of 9? How would you write such a number using your beloved notation? Would you contend that such a number does not exist?

0.999...9 - 0.000...1 = 0.999...8 Weeee, wasn't that fun?

Rewrite that in your limited notation. 0.999... is a trick phrase. It is used to confuse and deceive. It eliminates infinite possibilities and generates nothing but circular arguments.

Does 0.111... = 0.112... or 0.110...? What is the limit of 0.111...?

Lol the professor bit made me chuckle, it took me a couple of takes to get it!

Yes Wayne i'm inclined to agree with you on this.

Hahahaha,owwwch...my brain hurts,even though that's impossible...I love paradoxes.

Originally Posted by William E. Davenport

Even though .999.... is not a counting number while 1 is. If equality is only concerned about the property of value then your statement is true, but for complete equality it is false.

Equality, in math, is usually only concerned with value. For instance, 4/3 is clearly not the same thing as 8/6, but they are equal, in math, because they have the same value.

Your objection about the counting numbers vs. non-counting numbers would hold for 1 and 1.0 as well. I hope you would not be arguing that 1 does not equal 1.0?

"Is 0.999... equal to one?" is too vague for a definitive answer.

I would say we can reasonably limit it to just the consideration of value. So, it seems that it is your opinion that they are equal in value then?