Thread: Does 0.9 repeated = 1?

0.9 still has 0.1 left before 1.
0.99 still has 0.01 left before 1.
0.999 still has 0.001 left befor 1...........
0.9999999999999999999999999 still has 0.0000000000000000000000001 left before 1.
0.(9*n) still has 0.(0*n)1 left before 1

is that enough proof that 0.9 reoccuring does not equal 1?
of course for a machine it is impossible for it to display an infanite amount of nines, so of course it will round it to 1, but in real life, it has to be infanate, just like numbers itself is infanate.

of course for a machine it is impossible for it to display an infanite amount of nines, so of course it will round it to 1, but in real life, it has to be infanate, just like numbers itself is infanate.

Obviously you are thinking in terms of a calculator, not in terms of calculus.

is that enough proof that 0.9 reoccuring does not equal 1?

No, because in the case of an infinite sucession of 9s, it will be 0.0*(infinity)1 less than 1; 0.0*(infinity)1 is 0, so it is equal to 1, ie. no difference at all.

Originally Posted by spockman

Obviously you are thinking in terms of a calculator, not in terms of calculus.

did you not read the rest of my post?

Given that 1 = 0.999~, then
1 = 2 - 0.999...
= 1.000...(∞)...0001

Therefore the zeros in 1.000... are (can be) just as funky as the nines in 0.999...

But I think the main point of all this is that when you add repeating 9s to a 0.9 its not really a decimal number any more. It's series written using the decimal system, in order to convey what the actual number (at the end of an infinite series) is. Hence the endless comparisons with 0.33...

did you not read the rest of my post?

I didn but it doesn't implement the rules of calculus, rather it denies basic mathematic principles.

.9~=1. It's been proven without a shadow of a doubt. There is not the slightest room for debate or argument for something that has been proven and is mathematical law.

Originally Posted by RedfishBluefish

Given that 1 = 0.999~, then
1 = 2 - 0.999...
= 1.000...(∞)...0001

Therefore the zeros in 1.000... are (can be) just as funky as the nines in 0.999...

Yes, the 1 at the end of infinity is part of the same paradox. But infinite 0's without the 1 at the end of infinity is not paradoxical.

Originally Posted by RedfishBluefish

But I think the main point of all this is that when you add repeating 9s to a 0.9 its not really a decimal number any more. It's series written using the decimal system, in order to convey what the actual number (at the end of an infinite series) is. Hence the endless comparisons with 0.33...

I am glad you get the paradox, and I think you have the best explanation we have seen in here yet. What we might be dealing with is a flaw in our mathematical language system.

Originally Posted by Universal Mind

No. Again, the 0's do not approach a number forever.

Yes they do. They approach 1.000 ... 1

But, since there are no numbers in between those two, they can be said to be the same number.

Just like with .999... and 1. There is no number you could put in between the two, because they are the same number. There are no boundaries between them to distinguish them as separate numbers. This is reflected in mathematics.

Understand?

Originally Posted by A Roxxor

Yes they do. They approach 1.000 ... 1

Did you read my last post? I explained the distinction between two written forms of 1.

Originally Posted by A Roxxor

But, since there are no numbers in between those two, they can be said to be the same number.

Originally Posted by A Roxxor

Just like with .999... and 1. There is no number you could put in between the two, because they are the same number. There are no boundaries between them to distinguish them as separate numbers. This is reflected in mathematics.

Understand?

There is an infinitely small boundary between them. If two rectangles share a side, there is nothing separating them, but they are still separate rectangles. The fact that nothing separates two numbers does not prove that they are the same number.

Your point does not explain how the infinite series of 9's can lead to a whole number as if there is something on the other side of infinity, which is the bizarre concept that comes with the fact that the two numbers are the same number.

But he presents a good argument. How can two numbers be different numbers if there are no numbers in-between them?

two numbers can be different if there is no number inbetween them.
that would mean that the number before 0.9..... equals 0.9..... since there is no number between them. and the number before that equals that, and so on and so on. and that would mean that every number equals the same thing.

Originally Posted by spockman

But he presents a good argument. How can two numbers be different numbers if there are no numbers in-between them?

Did you see what I said about rectangles that share a side?

Besides, even if he gave us another proof that the two numbers are equal, it does not resolve the specific paradox I have been talking about.

I think it was the Greeks who defined equal numbers as meaning 'there existing no number greater than one number and less than the other'. It's basically axiomatic.

It was further explored by Cantor and is a matter of analysis, which is ridiculously complicated.

ive just had a think about this, and ive changed my mind on this, i think 0.9 reoccuring does infact equal 1, this is my reson:

first of all, 1/9 = 0.1 recurring

and 9/9 = 0.1 recurring * 9

9/9 = 1
0.1 recurring * 9 = 0.9 recurring

so 9/9 = 0.9 recurring and 9/9 = 1

therefore 0.9 recurring = 1


i have no idea if i wrote that right, i was trying to make it understandable, but i might have lost track myself. but i hope you understand what im getting at.

Originally Posted by slash112

ive just had a think about this, and ive changed my mind on this, i think 0.9 reoccuring does infact equal 1, this is my reson:

first of all, 1/9 = 0.1 recurring

and 9/9 = 0.1 recurring * 9

9/9 = 1
0.1 recurring * 9 = 0.9 recurring

so 9/9 = 0.9 recurring and 9/9 = 1

therefore 0.9 recurring = 1


i have no idea if i wrote that right, i was trying to make it understandable, but i might have lost track myself. but i hope you understand what im getting at.

How old are you?

Originally Posted by drewmandan

How old are you?

why the hell does that matter, but im 16.7 recurring years old

Originally Posted by slash112

two numbers can be different if there is no number inbetween them.
that would mean that the number before 0.9..... equals 0.9..... since there is no number between them. and the number before that equals that, and so on and so on. and that would mean that every number equals the same thing.

There are an infinite amount of numbers between any two given numbers.

There is an infinitely small boundary between them. If two rectangles share a side, there is nothing separating them, but they are still separate rectangles. The fact that nothing separates two numbers does not prove that they are the same number.

Yes it does.

In order to get 1 from .9... You have to add .0...1 to it, i.e. Zero. Therefore, they are the same number.

Your point does not explain how the infinite series of 9's can lead to a whole number as if there is something on the other side of infinity, which is the bizarre concept that comes with the fact that the two numbers are the same number.

No.

There is nothing on the other side of infinity, or anyhting like that, .9... is just an alternate way of writing 1. Just like 2/4 is the same number as 1/2 and 4/8. They are interchangeble, thus they are the same number.

Did you see what I said about rectangles that share a side?

Besides, even if he gave us another proof that the two numbers are equal, it does not resolve the specific paradox I have been talking about.

Your analogy isn't valid because there is no border at all. A shared side is a border.

What paradox?

Originally Posted by A Roxxor

There are an infinite amount of numbers between any two given numbers.

Perhaps that is not always true.

Devil's advocate there. I am convinced that 1 = 0.999..., and my part in this thread isn't about whether or not the two figures are equal. I am just saying that it is a paradox that has not been completely explained.

Originally Posted by A Roxxor

Yes it does.

In order to get 1 from .9... You have to add .0...1 to it, i.e. Zero. Therefore, they are the same number.

What is 0.00000...1? The figure suggests that there is something on the other side of infinity. Does it not? It definitely suggests that there is something, which is not nothing. That is part of the paradox. Yes, your math is accurate, but notice the paradox.

Originally Posted by A Roxxor

No.

There is nothing on the other side of infinity, or anyhting like that, .9... is just an alternate way of writing 1.

You are just reasserting your conclusion and not dissecting the paradox. I am calling into question certain aspects of the fact that the two figures are equal, not asking you to state again that they are equal. Telling me that one is a way of writing the other adds nothing to your conversation with me except a reassertion of the fact that I am trying to dissect.

Originally Posted by A Roxxor

Your analogy isn't valid because there is no border at all. A shared side is a border.

The border has no width, yet it is a border. The border between the rectangles is just as wide as the border between 0.999... and 1?

Devil's advocate again.

Originally Posted by A Roxxor

What paradox?

You are now officially on my troll watch list.

Just in case you or somebody else STILL doesn't get the paradox, I will explain it another way. And I am not calling into question whether the two figures are equal, so don't waste your time by telling me they are just two ways of writing the same number. That fact is exactly what I am trying to explain, not counter. Understand?

Okay, imagine the number 0.999... written on a piece of paper that goes forever. With every next digit, the number represented up to that digit is a little closer to 1. So how far along the number is there a point when the digits up to that point equal 1? A trillion light years? Quadrillion to the octillionth power light years? It never happens, ever, obviously. Infinity has no end. So the number can never reach 1, ever. Right? So the stretched out number that never can possibly get to 1 gets to 1 because that is the number that the entire number is. It can't reach 1, but it reaches 1.

Considering the very specific issue I have raised, why are the two figures representative of the same number, which they are? And please don't tell me what a converging geometric series is. That is the very thing I am calling into question.

Originally Posted by A Roxxor

There are an infinite amount of numbers between any two given numbers.

well, if the number already has a decimal place of infinity, then there cannot be numbers inbetween them.

It will never reach 1 if you try to write it out on paper because you will always end up writing a finite number of 9s. But 0.999~ does not represent a finite number of 9s, it represents an infinite number of 9s.

Perhaps that is not always true.

If you have any two real numbers which are not equal, then yes, there will be an infinite number of reals between those two numbers. You can't even count them (I can demonstrate if you want, it's quite interesting; there are infinity natural numbers, but there is an even greater infinity of real numbers).

Originally Posted by Xei

It will never reach 1 if you try to write it out on paper because you will always end up writing a finite number of 9s. But 0.999~ does not represent a finite number of 9s, it represents an infinite number of 9s.

If you have any two real numbers which are not equal, then yes, there will be an infinite number of reals between those two numbers. You can't even count them (I can demonstrate if you want, it's quite interesting; there are infinity natural numbers, but there is an even greater infinity of real numbers).

is there such a thing as infinity plus 1? if there is no such thing, then there are no numbers inbetween 0.9 recurring and 1.

what exactly are the numbers inbetween 0.9 recurring and 1? just name one for me. keeps in mind that infinity plus something probably doesnt exist, so dont say something like, 0.(9*infinity + 1)

That's the point. They are the same number because there are no numbers inbetween.

Originally Posted by Kushna Mufeed

That's the point. They are the same number because there are no numbers inbetween.

thats not the reason why i think 0.9 rec. = 1, i was just explaining that there are no numbers inbetween. i think this has nothing to do with it. but i explained in a few posts ago why it must be.

Originally Posted by Xei

It will never reach 1 if you try to write it out on paper because you will always end up writing a finite number of 9s. But 0.999~ does not represent a finite number of 9s, it represents an infinite number of 9s.

Can infinity be reached?

The 9's on the piece of paper never get to a point where the number has a value of 1, yet they do.

Originally Posted by Xei

If you have any two real numbers which are not equal, then yes, there will be an infinite number of reals between those two numbers. You can't even count them (I can demonstrate if you want, it's quite interesting; there are infinity natural numbers, but there is an even greater infinity of real numbers).

At least in every case you know of.

Originally Posted by slash112

is there such a thing as infinity plus 1?

No. So, is there such thing as 0.000...1?