People always get their panties in a bunch over this for some reason. But when you throw the Banach-Tarski paradox at them there never seems to be a counter-argument. Or, hell, an actual response.

Here's the thing: your brain sucks with infinities. I'm sorry, but it's true. When you start working with infinities all sorts of weird and counter-intuitive things start happening. But that's the beauty of math, it has to be logically consistent and it is.

Within the set of real numbers 0.999... is a short-hand notation for the following sum:

Stop trying to think of its value in terms of its partial sum. 0.999... is not "approaching" 1, it is exactly 1, because it can only have one single value. The only real number that makes sense to assign to that sequence in 1. Let's do a bit of a proof by contradiction, it won't really be a proof, but hopefully it should illustrate something:

First let me introduce a new notation: 0.9{n}. I will use this to indicate a decimal number with n 0s after the decimal point. As a corollary this is also true: 0.9{n} = 1 - 10^{-n}, so we can actually extend this to negative values for n, weird though that may be.

It should be clear that 0.9{n} < 0.999... for all natural numbers n.

Now, assume that 0.999... < 1. This is equivalent to saying that 0.999... = 1 - ε for any ε > 0.

Let's consider for a moment the equation: 0.9{n} > 1 - ε

Now we can rewrite this as: ε > 1 - 0.9{n}
ε > 10^-n

Now comes the actual argument: no matter how small we make ε, there will always be a finite integer n that will satisfy that the above inequality. That means that for every real number less than 1, there is a finite n that makes 0.9{n} bigger than that real number. Therefore, 0.9{n} is greater than every real number less than 1. So, 0.9{n} cannot be less than 1, because it is greater than every real number less than 1. So basically, this proves that 0.9... >= 1 (should be evident from here that it must be equal to 1).

This is essentially the same argument put forth earlier with the number line. Two real numbers are distinct if and only if there exists a real number between them.

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Never attribute to malice what can adequately be explained by incompetence.

I think you guys are just overthinking it, entirely.

0.9 repeating is 0.9 repeating.

1 is 1.

Plain and simple, they aren't the same, they aren't equal. Rather than thinking "hey, this math is flawed" you think "0.9 is equal to 1, because math can never be wrong!"... really?

Maybe the math we have come to understand only works to a reasonable extent before it falls apart, kind of like how the theory of relativity doesn't really work out so well with quantum mechanics, but is true for everything else.

They are not equal. They are mathematically equivalent under certain conditions. I dare you to go to class today and use .9 repeating in place of 1. I ****ing dare you. Then come back and tell me it's equal to one.

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My name is Ozymandias, king of kings. Look upon my works, ye mighty, and despair.

This whole argument is a bit ridiculous if you ask me. Just using logic, wouldn't the answer be:
0.999999... = 1 - (1/infinity) = 1 - (1/9999999999999999....)
= 0.9999... ? Just a thought

This whole argument is a bit ridiculous if you ask me. Just using logic, wouldn't the answer be:
0.999999... = 1 - (1/infinity) = 1 - (1/9999999999999999....)
= 0.9999... ? Just a thought

We're not talking about some value that approaches zero while another variable approaches infinity. We're talking about using the concept infinity itself here (because numbers are absolute, not equations) which makes it absolutely zero.

Which is the point I made earlier, or was taught, at least. There is no single number that is both smaller than any other number and non-zero. There is no 0.00000...1, because by the definition of infinity, there is no place for the one after the zeros.

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I'd like to think of myself as intelligent, but that, given the sheer variety and vastness of the cosmos that surrounds me, it is readily apparent that this is not the case.

We're not talking about some value that approaches zero while another variable approaches infinity. We're talking about using the concept infinity itself here (because numbers are absolute, not equations) which makes it absolutely zero.

Irrational numbers aren't absolute lol. And infinity is, as you said, a concept, not a number. Therefore, it and any equation involving infinity is not absolute but a concept

.999999... is not equal to one. It is approximately equal to one. It is called .9 repeating for a reason. Not one. Assuming you used some sort of calculator, the calculator rounded the result. The repeating is symbolic of being almost equal to the next number, but it isn't. Ususally, it is used in numbers that our base 10 number system cannot define truly. In base 3, one third is .1. Base 9, .3. In base 10, since 10 is not an multiple of 3 (3, 6, 9, 12, etc.), it must be represented as .33333333333... I think you get the idea.

BTW I didn't watch the video. The text gave me the impression that you were falsely proving something that cannot be proven, since it is false.

What you have to understand is that repeating number sequences are like that because they cannot be adequately defined in our base 10 number system. (This doesn't mean pi or e or anything, just point whatever repeating.)

Which is the point I made earlier, or was taught, at least. There is no single number that is both smaller than any other number and non-zero. There is no 0.00000...1, because by the definition of infinity, there is no place for the one after the zeros.

You mean, there's no single number comprehensible to the human mind that is smaller than any other number and non-zero.

It surely exists, just the concept of infinity is so hard for us to grasp that we really can't comprehend it.

Irrational numbers aren't absolute lol. And infinity is, as you said, a concept, not a number. Therefore, it and any equation involving infinity is not absolute but a concept

Yes they are. Even if you can't express them nicely (at least in base 10) they are still a single value. There are no variables involved, just a constant.

Edit: That's not to say 0.999... is even irrational. An irrational number is a non-terminating, non-repeating decimal number that cannot be written as a fraction. Because it repeats, we know that it is in fact rational and can be written as a fraction, which 3/3 (or just 1) satisfies.

We're not dealing with an equation, but a single number. 0.999... is also just a concept.

BTW I didn't watch the video. The text gave me the impression that you were falsely proving something that cannot be proven, since it is false.

That's actually what you're doing right now. This is a settled matter. Any mathematician will tell you that 0.999... is equal to 1 in much the same way that "three thirds" is.

That's actually what you're doing right now. This is a settled matter. Any mathematician will tell you that 0.999... is equal to 1 in much the same way that "three thirds" is.

From the very same wikipedia site:

...but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.

What you have to understand is that repeating number sequences are like that because they cannot be adequately defined in our base 10 number system. (This doesn't mean pi or e or anything, just point whatever repeating.)

It still baffles me how many non-mathemeticians out there are willing to so adamantly proclaim that something that basically every mathematician agrees is true (and for which there are a myriad of mathematical proofs) is actually not true. Go read the wikipedia article about it.

0.(9) is another way of writing 1. Both notations represent the same value. Some people have a hard time coming to grips with the idea that a value can be written in different ways, but you shouldn't The value 1/2 can also be written as 0.5 or 2/4.

The difference between 0.(9) and 1 would be something called an infinitesimal. In the set of real numbers, there are no infinitesimals that are not equal to zero. Since the difference between 0.(9) and 1 is equal to 0 in real numbers, that means they are equal by definition.

Get it through your heads. It's not an approximation or a rounding trick. 0.(9) is 1. Every integer other than 0 can be written, in decimal notation, either as the integer itself, or as it's absolute value minus one with an infinite number of 9's after a decial point.

Oh do tell me which number systems those are and what makes them different from the real numbers line. I'm going to guess you know less about number and set theory than I do, so have fun delving into bits of math the implications of which you haven't the slightest idea about.

Also good job ignoring the entire rest of the article and paying attention to only the one thing on there that makes you right, even though you know nothing of the conditions regarding this correctness.

It's talking about number systems that are not real numbers. Unless you're working in some obscure field of mathematics, You're using real (or maybe imaginary) numbers. There are some types of number theory that recognize non-zero infinitesimals, for example, where the notation 0.(9) represents a value that is not equivalent to 1. The difference here isn't that the math changes, the difference is that the notation itself 0.(9) represents a different value in that numbering system than it does in real numbers.

Okay, okay, I'm wrong. Sorry for expressing my (sadly) inaccurate opinion.

You were not expressing an opinion, you were expressing an untruth. Math is the single most objective field of knowledge there is. It is not open to opinion, merely proofs.

Which is the point I made earlier, or was taught, at least. There is no single number that is both smaller than any other number and non-zero. There is no 0.00000...1, because by the definition of infinity, there is no place for the one after the zeros.

Obviously in terms of 1 and 0.9 repeating, though. (Forgive me if I keep referring back to the 0 and 0.0000...1 example, I just like that example better.)

Now, why is it assumed that since it is infinitely repeating, it does not exist? Humans might not be able to comprehend such a number, but it definitely does exist. We can't just say "Oh, the number is too difficult to figure out... so it's not that number!" seems like a kind of silly way around it.

Also, why is 0.9 repeating not the number nearest to one? Just because it expands infinitely doesn't mean it's equal to the next greatest number. Sure, there might not be any "physical" number to represent the number nearest to one; by physical number, I mean a number that we can actually see from start to finish. Something like 0.9999.

However, 0.(9) is saying it's repeating forever, infinitely. 0.(9)is that infinitely repeating number that inches closer and closer to 1 but never reaches it. It's not a number we can actually observe, because it extends forever. 0.(9) is the nearest number to 1. This "infinite incomprehensible number" that is supposedly non-existent in the space between 1 and 0.9(8), is nicely shortened into the convenient 0.(9).

Sorry, this is a bit of a difficult concept to explain, when discussing concepts like infinity, it isn't too easy to find the right words to explain the point I'm making.

EDIT: I'm going to bring this point back again as well. Why do people assume that because mathematical equations say they're equal, they're equal? What if the math we know only works to a certain extent, like how relativity essentially fails on a quantum level. It's possible that the mathematics are flawed.

Here's the thing: your brain sucks with infinities. I'm sorry, but it's true. When you start working with infinities all sorts of weird and counter-intuitive things start happening. But that's the beauty of math, it has to be logically consistent and it is.

Within the set of real numbers 0.999... is a short-hand notation for the following sum:

http://www.wolframalpha.com/input/?i=sum 9/10^n for n from 1 to infinity

Stop trying to think of its value in terms of its partial sum. 0.999... is not "approaching" 1, it is exactly 1, because it can only have one single value. The only real number that makes sense to assign to that sequence in 1. Let's do a bit of a proof by contradiction, it won't really be a proof, but hopefully it should illustrate something:

First let me introduce a new notation: 0.9{n}. I will use this to indicate a decimal number with

n0s after the decimal point. As a corollary this is also true: 0.9{n} = 1 - 10^{-n}, so we can actually extend this to negative values for n, weird though that may be.It should be clear that 0.9{n} < 0.999...

for all natural numbers n.Now, assume that 0.999... < 1. This is equivalent to saying that 0.999... = 1 - ε for any ε > 0.

Let's consider for a moment the equation: 0.9{n} > 1 - ε

Now we can rewrite this as: ε > 1 - 0.9{n}

ε > 10^-n

Now comes the actual argument: no matter how small we make ε, there will always be a

finite integern that will satisfy that the above inequality. That means thatfor every real number less than 1, there is a finite n that makes 0.9{n} bigger than that real number. Therefore, 0.9{n} is greater thanevery real number less than 1. So, 0.9{n} cannot be less than 1, because it is greater thaneveryreal number less than 1. So basically, this proves that 0.9... >= 1 (should be evident from here that it must be equal to 1).This is essentially the same argument put forth earlier with the number line. Two real numbers are distinct if and only if there exists a real number between them.

0.9 repeating is 0.9 repeating.

1 is 1.

Plain and simple, they aren't the same, they aren't equal. Rather than thinking "hey, this math is flawed" you think "0.9 is equal to 1, because math can never be wrong!"... really?

Maybe the math we have come to understand only works to a reasonable extent before it falls apart, kind of like how the theory of relativity doesn't really work out so well with quantum mechanics, but is true for everything else.

My name is Ozymandias, king of kings. Look upon my works, ye mighty, and despair.

0.999999... = 1 - (1/infinity)

= 1 - (1/9999999999999999....)

= 0.9999...

? Just a thought

Check out my GitHub and Website!

Mostly moved on. May check back a few times a year.

It is infinitely small, but not 0

Check out my GitHub and Website!

iszero.We're not talking about some value that approaches zero while another variable approaches infinity. We're talking about using the concept infinity itself here (because numbers are absolute, not equations) which makes it absolutely zero.

Mostly moved on. May check back a few times a year.

Which is the point I made earlier, or was taught, at least. There is no single number that is both smaller than any other number and non-zero. There is no 0.00000...1, because by the definition of infinity, there is no place for the one after the zeros.

Irrational numbers aren't absolute lol. And infinity is, as you said, a concept, not a number. Therefore, it and any equation involving infinity is not absolute but a concept

Check out my GitHub and Website!

approximatelyequal to one. It is called .9 repeating for a reason. Not one. Assuming you used some sort of calculator, the calculatorroundedthe result. The repeating is symbolic of beingalmostequal to the next number, but it isn't. Ususally, it is used in numbers that our base 10 number system cannot define truly. In base 3, one third is .1. Base 9, .3. In base 10, since 10 is not an multiple of 3 (3, 6, 9, 12, etc.), it must be represented as .33333333333... I think you get the idea.BTW I didn't watch the video. The text gave me the impression that you were falsely proving something that cannot be proven, since it is false.

What you have to understand is that repeating number sequences are like that because they cannot be adequately defined in our base 10 number system. (This doesn't mean pi or

eor anything, just point whatever repeating.)You mean, there's no single number comprehensible to the human mind that is smaller than any other number and non-zero.

It surely exists, just the concept of infinity is so hard for us to grasp that we really can't comprehend it.

Edit: That's not to say 0.999... is even irrational. An irrational number is a non-terminating,

non-repeatingdecimal number that cannot be written as a fraction. Because it repeats, we know that it is in fact rational and can be written as a fraction, which 3/3 (or just 1) satisfies.We're not dealing with an equation, but a single number. 0.999... is also just a concept.

Mostly moved on. May check back a few times a year.

That's actually what you're doing right now. This is a settled matter. Any mathematician will tell you that 0.999... is equal to 1 in much the same way that "three thirds" is.

From the very same wikipedia site:

0.(9) is another way of writing 1. Both notations represent the same value. Some people have a hard time coming to grips with the idea that a value can be written in different ways, but you shouldn't The value 1/2 can also be written as 0.5 or 2/4.

The difference between 0.(9) and 1 would be something called an infinitesimal. In the set of real numbers, there are no infinitesimals that are not equal to zero. Since the difference between 0.(9) and 1 is equal to 0 in real numbers, that means they are equal by definition.

Get it through your heads. It's not an approximation or a rounding trick. 0.(9) is 1. Every integer other than 0 can be written, in decimal notation, either as the integer itself, or as it's absolute value minus one with an infinite number of 9's after a decial point.

1 = 0.(9)

1534 = 1533.(9)

-256 = -255.(9)

1000000000 = 999999999.(9)

This is the case for every integer other than 0.

Oh do tell me which number systems those are and what makes them different from the real numbers line. I'm going to guess you know less about number and set theory than I do, so have fun delving into bits of math the implications of which you haven't the slightest idea about.

Also good job ignoring the entire rest of the article and paying attention to only the one thing on there that makes you right, even though you know nothing of the conditions regarding this correctness.

It's talking about number systems that are not real numbers. Unless you're working in some obscure field of mathematics, You're using real (or maybe imaginary) numbers. There are some types of number theory that recognize non-zero infinitesimals, for example, where the notation 0.(9) represents a value that is not equivalent to 1. The difference here isn't that the math changes, the difference is that the notation itself 0.(9) represents a different value in that numbering system than it does in real numbers.

You were not expressing an opinion, you were expressing an untruth. Math is the single most objective field of knowledge there is. It is not open to opinion, merely proofs.

What BairSaysHi said,

Obviously in terms of 1 and 0.9 repeating, though. (Forgive me if I keep referring back to the 0 and 0.0000...1 example, I just like that example better.)

Now, why is it assumed that since it is infinitely repeating, it does not exist? Humans might not be able to comprehend such a number, but it definitely does exist. We can't just say "Oh, the number is too difficult to figure out... so it's not that number!" seems like a kind of silly way around it.

Also, why is 0.9 repeating not the number nearest to one? Just because it expands infinitely doesn't mean it's equal to the next greatest number. Sure, there might not be any "physical" number to represent the number nearest to one; by physical number, I mean a number that we can actually see from start to finish. Something like 0.9999.

However, 0.(9) is saying it's repeating forever, infinitely. 0.(9)

isthe nearest number to 1. This "infinite incomprehensible number" that is supposedly non-existent in the space between 1 and 0.9(8), is nicely shortened into the convenient 0.(9).Sorry, this is a bit of a difficult concept to explain, when discussing concepts like infinity, it isn't too easy to find the right words to explain the point I'm making.

EDIT: I'm going to bring this point back again as well. Why do people assume that because mathematical equations say they're equal, they're equal? What if the math we know only works to a certain extent, like how relativity essentially fails on a quantum level. It's possible that the mathematics are flawed.