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He just divided by zero

Needs a lot more crazy to measure up to one of Gene Scott's lectures.

/used to get baked and watch his show late at night in the 80's

Snapper Carr: [img846.imageshack.us image 345x249]

Needs a lot more crazy to measure up to one of Gene Scott's lectures.

/used to get baked and watch his show late at night in the 80's

Same here. Loved his chrome plated "Patton" helmet.

If the donations weren't coming in, he'd just sit there and sulk, and play his damn horse show videos.

What a maroon.

I counted to infinity twice yesterday......

/It was quicker the second time.....

But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

Just because you can't match 1:1 with irrationals:rationals, well, so what -- there's always one more of each. This doesn't seem like math to me. It seems like broken stoner philosophy. But I'm probably missing something.

Anyway, all of this seems like a new-age way to apply some "whoa man" to something that's really arbitrary. Humans invented counting...generally, we count by 1. Just because you split that up in a way (counting by thirds, for example) that makes the general method seem flawed doesn't mean you're not still arguing about arbitrary-ness. The millimeter marks on the top side of my ruler don't lessen the eighth-inch marks on the bottom side.

/but what do I know...English major

Scrotastic Method: But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

Just because you can't match 1:1 with irrationals:rationals, well, so what -- there's always one more of each. This doesn't seem like math to me. It seems like broken stoner philosophy. But I'm probably missing something.

Anyway, all of this seems like a new-age way to apply some "whoa man" to something that's really arbitrary. Humans invented counting...generally, we count by 1. Just because you split that up in a way (counting by thirds, for example) that makes the general method seem flawed doesn't mean you're not still arguing about arbitrary-ness. The millimeter marks on the top side of my ruler don't lessen the eighth-inch marks on the bottom side.

/but what do I know...English major

Interesting. My Granddad had this "Fisherman's Ruler" where one side was a normal ruler but the other side was sort of a "Liar's Ruler" with smaller inches so you could joke that your fish was bigger. The ruler has a set length, but one side simply had more divisions. I guess what you're stating is how math decides which notches in a line to put labels on. But the video stated that they were talking about whole numbers, which is a defined concept. It is a freaky concept, and I'm trying to get my brain around it. Anyway, if you pulled the Fisherman's ruler apart, there was a filet knife inside.

Interesting. Clear. Instructive.

Utterly unconnected to the work of Dr. Gene Scott.

/play I Wanna Know
//I want to know
///I want to know that he will welcome me there

I was going to watch the NeverEnding Story, but really...who's got that kind of time?

busy chillin': I was going to watch the NeverEnding Story, but really...who's got that kind of time?

If what I learned from that video is correct, and it is not, then it would take half as long to watch that movie twice.

Nifty.

Scrotastic Method: But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

Just because you can't match 1:1 with irrationals:rationals, well, so what -- there's always one more of each. This doesn't seem like math to me. It seems like broken stoner philosophy. But I'm probably missing something.

Here's an example of why it might be relevant, to a mathematician at least. Let's say a number is 'computable' if you can feed a computer some program and it spits out the decimal expansion of that number to whatever precision you want. So then all integers are computable (obviously), so are all rationals (since they repeat), also some irrationals (e.g. pi). But there are some irrationals that are not computable, because the number of programs is less than the number of irrationals.

If you have a number that you can't describe unambiguously on a piece of paper or in a computer program, that's makes it hard to do math on that number.

Prefers Dr. Everett Scott

Snapper Carr: used to get baked and watch his show late at night in the 80's

Once or twice anyway.

murray208: Scrotastic Method: But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

Just because you can't match 1:1 with irrationals:rationals, well, so what -- there's always one more of each. This doesn't seem like math to me. It seems like broken stoner philosophy. But I'm probably missing something.

Here's an example of why it might be relevant, to a mathematician at least. Let's say a number is 'computable' if you can feed a computer some program and it spits out the decimal expansion of that number to whatever precision you want. So then all integers are computable (obviously), so are all rationals (since they repeat), also some irrationals (e.g. pi). But there are some irrationals that are not computable, because the number of programs is less than the number of irrationals.

If you have a number that you can't describe unambiguously on a piece of paper or in a computer program, that's makes it hard to do math on that number.

I get that, that makes sense. But this talk was more like, "hey, when we impose these methods of organization on large sets of numbers, sometimes we find things that don't seem to make sense within the constraints of the rules we just imposed."

Advanced math is a lot like philosophy to me...sure, there are neat ideas bumping around, but who can really argue about this kind of stuff when it's all so...ethereal. Is a video like this one ever going to help build anything? Help cure something? And as I mentioned earlier, I'm an English major, so I fully understand the irony of calling out one portion of academia for needlessly berating small and possibly useless-in-the-grand-scheme ideas. :)

Snapper Carr: [img846.imageshack.us image 345x249]

Needs a lot more crazy to measure up to one of Gene Scott's lectures.

/used to get baked and watch his show late at night in the 80's

Not that it really relates to anything, but I always had a huge crush on his widow Melissa. Rawr.

Scrotastic Method: murray208: Scrotastic Method: But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

Just because you can't match 1:1 with irrationals:rationals, well, so what -- there's always one more of each. This doesn't seem like math to me. It seems like broken stoner philosophy. But I'm probably missing something.

Here's an example of why it might be relevant, to a mathematician at least. Let's say a number is 'computable' if you can feed a computer some program and it spits out the decimal expansion of that number to whatever precision you want. So then all integers are computable (obviously), so are all rationals (since they repeat), also some irrationals (e.g. pi). But there are some irrationals that are not computable, because the number of programs is less than the number of irrationals.

If you have a number that you can't describe unambiguously on a piece of paper or in a computer program, that's makes it hard to do math on that number.

I get that, that makes sense. But this talk was more like, "hey, when we impose these methods of organization on large sets of numbers, sometimes we find things that don't seem to make sense within the constraints of the rules we just imposed."

Advanced math is a lot like philosophy to me...sure, there are neat ideas bumping around, but who can really argue about this kind of stuff when it's all so...ethereal. Is a video like this one ever going to help build anything? Help cure something? And as I mentioned earlier, I'm an English major, so I fully understand the irony of calling out one portion of academia for needlessly berating small and possibly useless-in-the-grand-scheme ideas. :)

Sure, this is philosophy of mathematics, so it's math for mathematicians in the same way that there is art for artists. It's related to a lot of foundational questions in mathematical logic. The fact that we find things that don't make sense was actually a big surprise and a tremendous disappointment. Mathematicians used to believe that any well-formed question would be provably true or false.

The value of cantor's proof is more indirect. There is lots of more practical math that is built on top (e.g. Probability theory and statistics are built from Borel sets). Also the example I gave was one of the main results from Turing's paper -- the one where he is credited for inventing the computer.

Saw a good BBC documentary a couple of years ago about Cantor, Gödel, Ludwig Boltzmann, and Alan Turing.
All insanely brilliant. All committed suicide.

"Dangerous Knowledge" Part 1

"Dangerous Knowledge" Part 2

/Total running time 89 minutes, if you're into this kind of stuff

murray208: smart stuff

Thanks. I guess that's what I was getting at: some sort of practicality to be gained from all this. Says me, a guy with -- showing your analogy was spot-on -- tens of thousands of dollars of art in his house. I can't, say, eat that come hard times.

Mr. Potatoass: Saw a good BBC documentary a couple of years ago about Cantor, Gödel, Ludwig Boltzmann, and Alan Turing.
All insanely brilliant. All committed suicide.

"Dangerous Knowledge" Part 1

"Dangerous Knowledge" Part 2

/Total running time 89 minutes, if you're into this kind of stuff

Thanks, I'll be watching those later. Though, Turing's suicide was brought on by a lot more than just madness/genius. That guy got the rawest deal of all raw deals. What a waste; what a loss.

Mr. Potatoass: Saw a good BBC documentary a couple of years ago about Cantor, Gödel, Ludwig Boltzmann, and Alan Turing.
All insanely brilliant. All committed suicide.

"Dangerous Knowledge" Part 1

"Dangerous Knowledge" Part 2

/Total running time 89 minutes, if you're into this kind of stuff

Crap... not on Netflix. Got a youtube link? My DVD player can get youtube. Not Google video though (which I thought was killed off)

downstairs: Mr. Potatoass: Saw a good BBC documentary a couple of years ago about Cantor, Gödel, Ludwig Boltzmann, and Alan Turing.
All insanely brilliant. All committed suicide.

"Dangerous Knowledge" Part 1

"Dangerous Knowledge" Part 2

/Total running time 89 minutes, if you're into this kind of stuff

Crap... not on Netflix. Got a youtube link? My DVD player can get youtube. Not Google video though (which I thought was killed off)

/Enjoy!

Scrotastic Method: Thanks, I'll be watching those later. Though, Turing's suicide was brought on by a lot more than just madness/genius. That guy got the rawest deal of all raw deals. What a waste; what a loss.

Wholeheartedly agreed.

To Infinity and Beyond

That video seemed to go on for ever.

TXEric: If the donations weren't coming in, he'd just sit there and sulk, and play his damn horse show videos.

Or yell at the band, "PLAY IT AGAIN!"

Why do we give so much importance to the concept of infinity?
I'm being serious here. What, in the set of all things which actually exist, truly equals infinity?
Numbers are abstractions, which means they are only relevant in the context of describing representations of things.
At some point you are going to run out of representations to count.
I always understood the concept of infinity as not a number that goes on forever, but as an indicator that somewhere your logic was flawed and you attempted to calculate something which is in actuality impossible.
For instance, having one piece of pie from a pie with no pieces.
I.E. divide by zero: 1/0.
Zeno's paradox is another example. If there was such a thing as a infinitely small unit of measurement, then we would never be able to travel from point a to b. But we CAN travel from a to b, so what does that tell us about infinitely small units of distance? The only conclusion I can draw from that is there is a finite distance by which nothing is smaller. The concept of infinity here is irrelevant.
The more we spread videos around like this, the more confusion it causes.
What we really need is a video showing why the concept of infinity is irrelevant.

Scrotastic Method: I get that, that makes sense. But this talk was more like, "hey, when we impose these methods of organization on large sets of numbers, sometimes we find things that don't seem to make sense within the constraints of the rules we just imposed."

Same. I've had people try to explain the whole "some infinities are larger than others!" to me before, and clearly to them it was an amazing, miraculous thing, but... either I'm too dumb to understand it, or it's nonsense. If there's a number that's bigger than it, it isn't farking infinity. That's the point. The first half of this video was trivially simple, then it stops making sense. I don't mean the concepts become hard for me to grasp in the same way special relativity is hard to grasp, I mean I can't derive any meaning from what he's saying.

It's frustrating, because clearly it's an incredible, interesting discovery to people who get it, and I'd like to be able to share in that, but ...it's nonsense. The words he's saying don't mean anything.

Gunther: Same. I've had people try to explain the whole "some infinities are larger than others!" to me before, and clearly to them it was an amazing, miraculous thing, but... either I'm too dumb to understand it, or it's nonsense. If there's a number that's bigger than it, it isn't farking infinity. That's the point. The first half of this video was trivially simple, then it stops making sense. I don't mean the concepts become hard for me to grasp in the same way special relativity is hard to grasp, I mean I can't derive any meaning from what he's saying.

Well take the first example from the video. If we take all the whole numbers (1,2,3,4..) and count, then it's going to add up to infinity. But if we also take just the even numbers (2,4,6,8...) and start counting, it's also going to add up to infinity.

It's not that one infinity is larger that the other. It's that it's clearly obvious that there are more numbers in the first infinity than the second infinity. Even though they're both infinity. Which is totally f*cked up dude.

Scrotastic Method: But isn't this decimal vs. rationals "discovery" not really a discovery, but merely his realizing that his "sets" observation from the first 3 minutes of the video didn't really tell us anything about infinity in the first place?

No- there is a deeper story that the video doesn't really touch on. It's the story of human understanding of our world, and how our simple observations give rise to really fantastic realities.

Let's start with the most basic sets: finite sets. A finite set can be arbitrarily large, but it is not infinite. Examples of finite sets would be "three apples", "four oranges", or "6 billion people". Suppose you have two finite sets and you want to know whether they are the same size. As in the video, you try to come up with some matching so that every item in one set is paired with exactly one item in the other set.

Consider the three apples and the four oranges above. If you limit yourself to 1-1 matchings, you will always have one orange left over. In the context of finite sets, the 1-1 matching idea is a very robust and accurate way to determine if two sets are the same size. Indeed, as the video points out, you can even use this method to compare very large sets, even though you may not know (or it would be impractical) to count out how many members are in each set. Thus, we can compare any two finite sets and we know that at the end of the day we will find that one is larger than the other, or they are the same size.

The key idea here is that the idea of 1-1 matchings is a very obvious and simple thing to do in the very basic finite-set case BUT, when we apply our very obvious and simple 1-1 matching idea to things that are outside our basic realm of experience we get wholly unexpected results. In the case of infinite sets, we get that the set of even numbers is the same size as the set of whole numbers. You can say that the 1-1 matching method is not a valid way of comparing infinite set sizes, but the whole point was that it is a very straightforward generalization of how we treat finite sets. In doing so, you would be claiming that we are not allowed to reason about special infinite sets the same way we reason about regular finite sets, despite the fact that most infinite sets we think about are very simple generalizations of finite sets (e.g. the whole numbers).

The even more fundamental quandary is how we got here in the first place. If you know anything about formal reasoning then you know that you must always start with some assumption about how the world works. Mathematicians do this too, and the fundamental assumptions underlying set theory are extremely basic and designed to reflect "obvious" fundamental realities. Despite that, this whole infinite-set brouhaha is a logical consequence of our "obvious" fundamental realities. These basic assumptions are so ingrained to how the world works that mathematicians did work for thousands of years before anyone realized what an impact they had. This "axiomitization" of mathematical set theory actually happened in the same time-frame of the people in the video, in part due to people like you saying that the surprising conclusions of infinite sets was simply a consequence of some fundamental mistake of understanding.*

The uncomfortable thing here is that these are consequences of the reality that we live in.

*This kind of situation happened before, with the introduction and realization of Non-Euclidian geometries. Some people said that Non-Euclidian geometry it was absurd, but this theory now has important applications in the world today.

Joshka: Zeno's paradox is another example. If there was such a thing as a infinitely small unit of measurement, then we would never be able to travel from point a to b. But we CAN travel from a to b, so what does that tell us about infinitely small units of distance? The only conclusion I can draw from that is there is a finite distance by which nothing is smaller. The concept of infinity here is irrelevant.

This is false. The classical way of resolving Zeno's paradox is to say that infinitesimal distances DO exist, but the trick is that they take an infinitesimal amount of time to traverse.

Suppose you move at one meter per second, and you want to move one meter.

It takes you 1/2 a second to move halfway to your goal.

It takes you 1/4 of a second to cut the distance in half again.

The next half takes you 1/8 of a second. Etc.

The time it takes you to reach your goal can be described by the infinite sum from 1 to infinity of 1 / 1 + n. This is an infinite series, but we know that it's value is equal to exactly one. Even though it seems like there is always some portion of distance left to go, you can get there in a finite amount of time if you move fast enough.

Scrotastic Method

busy chillin': I was going to watch the NeverEnding Story, but really...who's got that kind of time?

If what I learned from that video is correct, and it is not, then it would take half as long to watch that movie twice.

I will have to watch the video...I have wanted to make that joke for a while and this thread seemed most appropriate.

John Nash: Gunther: Same. I've had people try to explain the whole "some infinities are larger than others!" to me before, and clearly to them it was an amazing, miraculous thing, but... either I'm too dumb to understand it, or it's nonsense. If there's a number that's bigger than it, it isn't farking infinity. That's the point. The first half of this video was trivially simple, then it stops making sense. I don't mean the concepts become hard for me to grasp in the same way special relativity is hard to grasp, I mean I can't derive any meaning from what he's saying.

Well take the first example from the video. If we take all the whole numbers (1,2,3,4..) and count, then it's going to add up to infinity. But if we also take just the even numbers (2,4,6,8...) and start counting, it's also going to add up to infinity.

It's not that one infinity is larger that the other. It's that it's clearly obvious that there are more numbers in the first infinity than the second infinity. Even though they're both infinity. Which is totally f*cked up dude.

But there aren't "more." They both go on forever...without an end, without a way to eventually do a count, you can't say there are more or less of anything. The whole numbers/even numbers thing isn't contradictory, it's just two sets that don't have upper limits. Because "infinity" is not a quantifiable thing.

Like, say I had infinite quarters. Then I got the same dollar amount delivered, only in nickels. You would assume there are more nickels, right, except, infinite is infinite. So there's infinite of both. Since it's not a real thing, infinity, we're back to what I was saying -- don't invent something (infinity), impose a constraint on it (makeup of sets), and then freak when you find something that breaks your own rule (even/whole). Because that's just...nothing.

/again, I know nothing of real math
//but I'm having fun
///and hopefully not irritating the math guys that clearly respect this stuff

Fubini: Joshka: Zeno's paradox is another example. If there was such a thing as a infinitely small unit of measurement, then we would never be able to travel from point a to b. But we CAN travel from a to b, so what does that tell us about infinitely small units of distance? The only conclusion I can draw from that is there is a finite distance by which nothing is smaller. The concept of infinity here is irrelevant.

This is false. The classical way of resolving Zeno's paradox is to say that infinitesimal distances DO exist, but the trick is that they take an infinitesimal amount of time to traverse.

Suppose you move at one meter per second, and you want to move one meter.

It takes you 1/2 a second to move halfway to your goal.

It takes you 1/4 of a second to cut the distance in half again.

The next half takes you 1/8 of a second. Etc.

The time it takes you to reach your goal can be described by the infinite sum from 1 to infinity of 1 / 1 + n. This is an infinite series, but we know that it's value is equal to exactly one. Even though it seems like there is always some portion of distance left to go, you can get there in a finite amount of time if you move fast enough.

I get that, the problem is essentially that you have infinite units of distance between a and b. So, if you want to get from a to b in one second, you are moving at infinite units of distance per second. Which is ridiculous.
If you are moving at infinite units per second you'd easily pass the furthest object in our universe. The model does not work.
Why doesn't it work? because the premise is flawed. There is no such thing as something which is infinite.

Joshka: Fubini: Joshka: Zeno's paradox is another example. If there was such a thing as a infinitely small unit of measurement, then we would never be able to travel from point a to b. But we CAN travel from a to b, so what does that tell us about infinitely small units of distance? The only conclusion I can draw from that is there is a finite distance by which nothing is smaller. The concept of infinity here is irrelevant.

This is false. The classical way of resolving Zeno's paradox is to say that infinitesimal distances DO exist, but the trick is that they take an infinitesimal amount of time to traverse.

Suppose you move at one meter per second, and you want to move one meter.

It takes you 1/2 a second to move halfway to your goal.

It takes you 1/4 of a second to cut the distance in half again.

The next half takes you 1/8 of a second. Etc.

The time it takes you to reach your goal can be described by the infinite sum from 1 to infinity of 1 / 1 + n. This is an infinite series, but we know that it's value is equal to exactly one. Even though it seems like there is always some portion of distance left to go, you can get there in a finite amount of time if you move fast enough.

I get that, the problem is essentially that you have infinite units of distance between a and b. So, if you want to get from a to b in one second, you are moving at infinite units of distance per second. Which is ridiculous.
If you are moving at infinite units per second you'd easily pass the furthest object in our universe. The model does not work.
Why doesn't it work? because the premise is flawed. There is no such thing as something which is infinite.

Sorry dude, but you're wrong. Look up infinite summations. You sum together an infinite number of positive values, yet the total sum is finite (meaning, in this case, there exists an upper bound).

Your fallacy is in saying that moving through an infinite number of spaces requires infinite amount of time. As demonstrated above, you can easily move through an infinite number of motions in a finite amount of time, so long as (essentially) the motions are successively smaller and smaller.

backs slowly away....

Fubini: Joshka: Fubini: Joshka: Zeno's paradox is another example. If there was such a thing as a infinitely small unit of measurement, then we would never be able to travel from point a to b. But we CAN travel from a to b, so what does that tell us about infinitely small units of distance? The only conclusion I can draw from that is there is a finite distance by which nothing is smaller. The concept of infinity here is irrelevant.

This is false. The classical way of resolving Zeno's paradox is to say that infinitesimal distances DO exist, but the trick is that they take an infinitesimal amount of time to traverse.

Suppose you move at one meter per second, and you want to move one meter.

It takes you 1/2 a second to move halfway to your goal.

It takes you 1/4 of a second to cut the distance in half again.

The next half takes you 1/8 of a second. Etc.

The time it takes you to reach your goal can be described by the infinite sum from 1 to infinity of 1 / 1 + n. This is an infinite series, but we know that it's value is equal to exactly one. Even though it seems like there is always some portion of distance left to go, you can get there in a finite amount of time if you move fast enough.

I get that, the problem is essentially that you have infinite units of distance between a and b. So, if you want to get from a to b in one second, you are moving at infinite units of distance per second. Which is ridiculous.
If you are moving at infinite units per second you'd easily pass the furthest object in our universe. The model does not work.
Why doesn't it work? because the premise is flawed. There is no such thing as something which is infinite.

Sorry dude, but you're wrong. Look up infinite summations. You sum together an infinite number of positive values, yet the total sum is finite (meaning, in this case, there exists an upper bound).

Your fallacy is in saying that moving through an infinite number of spaces requires infinite amount of time. As demonstrated above, you can ...

You're both wrong, because neither of you have mentioned God, Jesus, or the Bible EVEN ONCE.

.999... = 1

Discuss.

the opposite of charity is justice: .999... = 1

Discuss.

Fark I HATE that thing. Some nerd did that on a back of a bar napkin and it drove me crazy. How'd that go again?

Scrotastic Method: the opposite of charity is justice: .999... = 1

Discuss.

Fark I HATE that thing. Some nerd did that on a back of a bar napkin and it drove me crazy. How'd that go again?

With the real numbers, you can say they are different numbers if there exists some number 0.999...
To show that two numbers are different, there must be some difference between them (subtraction, yay!). 1/2 = 2/4 because you can use subtraction identity; 1/2 - 2/4 = 0 therefore 1/2 = 2/4. So, what does 1 - 0.999..... equal? 0.0000.....1? Can't happen, because there are infinite repeating zeros so that 1 never shows up at the end. So, 1 - 0.999.... = 0.0000.... which is still zero.

Or you can say that 1/3 = 0.333..., 2/3=0.666..., so 3/3=0.999...=1

John Nash

Well take the first example from the video. If we take all the whole numbers (1,2,3,4..) and count, then it's going to add up to infinity. But if we also take just the even numbers (2,4,6,8...) and start counting, it's also going to add up to infinity.

It's not that one infinity is larger that the other. It's that it's clearly obvious that there are more numbers in the first infinity than the second infinity. Even though they're both infinity. Which is totally f*cked up dude.

No, both those infinities have cardinality of Aleph nought. They are the same size. Can you create a map from one to the other? Yes, so they're the same size. You cannot create a map from the integers to the reals, so the reals are larger than the integers.

Counting the number of things is sort of a primitive way of comparing the size of sets. Mappings are the right way. Consider a bunch of people in a theater. Are there more people or more seats? You can count them all up and compare, or you can tell everyone to sit down. If there are people left over, then there are more people. If there are seats left over, then there are more seat. If they fit, they're the same size.

We can do this with, say, the positive integers and the positive even integers. Intuition will say that the positive integers are more, but we can make a one-to-one map between the two, namely that if you double an integer you get a unique even integer, and if you divide an even integer by two you get a unique integer.

Cantor proved that the reals are larger than the integers with his diagonal proof. It's pretty clever and should help explain. I suggest looking it up.

Well, I farked up that formatting, oops.

ykarie: Scrotastic Method: the opposite of charity is justice: .999... = 1

Discuss.

Fark I HATE that thing. Some nerd did that on a back of a bar napkin and it drove me crazy. How'd that go again?

With the real numbers, you can say they are different numbers if there exists some number 0.999...
To show that two numbers are different, there must be some difference between them (subtraction, yay!). 1/2 = 2/4 because you can use subtraction identity; 1/2 - 2/4 = 0 therefore 1/2 = 2/4. So, what does 1 - 0.999..... equal? 0.0000.....1? Can't happen, because there are infinite repeating zeros so that 1 never shows up at the end. So, 1 - 0.999.... = 0.0000.... which is still zero.

Or you can say that 1/3 = 0.333..., 2/3=0.666..., so 3/3=0.999...=1

I never found these arguments very convincing. In my head it sounds like 'things wouldn't work out nice if 0.99999 didn't equal 1 so it must be so'. I find it better to take it from the definition of a decimal expansion. The number 0.999999..., is defined to be the limit of the infinite sequence 9/10 + 9/100 + 9/1000 + 9/10000 ... , and that limit is 1.

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