.999... = 1

[SultanOfSurreal]

truth

[jonesthecurl]

and .11111recurring = one ninth, which answers Xeno and his effin tortoise.

[Frigidus]

I remember arguing with my friends about how that was total bull shit in 2nd grade...I thought it was a mind trick or something.

[InkL0sed]

http://en.wikipedia.org/wiki/0.999...

[StiffMittens]

Doesn't this assume that space-time is continuous (i.e. that it is infinitely divisible)? Isn't the jury still out on that one?

[jonesthecurl]

OK: those who understand Pythagoras' theorem stand by.

The rest of you go and study a bit.

Imagine a right-angled isosceles triangle where the equal sides are exactly 1".

The hypotenuse will be exactly (square root of 2) inches long.

This is not a number that can be expressed precisely.

It cannot be measured exactly in inches.
.

But let's use that length. Let's define that length, which is right in front of us, as exactly 1 jones.

Now, construct a right-angled isosceles triangle where the equal sides are exactly 1 jones.

For the same reason as above, the hypotenuse cannot be measure precisely in joneses

It will be (square root of 2)joneses long.
.

This is not a number that can be expressed precisely.

It cannot be measured exactly in joneses.
.

But it will be precisely 2".



Now imagine a square 2" on a side. Draw the diagonals. I'll leave the rest to your imagination.


And you thought .999...was a mind-mucker.

[Mr Scorpio]

I guess this would be easier to accept if you thing of .999... as a geometric sequence and not a number.

[TheProwler]

.9999recurring approaches 1, in my opinion.

Just like the difference between 1 and .9999recurring (ie. 1 minus .9999recurring) approaches 0.

People will throw equations like this at you:

x = .9999recurring

so

10x = 9.999recurring
-x = 0.9999recurring
---------------------------
9x = 9
x = 1

But don't believe it. Because 9.9999recurring is approaching 10 just a wee bit slower than 0.9999recurring is approaching 1.

For all intense purposes, though, you can consider .9999recurring to equal 1. We just aren't that precise yet. But when you are reaching for the last donut, you'd better hope you are the guy that reaches it in .9999recurring of a second and not a full second. Slow poke.

[owheelj]

No, 0.9 recurring does actually equal 1.

1/3 + 1/3 + 1/3 = 1

1/3= 0.3 recurring

0.3 recurring + 0.3 recurring + 0.3 recurring = 0.9 recurring.

Therefore 0.9 recurring = 1.

Maths isn't it a matter of opinions. If you can show something to be the case with in the system of mathematics then it is the case, no matter how counter-intuitive.

[TheProwler]

No, 0.9 recurring does actually equal 1.

1/3 + 1/3 + 1/3 = 1

1/3= 0.3 recurring

0.3 recurring + 0.3 recurring + 0.3 recurring = 0.9 recurring.

Therefore 0.9 recurring = 1.

Maths isn't it a matter of opinions. If you can show something to be the case with in the system of mathematics then it is the case, no matter how counter-intuitive.

You can't add recurring numbers up like that. Imagine adding these three numbers:

0.3333333333333333333333333333333333333333333333333333
0.3333333333333333333333333333333333333333333333333333
0.3333333333333333333333333333333333333333333333333333
---------------------------------------------------------------------------
0.9999999999999999999999999999999999999999999999999999

That is correct.

But, way over there on the right....see that 3+3+3=9? That is correct (because we are not dealing with .333recurring). And the next digit over, it would work the same way (ie. 3+3+3=9).

But (and it's a big but), if these numbers were all recurring (so equal to 1/3), that addition way over there on the right would look like this:

3.333recurring + 3.333recurring + 3.333recurring = 10

And the addition of the next digit would be 3+3+3+1(the 1 was "carried over")=10
And the addition of the next digit would be 3+3+3+1=10
And the addition of the next digit would be 3+3+3+1=10
etc.
And the last, and most spectacular, little bit of arithmetic that you'd have to do is carry over that last 1. To get an answer of: 1.

And, hence, you do not get 0.333recurring+0.333recurring+0.333recurring equals 0.999recurring. That is incorrect.

I used the word "opinion" because there are mathematicians would do not agree (but there are others who do agree). I was only being respectful to those who are, unfortunately, wrong.

[owheelj]

You are wrong. Mathematicians accept completely that 0.9 recurring is equal to 1.

From the wikipedia link;

This equality has long been accepted by professional mathematicians and taught in textbooks.

I know this because I was taught it by my maths teacher in high school. Only non-mathematicians ever claim that it doesn't.

Also if you correctly add up 0.3 recurring x 3 then you would never round over that last digit. You would just keep adding up the digits forever because they are number that go on forever. Rounding up that last digit can only happen with non recurring numbers that give you a point to stop adding 3+3+3.

[TheProwler]

You are wrong. Mathematicians accept completely that 0.9 recurring is equal to 1.

From the wikipedia link;

This equality has long been accepted by professional mathematicians and taught in textbooks.

I know this because I was taught it by my maths teacher in high school. Only non-mathematicians ever claim that it doesn't.

I studied this sometime in 1st or 2nd or 3rd or 4th year university (it was so long ago!). I was in Applied Math with Computer Science (an honours mathematics program) at the University of Waterloo - one of the top mathematics universities in the world. No offense, but your math teacher probably couldn't have got in the front door.

You got this part in while I was typing:

Also if you correctly add up 0.3 recurring x 3 then you would never round over that last digit. You would just keep adding up the digits forever because they are number that go on forever. Rounding up that last digit can only happen with non recurring numbers that give you a point to stop adding 3+3+3.

Think right to left. They might not teach it this way anymore, but we used to add the right-most digit first, carry over anything over 9, then do the next digit, etc.. Your first digit on the right (however far you'd want to go) would equal 10, not 9.999recurring. It's okay man, I think the motivation behind this "agreeing that 0.999recurring equals 1" is that mathematicians want to solve math problems, not discuss philosophy. But the top minds know. The difference is not significant in any practical application, so they make it easy so we can all agree and move on. Even if we know the truth.

[owheelj]

It doesn't matter if you start left to right, right to left, top to bottom or diagonally. 3+3+3=9. You don't round up until you finish the equation. With infinite numbers you never do. Rounding up is a convention, it's not something you need to do.

Nice argumentum ad verecundium. I was once ranked in the top 99.97% of maths students in Australia (in my year group). I was doing university maths when I was in 13. I don't know about my maths teacher, but I think I could have got in to your little university.


http://en.wikipedia.org/wiki/0.999

Skepticism in education

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[33]
Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[34]
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.[35]
Some students regard 0.999… as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.
Some students believe that the value of a convergent series is at best an approximation, that .
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999….
Many of these explanations were found by professor David O. Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".[36]
Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.[37] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.[38] Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.
Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."[39]
As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.[40]

[Frigidus]

Oh, yeah? Well, the things I do with numbers are akin to sorcery. Those who witness my mad math skills weep at the sight, lain low by my brilliance. I'm the Michael Jordan of math. That said, .999... = 1, with a +/- 47% margin of error. And now, gentlemen, I bid you farewell.

[TheProwler]

Hey, you talked about your math teacher. I tried to give you some insight about the level of my teachers.

You can quote wikipedia all you want. If the current mathematical community has decided that this argument needed to be settled with a simple explanation so students can move on a learn new things, so be it.

I understand that this may be difficult to grasp. But when you are talking about adding 3+3+3, you are missing a key element. Look at my first explanation again. If you were to write out a thousand 3's, or a million 3's, or a billion 3's, your first addition on the far right would still be 3.333recurring+3.333recurring+3.333recurring which does not equal 9 (as you mistakenly are stating). It equals 10. And that means you have to carry the one. And so on and so on and so on.

You are missing the idea that you are trying to explain theoretical numbers with real numbers. At some point, you are adding up 3 theoretical numbers, not 3 real numbers. Or, you could even do better than that. At some point you can add up three real numbers: 3 1/3 + 3 1/3 + 3 1/3 which equals 10. And that means you have to carry the one. And so on and so on and so on.

You can read this if you want to read some opinions of thinkers, not just repeaters:
http://www.boards.ie/vbulletin/archive/ ... 39461.html

[owheelj]

I never claimed 3.333 recurring x 3 = 9 and not 10. I claimed that 3+3+3=9. Do you disagree with this?

If the current mathematical community has decided that this argument needed to be settled with a simple explanation so students can move on a learn new things, so be it.

There have been proofs of this for well over 200 years, probably a lot longer.

Watch this;


3*3=9
3.3*3=9.9
3.33*3=9.99
3.333*3=9.999
3.3333*3=9.9999
3.33333*3=9.99999
3.333333*3=9.999999
3.3333333*3=9.9999999
3.33333333*3=9.99999999
3.333333333*3=9.999999999
3.3333333333*3=9.9999999999
3.333... 100 3s... 333*3=9.999...100 9s... 999

therefore
3.333... infinite 3s... 333*3=9.999... infinite 9s... 999

[SultanOfSurreal]

Hey, you talked about your math teacher. I tried to give you some insight about the level of my teachers.

You can quote wikipedia all you want. If the current mathematical community has decided that this argument needed to be settled with a simple explanation so students can move on a learn new things, so be it.

I understand that this may be difficult to grasp. But when you are talking about adding 3+3+3, you are missing a key element. Look at my first explanation again. If you were to write out a thousand 3's, or a million 3's, or a billion 3's, your first addition on the far right would still be 3.333recurring+3.333recurring+3.333recurring which does not equal 9 (as you mistakenly are stating). It equals 10. And that means you have to carry the one. And so on and so on and so on.

You are missing the idea that you are trying to explain theoretical numbers with real numbers. At some point, you are adding up 3 theoretical numbers, not 3 real numbers. Or, you could even do better than that. At some point you can add up three real numbers: 3 1/3 + 3 1/3 + 3 1/3 which equals 10. And that means you have to carry the one. And so on and so on and so on.

You can read this if you want to read some opinions of thinkers, not just repeaters:
http://www.boards.ie/vbulletin/archive/ ... 39461.html

lmao

ok, your "i am such a brilliant mathematical expert, i can only understand addition in terms of how arithmetic is taught to first graders" argument aside, please show me a single real scholar who claims .999... is not 1

[Frigidus]

Hey, you talked about your math teacher. I tried to give you some insight about the level of my teachers.

You can quote wikipedia all you want. If the current mathematical community has decided that this argument needed to be settled with a simple explanation so students can move on a learn new things, so be it.

I understand that this may be difficult to grasp. But when you are talking about adding 3+3+3, you are missing a key element. Look at my first explanation again. If you were to write out a thousand 3's, or a million 3's, or a billion 3's, your first addition on the far right would still be 3.333recurring+3.333recurring+3.333recurring which does not equal 9 (as you mistakenly are stating). It equals 10. And that means you have to carry the one. And so on and so on and so on.

You are missing the idea that you are trying to explain theoretical numbers with real numbers. At some point, you are adding up 3 theoretical numbers, not 3 real numbers. Or, you could even do better than that. At some point you can add up three real numbers: 3 1/3 + 3 1/3 + 3 1/3 which equals 10. And that means you have to carry the one. And so on and so on and so on.

You can read this if you want to read some opinions of thinkers, not just repeaters:
http://www.boards.ie/vbulletin/archive/ ... 39461.html

lmao

ok, your "i am such a brilliant mathematical expert, i can only understand addition in terms of how arithmetic is taught to first graders" argument aside, please show me a single real scholar who claims .999... is not 1

There aren't any, but that's only because the athiests running academia want you to think that they're the same, in an attempt to disprove logic and God. Fortunately, us non-sheeple see it for the lie it is.

[Almo]

http://en.wikipedia.org/wiki/0.999...

how does jimbo wales' cock taste you doughy ass rainbow flag flying nerd

[jonesthecurl]

Oncer again the big brains weigh in.

[AAFitz]

I studied this sometime in 1st or 2nd or 3rd or 4th year university (it was so long ago!). I was in Applied Math with Computer Science (an honours mathematics program) at the University of Waterloo - one of the top mathematics universities in the world.

I was once ranked in the top 99.97% of maths students in Australia (in my year group). I was doing university maths when I was in 13. I don't know about my maths teacher, but I think I could have got in to your little university.

I enjoyed that.

Now, I have some university math, but I dont enjoy it. However, it seems to me that arguing that .999recuring not equaling one, is as useful as arguing that 1 does not equal 1. It is true on a basic level at times, but not extremely practical.

However, in 10,000 years, if we have matter transportation capability, and my starship is 500000000000000000000000000.999recurring miles away, and you have to beam me onto it, I do hope you dont just round up, and fuse me into the hull. (though at some point you have to stop the computation, otherwise the number will never be displayed, so it cant be used in the equation, since 9's will continue to be added to it forever.)

However, If I buy a soda, and it comes out to 1.999recuring cents....keep the change please

[Snorri1234]

I studied this sometime in 1st or 2nd or 3rd or 4th year university (it was so long ago!). I was in Applied Math with Computer Science (an honours mathematics program) at the University of Waterloo - one of the top mathematics universities in the world.

I was once ranked in the top 99.97% of maths students in Australia (in my year group). I was doing university maths when I was in 13. I don't know about my maths teacher, but I think I could have got in to your little university.

I enjoyed that.

Now, I have some university math, but I dont enjoy it. However, it seems to me that arguing that .999recuring not equaling one, is as useful as arguing that 1 does not equal 1. It is true on a basic level at times, but not extremely practical.

However, in 10,000 years, if we have matter transportation capability, and my starship is 500000000000000000000000000.999recurring miles away, and you have to beam me onto it, I do hope you dont just round up, and fuse me into the hull. (though at some point you have to stop the computation, otherwise the number will never be displayed, so it cant be used in the equation, since 9's will continue to be added to it forever.)

It's not round up, .999 recurring is 1.

[jbrettlip]

I thought this was about Money market funds.

[xelabale]

[quote="wikipedia]With the rise of the Internet, debates about 0.999… have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999… via 1⁄3 and limits, saying of misconceptions,

The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense.[42]

The Straight Dope cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company issued a "press release" on April Fools' Day 2004 that it is 1:

We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.[43]

Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:[44]

Q: How many mathematicians does it take to screw in a lightbulb?

A: 0.999999….[/quote]

Darn, someone got there first...

[john9blue]

I was once ranked in the top 99.97% of maths students in Australia (in my year group). I was doing university maths when I was in 13. I don't know about my maths teacher, but I think I could have got in to your little university.

I was once ranked in the top 99.99999999999999% of students in my high school.

Of course, according to Prowler I would not have been valedictorian.